Stockhausen and the Serial Shaping of Space

by

Paul Miller

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Dave Headlam

Department of Music TheoryEastman School of Music

University of RochesterRochester, New York

2009

Part 1Text

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Dedication

This dissertation is dedicated to my parents, Joseph S. and Rosemary Miller.

Without their support, the present work would not have been possible.

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Curriculum Vitae

Paul Miller was born in Gloversville, New York on 9 April 1976. After attending

Arlington High School in LaGrangeville, New York, he graduated in 1998 with an A.B.

with honors in music from Vassar College in nearby Poughkeepsie. Along with studies

at Vassar, Paul attended the New England Conservatory in Boston, MA for a year where

he met and subsequently studied privately with Carl Schachter for two summers.

Thereafter, Paul earned Master’s degrees in Music Theory (2000) and Viola Performance

(2006) from the Eastman School of Music in Rochester, New York. Thanks to grants

from the Deutsche Akademische Austauschdienst (DAAD) and the Presser Foundation,

the research for the present dissertation was successfully completed in 2006 and 2007.

Paul attended the Stockhausen Summer Courses in Kürten, Germany seven times. In

2005 he gave the world premiere of the viola version of Stockhausen’s IN

FREUNDSCHAFT for viola in Kürten. In addition, Paul attended the Darmstadt

Vacation Courses in New Music three times, the second time winning a Stipendiumpreis

for viola and viola d’amore performance.

Along with his duties as Adjunct Professor of Music at the Boyer School of Music at

Temple University in Philadelphia, Pennsylvania, Paul performs violin, viola and viola

d’amore regularly with baroque ensembles up and down the east coast. He appeared

notably in 2008 at the Metropolitan Museum of Art in New York City where he lectured

and performed on the viola d’amore. In fall 2009, Paul will join the faculty at the

University of Colorado in Boulder.

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Acknowledgments

There are many people without whom the present work would not have been possible.

First, thanks are due to Karlheinz Stockhausen, whose work forms the subject of this

dissertation, and whose muse continues to inspire me and many others. Although

Stockhausen will never see this work in its final form, he was aware of some of its

conclusions during his lifetime. I hope he would be as happy to read it as he was during

the rich autumn of his life in Kürten.

Two ladies associated with Stockhausen deserve special mention. First, Suzanne

Stephens helped to make the extensive archives of the Stockhausen archive open to my

study in 2005, 2006 and 2007. Her support was indispensable to my work. Second,

Maria Lukas, archivist of the Stockhausen foundation, assisted me in countless ways

and shared a good many cups of Rozenbottel tea in Kürten.

My studies at the Stockhausen archive would not have been possible without the

incredibly generous assistance provided by Gudrun and Gerd Papier, in whose house I

spent many productive weeks.

My dissertation adviser, Dave Headlam, helped me to see this project through to the end.

For his kind and very practical assistance I will always be grateful. Aside from Professor

Headlam, Jerome Kohl earned my gratitude and respect for helping to give final form to

the dissertation. Kohl’s knowledgeable, humorous, and immensely helpful criticism

helped to polish my work. Ciro Scotto also assisted in helping me to prepare the

dissertation. I am very grateful to all three of these gentlemen.

Assisting in many other ways was Stephen Zohn, who from his perch at the Boyer

School of Music at Temple University provided much useful practical advise and

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encouragement. Renee Barrick, at the Metropolitan Museum of Art in New York City

read through a draft of the dissertation, providing many helpful suggestions.

Generous financial assistance provided by the Presser Foundation and the Deutsche

Akademische Austauschdienst (German Academic Exchange Service) allowed me to

complete research for the dissertation.

I am grateful to Karli Molter, my neighbor, who was often called upon to use her

borrowing privileges at the University of Pennsylvania Libraries to retrieve books and

scores for me. Doris Dabrowski and Richard Stoy, neighbors and fellow music-lovers,

fed and encouraged me in the city of Brotherly Love. Katharine Chandler helped out in

various ways, as well.

But my deepest gratitude goes to my parents, Joseph and Rosemary Miller, to whom

this dissertation is dedicated.

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Abstract

In countless lectures, essays, and interviews, Karlheinz Stockhausen (1928-2007)

emphasized the importance of spatial movement in his music. Throughout his over 50-

year compositional career, Stockhausen developed sophisticated techniques for

spatialization in both his electronic and instrumental music. From early works such as

GESANG DER JÜNGLINGE (1955-56) to his last major piece, COSMIC PULSES

(2007), one can find extensive sketch material relating to spatial movement, as well as

detailed indications of spatial motion in his scores. Although scholars such as Harley

(1994), Nauck (1997), Misch (1999), Overholt (2006) and Hofmann (2008) have

investigated spatialization in contemporary music, none have developed an analytical

methodology for quantifying and measuring the speeds, shapes, densities, and structure of

spatial movement in Stockhausen’s works, or any other music. Yet, in interviews with

Jonathan Cott (1973) and others, Stockhausen suggested that such a methodology was

necessary to understanding his achievements.

In the dissertation, I develop a method for precisely locating the pathway of movement in

spatial music. Then, drawing from the music-theoretic tradition of set theory and

transformation theory, and the mathematical field of graph theory, I developed techniques

to identify spatial structures and relate them according to mathematical transformations. I

then characterized the symmetries that might be present in physical musical space using

mathematical groups. Thanks to several computer algorithms I developed, I was able to

find ways to calculate the statistical distribution of sounds within a performance space, as

well as the speed and density of musical material.

Applying this array of analytical tools to two major late works of Stockhausen,

OKTOPHONIE (1991-2) and LICHTER-WASSER (1999), I uncovered spatial structures

which were as subtle and intricate as Stockhausen’s formal, rhythmic, timbral or pitch

languages. Apart from providing a wealth of new data relating to Stockhausen’s

idiosyncratic compositional process, my analyses suggest new ways of hearing these

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works which Stockhausen himself might not have been aware of. These techniques were

then extended further, to assist in analyzing and better understanding the “sonic storm” in

Xenakis’s Terretektorh (1965/66). Finally, I use the methodology to argue for a spatial

structure that significantly enhances the sophisticated multi-choir texture in Thomas

Tallis’s (c. 1505 – 1585) famous 40-part motet Spem in alium.

My analyses provide a new and provocative glimpse into a musical domain which has

figured greatly into contemporary musical practice, but has so far proved elusive to

methods of quantitative analysis. The analytical techniques I developed are flexible

enough to provide theorists and performers with tools for a wide range of music, not just

Stockhausen’s. Because of this, my work opens up many new paths of development in

the fields of music cognition, music analysis, and musical spatialization.

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Table of Contents

Chapter 1. The Need to Confront Space ................................................. 1

1.1. Introduction 1

1.2. Spatialization in Stockhausen’s Music 3

1.3. Spatialization in Stockhausen’s Essays, Interviews and Sketches 20

1.4. Literature on Spatialization and its Role in Stockhausen’s Music 28

1.5. Argument for more analytical rigor 36

Chapter 2. Analytical Methodologies ...................................................... 41

2.1. Analytical Orientation 42

2.2. Locating a sound in space 46

2.3. The Shape of Space in Spatial Music 50

2.4. Sets and Graphs in Spatial Music 54

2.5. Transformations in Space 67

2.6. Groups 71

2.7. Statistics 73

2.8. Conclusions 74

Chapter 3. Stockhausen’s OKTOPHONIE ............................................ 77

3.1. OKTOPHONIE and LICHT 77

3.2. Elements of Sketches and Score that Pertain to Spatialization 83

3.3. Shape of the space in OKTOPHONIE 86

3.4. Spatial shapes in OKTOPHONIE 88

3.5. Other Spatial Relationships in OKTOPHONIE 93

3.6. Conclusions 95

Chapter 4. Stockhausen’s LICHTER-WASSER .................................. 97

4.1. Introduction 97

4.2. Elements of the score that pertain to spatial movement 104

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4.3. Shape of the space in LICHTER-WASSER 106

4.4. Instrumental Motion 113

4.5. Vocalist Motion 127

4.6. Trends of Spatial Movement in LICHTER-WASSER 129

4.7. Relation of shapes in LICHTER-WASSER to each other 134

4.8. Group Structure 138

4.9. Conclusions 141

Chapter 5. Further Applications of Spatial Analysis and Performance 143

5.1. Directions for research in spatial music 143

5.2. Other Stockhausen Works 144

5.3. Xenakis and Terretektorh 151

5.4. Spatialization in other musical repertoires: Tallis’s Spem in alium 156

5.5. Electronics and Spatialization 163

5.6. Concluding Remarks 165

Afterword: Spatial “Serialism” ............................................................... 169

Bibliography ............................................................................................. 171

Figures and Examples .............................................................................. 183

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List of Figures

Chapter 1

1.2.1a. Seating Plan for GRUPPEN

1.2.1b. Spatial Setup in KONTAKTE

1.2.1c. Spatial Setup in CARRÉ

1.2.1d. Four Box Diagrams in CARRÉ

1.2.1e. Spatialization of chords in HYMNEN

1.2.1f. Possible Rotation Table Configurations in HYMNEN

1.2.2a. Spatialization in Osaka

1.2.2b. Spatialization in SIRIUS

1.2.2c. Speaker setup for SIRIUS

1.2.2d. Podium design in SIRIUS

1.2.2e. Rotation loudspeaker for SIRIUS

1.2.3a. Spatialization in UNSICHTBARE CHÖRE

1.2.3b. Spatialization in KATHINKAS GESANG

1.2.3c. Spatialization in DER KINDERFÄNGER/ENTFÜHRUNG

1.2.3d. Spatialization in DER KINDERFÄNGER/ENTFÜHRUNG (Tonszene 36)

1.2.3e. Spatial setup in OKTOPHONIE

1.2.3f-h. Three Speaker Arrangements in FREITAG-VERSUCHUNG

1.4.3a. Nauck’s Hörpartitur

1.4.3b. Nauck’s Typologie

Chapter 2

2.2.2a. Spatialization in a stereo panorama

2.2.2b. Calculation of the location of the sound between two loudspeakers

2.2.2c. Calculation of the location of a sound in quadrophonic panorama

2.2.2d. Final analysis of location

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2.4.1. The shape of sets in spatial music

2.4.3a. Distance between points in space

2.4.4.1. Basic Terminology and Properties of Graphs

2.4.4.2a. Isomorphism

2.4.4.2b. Isomorphism, Automorphism

2.4.4.3a. Path, Cycle

2.4.4.3b. Cutvertex, bridge, separator

2.4.4.4. Digraph

2.4.4.5. Tree, branches, root, leaves.

2.4.6.1a. “Graph 1”

2.4.6.1b. “Graph 2”

2.4.6.1c. “Graph 3”

2.4.6.3a. Adjacency Matrix A for Graph 1

2.4.6.3b. Adjacency Matrix A2 for Graph 1

2.4.6.3c. Analysis of path outflow for v13 in Graph 1

2.4.6.3d. The 64 sets of cardinality 3 in Graph 1 for each vertex.

2.4.6.4a. Adjacency Matrix A for Graph 2

2.4.6.4b. Adjacency Matrix A2 for Graph 2

2.4.6.4c. Analysis of path outflow for v11 in Graph 2

2.4.6.4d. Analysis of path outflow for v16 in Graph 2

2.4.6.5a. Adjacency Matrix A for Graph 3

2.4.6.5b. Adjacency Matrix A2 for Graph 3

2.4.6.5c. Analysis of path outflow for v9 in Graph 3

2.4.6.6a. Hamiltonian cycles through Graphs 1, 2 and 3

2.4.7a. Spatial orchestration

2.4.7b. An alternative graph structure based on a placement of different instruments in

Graph 3.

2.5.1. Matrix representations of traditional transformations as applied to pitch-class

2.5.2a. Matrix representation of four basic transformations in a plane

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2.5.2b. The 8 spatial set classes of cardinality 3 with two moves, under rotational

equivalence

2.5.2c. The 5 spatial set classes of cardinality 3 with two moves, under rotational and

reflection equivalence

2.5.3a. A space with four sound sources

2.5.3b. Transformations in the space of Example 2.5.3a

2.5.3c. 9-vertex space (subgraph of graph 1)

2.5.3d. Directions of translation in Example 2.5.3c

2.5.3e. Mappings in Example 2.5.3c

2.5.3f. Multiplication/dilation in Graph 3

2.5a. Comparison of set-generating potential in Graphs 1, 2 and 3

2.5b. Use of spatial transformations to model listeners’ experiences of spatial music

2.5.2a. Group table of the dihedral group D4

2.6.3a. Group table of translation group

Chapter 3

3.1.1a. Nuclear formulas in Stockhausen’s LICHT

3.1.2. “Study for OKTOPHONIE”

3.1.3a. Formal Structure of OKTOPHONIE, Part 1

3.1.3b. Formal Structure of OKTOPHONIE, Part 2

3.2.1a. Sketch for the spatialization of OKTOPHONIE

3.2.1b-e. Methodology for measuring spatial motion in OKTOPHONIE (Same as

examples 2.2.2a-d)

3.2.2a. Bomb Analysis 1

3.2.2b. Bomb Analysis 2

3.2.2c. Bomb Analysis 3

3.2.2d. Combined Scatter Plot of Bombs in OKTOPHONIE

3.2.2e. Coordinates of Bombs

3.2.3a. General shot trajectories in different sections of OKTOPHONIE

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3.2.3b. Shot Analysis 1

3.2.3c. Shot Analysis 2

3.2.3d. Shot Analysis 3

3.2.3e. Combined Scatter Plot of Shots 1-48

3.2.3f. Coordinates of shots 1-48

3.2.3g. Shot analysis 1 for shots in the second half of OKTOPHONIE

3.2.3h. Shot analysis 2 for shots in the second half of OKTOPHONIE

3.2.3i. Combined Scatter Plot of Shots in the second half of OKTOPHONIE

3.2.3j. Coordinates of shots 1-10 in second half of OKTOPHONIE

3.2.3k. Coordinates of shots 11-18 in the second half of OKTOPHONIE

3.2.4a. Crash analysis

3.2.4b. Combined scatter plot of crashes

3.2.4c. Coordinates of the crashes

3.2.4d. Shape of a descending crash

3.3.1a. Alternative loudspeaker arrangement for OKTOPHONIE

3.4.2a. Landing points for the 65 bombs

3.4.2b. Landing points for the 65 bombs - angle & distance measurements

3.4.2c. Angles of 2 degrees

3.4.2d. Obtuse angles

3.4.2e. Plot of distances and angles of the 65 bombs

3.4.3a. Endpoints for the first 48 shots

3.4.3b. Endpoint data for the first 48 shots - angle & distance measurements

3.4.3c. Acute angles of 2 to 3 degrees

3.4.3d. Obtuse angles

3.4.3e. Change in distance and angle for the endpoints of the first 48 shots

3.4.3f. Endpoints of the 18 shots in Part 2

3.4.3g. Distances & angles of shots in 2nd half

3.4.3h. Correlation of shots 1-4 in Part 1 of OKTOPHONIE with shots 1-4 in Part 2

3.4.4a. Distance from preceding bomb to the next crash

3.5.1a. Overall density of bombs, shots and crashes

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3.5.2a. Spatial grouping of the first 48 shots

3.5.2b. Relationships between selected shots

Chapter 4

4.1.3a. The Kernformeln or “Nuclear Formulas” in LICHTER-WASSER (central pitches)

4.1.3b. Division of the 29 musicians into two “orchestras”

4.1.3c. Formal Structure and Pitch Structure of LICHTER-WASSER

4.2.1a. Various spatial shapes Stockhausen considered using in LICHTER-WASSER

4.3.3a. LICHTER-WASSER, mm. 139-147

4.3.3b. LICHTER-WASSER, mm. 398-401

4.3.3c. LICHTER-WASSER, mm. 249-253

4.3.3d. LICHTER-WASSER, mm. 336-340

4.3.3e. LICHTER-WASSER, mm. 324-328

4.3.4a. Sequence of instrumental moves in wave sections (page 1 of 5)

4.3.4b. Sequence of instrumental motions in wave sections (page 2 of 5)

4.3.4c. Sequence of instrumental moves in wave sections (page 3 of 5)

4.3.4d. Sequence of instrumental moves in wave sections (page 4 of 5)

4.3.4e. Sequence of instrumental moves in wave sections (page 5 of 5)

4.3.4f. Sequence of instrumental moves in bridges 5 and 6

4.3.4g. Coordinates of the 29 instrumentalists in LICHTER-WASSER

4.4.1a. Speed of motion in LICHTER-WASSER (page 1 of 2)

4.4.1b. Speed of motion in LICHTER-WASSER (page 2 of 2)

4.4.1c. Statistical results of movement analysis

4.4.3a. Basic cycles

4.4.3b. M1 and E1 waves

4.4.4a. Second block of waves: M2, M3 + E2

4.4.5a. Third block of waves: M4, M5 + E3, E4

4.4.6a. Fourth block of waves

4.4.8a. Fifth block of waves

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4.4.9a. Sixth block of waves

4.4.10a. Seventh block of waves

4.4.11a. One Possible Exit-Tree for the Ausgang

4.4.11b. Second Possible Exit-Tree for the Ausgang

4.5.2a. Movement of singers through all wave and bridge sections

4.5.2b. Comparison of instrumental and vocal motion

4.6.1a. Number of times each instrumentalist is used in LICHTER-WASSER (according

to data from Examples 4.3.4a-e)

4.6.2a. Amount of Time (in seconds) each instrumentalist is used in LICHTER-WASSER

(according to data from Examples 4.3.4a-e)

4.6.3a. Most common instrumental moves in LICHTER-WASSER

4.6.3b. The A Matrix for LICHTER-WASSER

4.6.3c. The A2 Matrix for LICHTER-WASSER

4.6.4d. Connectivity out, based on LW A2 Matrix

4.6.4e. Connectivity in, based on LW A2 Matrix

4.7.1a. Most common paths, related by rotation or reflection, which originate in each of

the four corner instruments

4.7.2a. M12 (black); basic M-cycle, rotated 90° clockwise (gray)

4.7.2b. Inner circle of Basic M-cycle detached and flipped in the horizontal axis

4.7.2c. Comparison of paths in E11 and E12

4.7.3a. Possible “background” structures in LICHTER-WASSER

4.8.1. Unrealized group structure of operations in LICHTER-WASSER

Chapter 5

5.2.1a - b. Spatial notation in the sketches of KONTAKTE

5.2.4a. Distribution of Loudspeakers in COSMIC PULSES

5.2.4b. Transfer of Stockhausen’s sketch data to “box diagrams”

5.2.4c. Number of times each loudspeaker group is used in COSMIC PULSES

5.3.1a. Front page of Xenakis's score to Terretektorh

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5.3.2a. Three spirals used by Xenakis in Terretektorh

5.3.2b. Maria Harley's analysis of spatial motion in mm. 1-74 of Terretektorh

5.3.2c. Coordinates of musicians in Terretektorh

5.3.2d. Coordinates of regions in Terretektorh

5.3.2e. Spatial movement in Terretektorh, measures 1-47

5.3.2f. Spatial movement in Terretektorh, measures 51-74

5.3.2g. Calculations of speed in Terretektorh, mm. 1-66

5.3.2h. Calculations of speed in Terretektorh, mm. 65-74

5.4.4a. Whittaker’s diagram of Tallis’s Spem in Alium

5.4.4b. Whittaker’s 1929 spatialization

5.4.4c. Detailed diagram of mm. 85-110, showing paired antiphonal passages

5.4.4d. Circular arrangement of choirs

5.4.5a. Principal spatial movements in Spem in Alium, mm. 1-85

5.4.5b. Principal spatial movements in Spem in Alium, mm. 86-end

5.4.5c. Operations that model spatial movements in a circular spatialization of

Spem in Alium

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Chapter 1.The Need to Confront Space

1.1. Introduction

Spatial music is music in which the movement or distribution of sounds in the space

around the listener play a role in the structure of the work itself. Among recent

composers who have spatialized their music, Stockhausen developed an extensive body

of theories and practices. He gave thought to spatial issues in nearly every one of his

works, from KREUZSPIEL (1951)1 – in which the arrangement of the musicians is

analogous to a registral “crossing”2 – to his last, incomplete work-cycle, KLANG (2005-

2007). Stockhausen’s writing on the subject consisted of many different articles and

essays over a lengthy span of time. Indeed, Stockhausen said that “instrumental and

electronic spatial composition have been my artistic mandate since 1951”.3

In Stockhausen’s compositions, the structure of some musical dimensions often affected

others in unusual ways. In connection with his work OKTOPHONIE (1990-91),

Stockhausen wrote

The simultaneous movements – in 8 layers... demonstrate how... a new dimensionof musical space-composition has opened. In order to be able to hear suchmovements – especially simultaneously – the musical rhythm has to bedrastically slowed down; the pitch changes must take place less often and only insmaller steps or with glissandi, so they can be followed; the composition ofdynamics serves the audibility of the individual layers – i.e. dependent on thetimbres of the layers and the speed of their movements; the timbre compositionprimarily serves the elucidation of these movements.4

This statement confirms that spatial movement and other musical dimensions, such as

1 Although it is not common practice to capitalize the names of musical works, Stockhausen did in almostall of his published works and analyses. I will follow this convention throughout the present study.

2 Texte 4, p. 49. See also the preface to the score to KREUZSPIEL.3 Nauck 1997, p. 173. “Instrumentale und elektronische Raumkomposition sind mein künstlerischer

Auftrag seit 1951”.4 Stockhausen 1994, p. xxviii. The German version can be found in Texte 8, p. 374. Original boldface &

italics.

2

pitch, rhythm and timbre, interact with one another in Stockhausen’s concept of his

music. The ways in which these musical dimensions affect each other is a complex topic.

But Stockhausen once said that

... any point in space should be precisely defined with respect to where the soundoccurs and how it travels from one point to any other.5

Here, Stockhausen suggests that even though space is related to other musical parameters,

it is still important to know what is going on in the spatial domain on its own terms by

measuring the shapes, speeds, densities and directions of spatial movement. There are

few, if any analytical tools to help us accomplish this, or precedents for doing so.6

Measuring events in the spatial domain will help us know better what is going in that

dimension, and will ultimately allow us better to understand how spatial elements relate

to pitch, rhythm, and form.

Although Stockhausen employed an idiosyncratic method of serialism throughout his life

to structure many aspects of his compositions, his spatial techniques evolved – for the

most part – independently of strict serialism. Still, any analysis of Stockhausen’s music

must be seen within the context of serial methods.7 The present study proposes first to

find ways to measure and find shapes, speeds, directions and densities of musically

relevant material Stockhausen’s spatialized music. Then, we will interpret the data.

Analytical tools to do this will be developed out of well-established methods of musical

set theory, transformation theory, and group theory, which have been fruitfully applied to

the analysis of serial music. In addition, graph theory – a field in mathematics which has

undergone immense development in the last 30 years but which has not yet been used in

5 Cott 1973, pp. 202-203.6 Roger Reynolds also lamented this fact, over thirty years ago: “The equipment and much of the

perceptual information that would allow an orderly examination of the geometry of sound alreadyexists; what is lacking is informed strategy.” (Reynolds 1978, p. 183)

7 Grant sees spatialization as increasing the experiential interest in serial compositions. The“impossibility of preparing oneself mentally for the sounds which will occur” in electronic music,which is an “aesthetic ideal of serial music per se” is enhanced by spatialization. Grant 2001, p. 99.

3

many musical studies – will play a role.8

My analyses, which interact with these branches of music theory, will show that spatial

motion in Stockhausen can not only be measured, but that interpreting the measurements

reveals musically relevant and structurally important aspects in the music. I will show in

detail the structure of spatial motion in two works of Stockhausen. Ultimately, I intend to

demonstrate that the compositional use of physical space can be musical in and of itself.

Through my work, I hope to show one composer’s approach to organizing physical

space, while at the same time laying the foundation for more detailed music-theoretic

study of spatial movement in music.

1.2. Spatialization in Stockhausen’s music

Throughout his compositional career, Stockhausen composed 370 individually

performable works.9 Most of these works have directions specifically relating to

spatialization in the score.10 I divide these works into three groups, based upon a

chronology that is determined both by the growth of technical means and the

development of Stockhausen’s spatial imagination.

1.2.1. Spatialization of early works. Although the seating plan of the musicians in

KREUZSPIEL (1951) was partially determined because of the way in which the

registral fabric crosses over on itself,11 Stockhausen’s first notable experiment with

8 One notable area in which graph theory has played a part is in the numerous studies of neo-Riemanniantheory. It has also been developed in relation to Klumpenhouwer networks. See Lewin 1990 and Cohn1997.

9 Stockhausen Verlag 2008.10 A survey which is less detailed but much broader than mine can be found in Conen 1991, pp. 26-33.11 Texte 9, p. 595. “Schon bei meinen ersten Kompositionen wie KREUZSPIEL und ZEITMASZE habe

ich Zeichnungen im Vorwort der Partitur drucken lassen, mit denen ich angebe, wie die Musiker sitzensollen.”

“Already in my first compositions like KREUZSPIEL and ZEITMASZE, I made indications in theintroduction to the score in which I indicated where the musicians should sit”. The seating arrangement

4

spatial music is his electronic work GESANG DER JÜNGLINGE (1955-56).

Originally this piece was conceived for six channels,12 but because of the difficulties

involved with this realization it was composed for five. Finally, the fifth channel was

mixed down and combined with the first channel.13 This is the version that is most often

heard in a concert performance.

Concerning the use of space in the composition, Stockhausen wrote:

In my “GESANG DER JÜNGLINGE”, I attempted to form the direction andmovement of sounds in space, and to make them accessible as a new dimensionfor musical experience. The work was composed for 5 groups of loudspeakers,which should be placed around the hall. From which side, by how manyloudspeakers at once, whether with rotation to left or right, whether motionless ormoving – how the sounds and sound-groups should be projected into space; allthis is decisive for the comprehension of this work.14

Despite Stockhausen’s relative inexperience in dealing with spatial music, the effect of

the work on audiences of the time seems to have been quite overwhelming.15 However,

the way Stockhausen spatialized this work suggests that the space is primarily used as aof musicians in KREUZSPIEL may have originated several years after Stockhausen wrote it.

12 Misch 1999a, p. 148.13 “The 5th channel was played back on a mono tape recorder which started synchronously (by hand) with

a 4-track tape recorder. Originally, this 5th channel should have been played back above the listenersover a loudspeaker hung at the center of the ceiling. However, even at the world premiere this was notpossible, and the loudspeaker for the 5th track stood at the front, in the middle of the stage. Thesynchronization of the two tape recorders was unsatisfactory, so I decided after the world premiere tosynchronize the 5th track onto the 4th track. Since then, GESANG DER JÜNGLINGE has beenprojected 4-track.” Stockhausen 2001b (CD 3), p. 136.

Although the original fifth channel for GESANG is completely lost physically, evidence of its materialcan also still be found in Stockhausen’s sketches. John Philipp Gather did important work separatingthe material of the original fifth channel from the first: although Stockhausen claimed the fifth channelwas combined with the fourth, it is actually combined with the first channel. Unfortunately, Gather’suseful work on GESANG did not make it into the final copy of his 2004 dissertation.

14 Stockhausen 1961, pp. 68-69.15 In his review of the first performance of Gesang, Heinz-Klaus Metzger wrote that: “...Until now, no

work has been so thoroughly conceived for the concert hall as Stockhausen’s Gesang der Junglinge,which counts five loudspeaker groups spread around the hall...” (Metzger 1956, p. 222). Other listenersreacted less favorably: “The performance provoked turbulent demonstrations in the hall as well asfrenetic applause. Doors were slammed, one appalled listener yelled, ‘that’s blasphemy!’” (Kurtz 1992,p. 84.)

5

way of clarifying serial texture, and not as an end in and of itself.16

At the same time Stockhausen was working on GESANG, he conceived another way or

organizing the space around listeners in his work GRUPPEN (1955-57). GRUPPEN

employs three large orchestras set up in unusual “horseshoe” shape around the audience,

an arrangement which is shown in Example 1.2.1a. Stockhausen took advantage of the

spatial arrangement of the orchestras in several ways. First, antiphonal passages “switch”

abruptly from one part of the performing space to another.17 In other passages (especially

the Groups 71-73) one particular timbre – in this case, the drum – “circulates” around the

audience in a way that suggests that a sound jumps from one area to another in

discontinuous motion. These types of spatial motion are already present in GESANG

DER JÜNGLINGE. But in GRUPPEN, a brass chord in Group 119 seems to “rotate”

around the audience like a searchlight. This chord dynamically “fades in” and out of

each orchestra as it is passed around the space, creating the illusion of continuous motion

– a technique that Stockhausen did not use in earlier compositions.18 Stockhausen further

developed the techniques for rotating sounds around the listener in a later

work, KONTAKTE.

KONTAKTE (1958-60) is for two soloists (piano and percussion) and an array of four

speakers set up around the audience. The arrangement is shown in Example 1.2.1b.

In KONTAKTE,

six forms of spatial movement, with differentiated velocities and directions,contact each other in ever changing ways: rotations, looping movements,alternations, disparate fixed sources (different sounds from each of the 4

16 Stockhausen 1961, pp. 69-70.17 There is, of course, ample precedent for this type of spatialization in works of Gabrieli, Tallis, Berlioz,

and others. Some of these effects will be explored later in §5.4.18 Although cross-fading also occurs in individual notes within a chord in ZEITMASSE (1957), especially

m. 187, the effect in this work is not as dramatic as in GRUPPEN.

6

loudspeakers), connected fixed sources (the same sounds in all the loudspeakers),isolated spatial points.19

The way in which Stockhausen accomplished the rotation movements in KONTAKTE

was through the use of a rotation table. A single speaker was mounted on a platform that

could be rotated. Four microphones were set up around the platform, and the music

played back through the rotating loudspeaker was recorded by these microphones.

Even though some other works from this period do not have specific instructions

pertaining to spatialization, Stockhausen urged performers to create some spatial

“enhancement” nevertheless. His comments regarding the works REFRAIN (1959) and

ZYKLUS (1959) make this position clear:

In REFRAIN AND KONTAKTE, all instruments are amplified and the electronicmusic is played over 4 groups of loudspeakers. ZYKLUS is also amplified. ForZYKLUS...the spatial projection...should sound – in the auditorium –approximately as the percussionist – surrounded by the instruments – hears thepiece, but the direct sound should always be slightly louder than the amplifiedsound.20

This attitude towards amplification means that the “sound projectionist”, who oversees

the spatialization, takes on a more significant role – almost that of a performer.21

A further development in the technique of spatialization can be heard in CARRÉ (1959-

60). It is composed for four orchestras and choirs spread around the concert hall in the

way shown in Example 1.2.1c. To “compose out” the musical details of CARRÉ,

19 Stockhausen 2001b (CD 3), p. 174.20 Stockhausen 1993b (CD 6), p. 128. Additionally in ZYKLUS, the instruments are arrayed in a circle

which corresponds to their most prominent positions in the graphic score.21 The role of the sound projectionist in Stockhausen’s works can also be found in MIXTUR (1964) where

the mixing console operator has a significant effect on the ultimate sonic realization of the work.Another piece featuring performers in the role of sound projectionist is MIKROPHONIE I (1965).Here, sounds from a large tam-tam are picked up with microphones, and then electronically transformedby a group of “performers” using electronic filters. Texte 3, p. 51-53, 57-58.

7

Stockhausen relied on the British composer Cornelius Cardew,22 who realized the

framework plan for the work that Stockhausen himself made.

Each of the 101 musical “groups” in CARRÉ contains a simple box diagram that shows

which choir and orchestra groups play. The first four box diagrams are reproduced in

Example 1.2.1d. These box diagrams became the standard way that Stockhausen

represented spatial movement in his later works. Between some groups, there are

musical “insertions” that have little or nothing to do with the main structure of the work.

Cardew wrote that “...the insertions are certainly the most ‘sensational’ aspect of the

piece.”23 The spatial motion in these insertions generally includes rotations around the

hall.

Cardew provided a considerable amount of information about the spatial aspect of

CARRÉ. The language of space in this work became complex enough that several

rotations could be superimposed:

[In Insertion 69X, for instance], a soprano D (a ninth above middle C) is passedaround the four choruses at the rate of 12 changes per minute – each chorussustains it for five seconds plus one second after the next chorus has taken it up.Simultaneously, the strings and woodwind revolve in the opposite direction at therate of 60 changes per minute...Soon all the percussive instruments (harp, piano,vibes, cymbalum) enter simultaneously with an extremely sharp ff chord, and theflimmer becomes a murmur as this dies away.24

But concerning the technique of composing “rotating music” for widely-spaced orchestral

groups, Cardew wrote that “our main problem – chiefly because it was the most

predictable one – was that the sounds would always proceed by jerks around the room,

22 Cardew 1961a and 1961b. The more interesting information about space is in 1961b. Directions onhow to realize the symbols that Stockhausen drew are in Cardew 1961a, p. 619. Cardew’s later falling-out with Stockhausen is expressed most vividly in Cardew 1974.

23 Cardew 1961b, p. 699.24 Ibid., p. 698.

8

for between each orchestra there was a considerable space.”25 If the rate of rotation was

faster, the effect was less discontinuous, but these quick movements opened the door for

performance error, which could annihilate the desired “flimmer” of rotating sound.

HYMNEN (1966-7/1969) exists in three different forms: as electronic and concrete

music (1966-7), as electronic and concrete music with four instrumental soloists (1966-7,

withdrawn), and as electronic music with orchestra (1969).26 The first, four-channel

electronic version is the most interesting from the perspective of music in space.

Concerning the spatialization of HYMNEN, Stockhausen wrote

The direction and movement of sounds through four-channel spatial projectionis as important in this work as melody, harmony, rhythm, dynamic, color andsemantic. One can discover many new techniques of sound-rotation, alternation,flow, and depth-layering – as a further development of my compositionKONTAKTE.27

Given Stockhausen’s remarks, it is particularly lamentable that there is no spatial

information in the otherwise very useful 1968 score. But in Stockhausen’s “Realization”

essay,28 he showed in detail how he used the Soviet anthem in Regions II and III of

HYMNEN as a kind of structural background.29 Each vertical chord in the anthem was

numbered. Then, the chords were synthesized electronically, filtered, and lengthened

considerably so that in some sections, a quarter-note lasted as long as 24 seconds. These

chords were the basic material that was spatialized. Stockhausen spatialized chords 1-22

in a way that makes them seem to rotate clockwise. Chords 23-47 were then rotated

counterclockwise. Chords 48-83, in the third group, cross over each other while those in

the last group (chords 84-112) move in a trajectory similar to the third group but rotated

90° counterclockwise. Braun’s examples, which diagram the shapes that Stockhausen

25 Ibid., p. 698.26 Stockhausen Verlag 2008, p. 21.27 Texte 7, p. 91.28 Ibid., pp. 131-159; 160-162.29 Ibid., p. 134.

9

created,30 are reprinted in my Example 1.2.1e. In addition, Braun went one step further

and showed exactly how these sounds move around the space by doing a waveform

analysis of the four-track tape.31 Stockhausen’s essay on HYMNEN revealed more

spatial shapes that he created, which are reproduced in Example 1.2.1f.

The techniques used for spatialization in HYMNEN demonstrate that the increasingly

sophisticated technology at Stockhausen’s disposal altered his music; in effect,

Stockhausen’s compositional techniques evolved along with technology. If at first

Stockhausen’s goal was to use space to clarify different layers of serial polyphony, it was

now having the effect of giving the listener the impression of flying around in space in a

way that was not physically possible.32

1.2.2. Second phase of spatialization. The year 1970 marked a turning point for

Stockhausen’s spatial music. For the 1970 Osaka exhibition, the architect Bornemann

designed a spherical auditorium for the German pavilion.33 Here, Stockhausen’s dreams

of making music in such an unusual space were finally realized. Fifty speakers were

mounted along the walls of the auditorium, as shown in the diagram reproduced in

Example 1.2.2a. From March to September, there were 183 daily performances of

Stockhausen’s music without one free day.34 The German company Siemens designed

two “rotation mills” which Stockhausen used to control the location of the sounds in

space.35 Among many other works, the process compositions SPIRAL (1968),

30 Braun 2004, pp. 55-56.31 This analysis is exactly like the kind of “score” Stockhausen later printed in the score to the

HELICOPTER QUARTET.32 Kohl 2002 argues that the spatialization in TELEMUSIK (1966) – another spatialized electronic work

from this period – is more like GESANG DER JÜNGLINGEN, and contrasts to KONTAKTE andHYMNEN. In my view, the idea of moving rapidly in a metaphorical space from one continent toanother draws a connection between the spatialization of TELEMUSIK and HYMNEN.

33 Texte 3, pp. 153-155. Stockhausen’s essay “Osaka-Project: Kugelauditorium EXPO 70” providesbackground information to this undertaking. His report on the setup can be read in Texte 3, p. 139.

34 The complete performance plan can be found in Texte 3, pp. 177-181.35 The rotation mills were turned as if they were “coffee mills”. Stockhausen 2000b, p. 62.

The extensive technical expertise needed to realize the project was provided not only by Bornemann

10

KURZWELLEN (1968) and POLE/EXPO (1969-70) were performed hundreds of

times.36 Although these works do not specifically have any set directions for

spatialization in their scores, Stockhausen improvised spatial motion at his console by

creating “horizontal circles, vertical circles, below-above, above-below. Or spiral

movements of all different loops”.37

During the late 1960s and early 1970s, Stockhausen did not just explore a new kind of

concert hall – the spherical auditorium – but also, he took music outside of the traditional

performing space of Western music. In FRESCO (1969), musicians performed in the

closets, auditoriums, and other connecting hallways of the Beethovenhalle in Bonn.38

The musicians, divided into four groups, played several hours of music. The goal was to

create “wandering sound” by arranging performers in lines corresponding to their

tessitura: low on the left, high on the right. Like FRESCO, STERNKLANG (1971) also

departs from the normal space of the concert hall. Here, Stockhausen moved the musical

experience outdoors, into a park.39 Five groups of musicians play, while the audience sits

on the ground or ambulates freely. The exploration of different preforming spaces in

FRESCO and STERNKLANG marks a further departure from Stockhausen’s earlier

works, which were typically performed inside concert halls.

For the spatialization of SIRIUS (1975-77), Stockhausen arranged his soloists in a way

but also by Fritz Winckel, director of the Electronic Music Studio at the Technical University ofBerlin, and Max Mengeringhausen. Winckel and Mengeringhausen designed a spherical controllerwhich manipulated not only sound throughout the hall but also light. However, Stockhausen seems tohave preferred using his own 10-channel rotary mill to the more sophisticated Berlin design. SeeFöllmer 1996, Sigel 2000, Custodis 2004, p. 180-181, and Stockhausen 2000b, p. 62.

Michael Custodis wrote a history of the entire OSAKA 1970 undertaking from a financial, political andartistic perspective (Custodis 2004, pp. 161-186). He remarks that in addition to Stockhausen’s music,which was performed in the afternoon, works by Bernd Alois Zimmermann, Herbert Eimert, BorisBlacher, Erhard Großkopf, Beethoven and J. S. Bach were heard in the morning (ibid., p. 181).

36 Hopp’s analysis of the theory, practice and performance of Stockhausen’s process compositions,especially KURZWELLEN, is exemplary. See Hopp 1998.

37 Cott 1973, p. 46.38 Maconie 2005, p. 321.39 Texte 4, pp. 170-180. A diagram of one possible park setup is in Texte 4, p. 171.

11

that is reminiscent of CARRÉ. In SIRIUS, four soloists stand on podia at each of the

cardinal points of the space. In addition, the score calls for eight loudspeakers around the

audience, through which electronic music is heard. Since the podia on which the soloists

stand may not be the same height as the loudspeakers, there is the potential for musical

interplay on different vertical levels.40 This marked the first time that Stockhausen

composed electronic music for an eight-speaker array. Additionally, he employed a

rotating loudspeaker to project some of the sounds at the beginning and end of SIRIUS.

This rotating loudspeaker was electrically driven and could rotate up to 25 times per

second.41 The spatial setup in SIRIUS is shown in Examples 1.2.2b-e.

In SIRIUS, the possibility of vertical as well as horizontal interplay, taken along with the

larger loudspeaker array is clearly a development from Stockhausen’s Osaka

experience.42 However, the quadrophonic arrangement of soloists hearkens back to

Stockhausen’s earlier days organizing four sound sources, symmetrically arranged around

the audience. By combining these two spatial “traditions” with the possibility of outdoor

performance, Stockhausen integrated a variety of earlier experiences in this work.

1.2.3. Spatialization in LICHT (1977-2002). Shortly after composing SIRIUS,

Stockhausen embarked on the largest-scale project of his compositional career: a cycle of

seven operas called LICHT (“Light”). In these operas, Stockhausen continued to explore

new ways of organizing the space around listeners. The entire cycle is based on a single

one-minute piece, called the Superformula.43 The structure of the Superformula is

expanded, or “projected” over 29 hours. While there are specific elements determining

pitch, rhythm, form, timbre and articulation in the Superformula, there is no spatial

40 Misch believes that SIRIUS includes more movement in the vertical plane, unlike earlier works such asKONTAKTE. Misch 1999a, p. 152.

41 Stockhausen 2000a, p. 43.42 Stockhausen chose many unusual spaces for performances of SIRIUS, including a cloister in France.

Other performances were given at the planetarium at the National Air and Space Museum and theAlbert Einstein Spacearium, both in Washington DC. See Texte 4, pp. 438-439, and pp. 601-604.

43 Sketches for the Superformula can be found in Texte 5, pp. 147-160. For analytical studies, see Kohl1990 and Texte 9, pp. 13-34 (“Die Tonhöhen der Superformel für LICHT”).

12

information. Thus, Stockhausen was free to construct the spatial elements for each opera

without reference to any preexisting plan. Accordingly, the spatial design of each opera

varies considerably. By examining the way Stockhausen treats space in LICHT we can

gain insight to his developing compositional practice in a way not possible by turning

attention solely to these other, determined musical dimensions.

THURSDAY was the first LICHT opera that Stockhausen composed. In it, Stockhausen

set the stage for the more consistent, thoroughgoing spatial planing that characterize all of

his later works. One part, called UNSICHTBARE CHÖRE (1979) is scored for

fourteen taped choir groups, which are projected through eight loudspeaker groups.44 The

loudspeakers are deployed in the same circular arrangement as in SIRIUS. However,

action in the opera occurs on stage, in front of the audience. UNSICHTBARE CHÖRE is

performed twice during the THURSDAY opera – first as “background music” for Act I,

and then Act III. During Act III, the channels are reversed, as shown in Example

1.2.3a.45 This unusual channel reversal causes sounds to move to corresponding locations

180° apart from their original places. However, it does not cause the choir motions to

change from clockwise to counterclockwise.

Stockhausen’s plans for this piece allow us to determine that the spatial movement in

UNSICHTBARE CHÖRE is usually clockwise by “step” – from one loudspeaker group

to the next adjacent one. Although one loudspeaker is usually associated with a single

choir at any one point, a choir is occasionally heard in two adjacent speaker groups. This

has the effect of emphasizing that choir’s music, as listeners will hear it coming from a

larger area of space (and presumably slightly louder, as well.) Once, a choir is heard

“split” over two non-adjacent speakers. Occasionally, there are leaps across the space,

44 According to Stockhausen, the work cannot be performed live because of the polyphonic complexity;only from tape. See Texte 5, pp. 204-205.

45 The channel reversal corresponds to a peculiar and noteworthy aspect of the THURSDAY opera: in ActIII scene II, there is a “Schattenspiel” during the section called VISION, where the actors’ and mimes’gestures are projected upside-down simultaneously. A photograph can be seen in Texte 5, plate 206.This “upside-down play” is analogous to the channel swapping in UNSICHTBARE CHÖRE.

13

from one loudspeaker group to a more distant one.

The essential pitch fluctuation in UNSICHTBARE CHÖRE is actually somewhat slower

than the rate of spatial change.46 Although Stockhausen realized the pitch material of

each choir in a way that makes it seem to be more rhythmically interesting, the spatial

movement is more active than the structural pitch material. In general, Stockhausen’s

spatialization procedure in this work helps to organize and isolate the complex

polyphonic structure of the piece into different spatial areas.

KATHINKAS GESANG (1983/2001) is the second scene of the next opera to be

composed in the LICHT cycle, SATURDAY.47 The 1983 version is for solo flute and six

percussionists. Each percussionist plays two sound plates (rectangular pieces of metal

that have a pitched sound) and an array of homemade instruments that are hung from the

their bodies. The percussionists are stationed symmetrically around the audience along

the walls of the performance space. Each is amplified and projected through

loudspeakers at his or her location. This spatial design is shown in Example 1.2.3b.

Towards the end of KATHINKAS GESANG, the flutist must run from one percussionist

to another, swapping some of the sound-plates. This swap has the effect of physically

moving some plates that contain the notes of the Eve-formula, a motive associated with

one of the opera’s protagonists.48 Toop theorizes that this is an “emanation of Eve,

guiding Lucifer (and potentially, anyone else,) to a possible rebirth.”49 The sound-plate

46 The essential pitch structure can be found in the sketch printed in Texte 5, p. 210.47 Although we focus here only on the work KATHINKAS GESANG from the SATURDAY opera, this

opera has many novel spatial ideas. In a sense, the opera represents a large-scale “opening up” ofspace. The GREETING (opening) is played by four brass choirs in the four corners of the hall. SceneIII, called LUZIFERS TANZ, is performed in a vertical array. Finally LUZIFERS ABSCHIED (sceneIV) takes place in a cathedral where the choir surrounds the audience. Towards the end, choir andaudience move outside, into the town square.

48 Toop 2005, pp. 101-128. The precise way in which the sound-plates are swapped is detailed in the1983 score, p. xiv, and again in Toop 2005, p. 126.

49 Toop 2005, p. 126.

14

swapping is also a way to “create a little spatial variety.”50 In this way, spatial movement

takes both a structural function and also a programmatic one within the opera.

Stockhausen’s spatial language in DER KINDERFÄNGER (1986) is evidence of

another development in the composer’s spatial design. This work, which is part of the

Monday opera, is often paired with the next piece, “ENTFÜHRUNG,” when it is not

performed in the context of the staged opera. For the Monday opera itself, Stockhausen

selected a number of musique concrète sounds and arranged them into 55 “Tonszenen”

(“sound-scenes”). The single-channel (mono) sounds were spatialized by using two

stereo controllers and an uni-potentiometer joystick. This joystick is a further

development of the rotation-table technique: Stockhausen wrote that it was “relatively

easy to rotate quickly”.51 The Tonszenen appear in certain places throughout the entire

Monday opera, and to make them easier to analyze, Stockhausen published diagrams of

the way in which each one was spatialized.52 The explanation of his “box diagrams” is

shown in Example 1.2.3c. In DER KINDERFÄNGER and ENTFÜHRUNG,

Stockhausen used the most Tonzenen in any one section of the opera.

During the 2002 Summer Courses in Kürten, Germany, Stockhausen made many

revealing comments about the way the spatial movements were composed. While the

movements are “not serial”, they are “statistically balanced” throughout the space. The

decision of how to spatialize each Tonszene came from the composer’s “imagination”: “I

found a movement that would fit”.53 The freedom with which Stockhausen manipulated

the spatial domain allows for striking effects, especially as many of the sounds are

naturally exciting.

50 Ibid.51 Texte 7, p. 446.52 Texte 7, pp. 436-463.53 These transcriptions were made by the author in 2002. Stockhausen’s comments suggest that adopting

analytical methods applied to serial music may not be wholly appropriate in some spatial music.

15

One Tonszene is reproduced in Example 1.2.3d. In Tonszene 36 there are four different

sounds: an airplane, machine gun, lion and shattering flower-pot. Stockhausen has

spatialized the first three sounds in a way that we might hear them in real life. The

airplane emerges from the horizon behind us and flies quickly overhead; after it

disappears in front of us, it is possible to imagine a machine-gunner appearing from the

horizon in the same place. We might also imagine seeing a lion – in front of us – at a

zoo. However, the sound of the shattering flower-pot has a spiral spatialization, which is

not realistic. In this as in many Tonszenen, it is evident that Stockhausen mixed his sense

of real with imaginary when spatializing the sounds in DER KINDERFÄNGER.

It is possible to see a programmatic justification for the spatial motion in these Tonszenen

in terms of the underlying plot of the pair of pieces DER KINDERFÄNGER and

ENTFÜHRUNG. The mixture of “realistic” spatializations – those that somehow reflect

the way we would expect a sound to move around us, such as the airplane flying

overhead – with the “imaginary” – freely composed movements that move in unrealistic

ways – blurs the line between reality and fantasy. Through this, the Pied Piper’s victims

eventually lose their ability to judge what is real and what is imaginary, making them

easier to lead away to whatever fate awaits them.

Although OKTOPHONIE (1990-91) and ORCHESTER-FINALISTEN (1995-96) are

from different operas – OKTOPHONIE is part of TUESDAY while ORCHESTRA-

FINALISTEN is from WEDNESDAY – they share the same basic spatial setup. In both,

Stockhausen set his speaker array in the shape of a cube. It is likely that he did this in

order to try to recreate the effect of the spherical auditorium in Osaka.54 In this unusual

octophonic setup, the loudspeakers are placed at the eight vertices of the cube, as shown

in Example 1.2.3e. Listeners sit in the middle of the bottom four speakers.

54 Unlike Osaka, sounds cannot be spatialized below the listeners. This causes the audience to focusattention on activity overhead, which matches the programmatic choice of “shots”, “bombs” and“crashes” that we analyze in chapter 3.

16

The TUESDAY opera is a dramatization of a conflict between two protagonists in

LICHT, Michael and Lucifer. This helps to explain some of the spatial activity in

OKTOPHONIE.55 Here, there are essentially two types of electronic music. Drones

slowly rotate around the listener in various planes, while the more active sounds consist

of “shots”, “crashes”, and “sound bombs”. Although the drones are a further

development of Stockhausen rotation-table paradigm, the other sounds are meant to

evoke the atmosphere of an aerial bombardment, possibly reminiscent of Stockhausen’s

experiences as a young boy in World War II. The shots, crashes and sound bombs create

much more vertical movement in the sound scene than is common in Stockhausen’s

earlier music. While the electronic sounds pass from ceiling to floor and vice-versa,

other scenes take place on stage. A thorough analysis of this complex spatialization will

be offered later, in Chapter 3.

The electronic and concrete music to ORCHESTER-FINALISTEN is in the

WEDNESDAY opera, which in contrast to TUESDAY is the “day of cooperation”

among the three protagonists: Eve, Michael and Lucifer. Although the octophonic

loudspeaker setup in ORCHESTER-FINALISTEN is the same as in OKTOPHONIE, the

way Stockhausen uses it is quite different. Instead of focusing attention on slow rotations

and rapid directional spatial movement, Stockhausen composed relatively static music for

each plane of the cube.56 In OKTOPHONIE, sounds move from point to point, allowing

the listener to localize them in the space, whereas in ORCHESTER-FINALISTEN,

listeners must be more attuned to hearing entire “planes” or walls of music. This is an

important aural adjustment one has to make, and shows different compositional

approaches to spatialization within the same loudspeaker setup.

55 A detailed explanation of the spatialization procedures, clarifying Stockhausen’s use of potentiometersand the QUEG, is in Stockhausen 1993a, pp. 150-170.

56 The swapping of layers that occurs in the three-dimensional space makes an important analogy to theswapping of layers in the special Wednesday-Superformula composed for the opera. Richard Tooppointed this out in a lecture at the 2006 Stockhausen Courses in Kürten.

17

Between OKTOPHONIE and ORCHESTER-FINALISTEN, Stockhausen composed

spatial music for the FRIDAY opera. Although the Tonszenen from FREITAG (1991-

94) share the same appellation as those in the Monday opera, there are not many

similarities. The music that is heard during the Friday Tonszenen is not sampled from a

sound archive or from everyday experience; rather, it is composed of complex melodic

lines derived from the Superformula.57 Moreover, there are only twelve Tonszenen in

Friday, instead of the 55 in Monday. Finally, the speaker placement is significantly

different than in the Monday opera.

In FREITAG, a bank of twelve speakers is ideally set up as a “pyramidal panorama” in

front of the audience. Examples 1.2.3f-h show three different ways of setting up the

loudspeakers. The standard way of arranging the loudspeakers is shown in Example

1.2.3f. This was the arrangement during the 2001 Stockhausen festival in Kürten. When

the work was performed at the Leipzig opera, fourteen loudspeakers had to be used

instead of twelve. This alternative arrangement is shown in Example 1.2.3g. In an

uncharacteristic move, Stockhausen allowed for an alternative speaker setup for concert

performance, where the speakers are arranged in a circle – but in this arrangement, some

speakers must be hung higher than others. The arrangement is shown in Example 1.2.3h.

These three different loudspeaker setups mean that multiple spatial experiences are

possible with the same music.

In one respect, spatialization in the Friday Tonszenen is used to clarify a dense

polyphonic texture. Each of the twelve loudspeakers is musically linked to a pair of

“dancer-mimes”. The dancer-mimes make twelve entrances during the opera.

Stockhausen created elaborate plans for the way in which these twelve groups of

performers enter. A new couple is introduced in each successive Tonszene, but the order

57 Toop 2005, p. 146ff.

18

of entry of each couple always changes.58 Because of the way the couples are linked to

loudspeakers, the spatial shapes created by their entry are always different for each

Tonszene. Following this, different combinations of the speakers are used; in the first

Tonszene, only one speaker is active, whereas at the end all twelve have music projected

through them simultaneously.

In another respect, spatialization is used to create unusual shapes in space. Not only are

the loudspeakers used to project the music of the twelve couples, but also they are used

for playback of the Friday electronic music. This music is projected through eight of the

twelve loudspeakers that are used to amplify the music of the dancer-mimes. The

channels are indicated by roman numerals in Examples 1.2.3f-h. Regarding one of the

densest polyphonic passages, Toop remarked that “even with the most sophisticated

spatial projection, there’s a point at which entropy has to set in, and it would make sense

not to proceed that far.”59 Despite this, Stockhausen insisted that “there are clearly

perceptible flight paths, forms of movement, and speeds in space”.60

In the seventh and final LICHT opera, SUNDAY, spatialization is focused on the

movements or shapes created by performers themselves. There are no purely electronic

tape compositions in the opera at all, though the last scene (“SONNTAGS-ABSCHIED”)

is scored for five electronic synthesizers. Although ENGEL-PROZESSIONEN (2000)

and LICHT-BILDER (2002-03) both have sophisticated spatial languages, the most

significant work of spatial music in the opera is LICHTER-WASSER (1999).

In LICHTER-WASSER, spatialization is used not only to clarify contrapuntal structure,

but also to paint various rotations, shapes, and forms in space. An orchestra of 29

58 See Toop 2005, p. 137 for an analysis of the order in which the couples enter for each Tonszene.Stockhausen’s sketches dated 27/28 December 1991 and 29 December 1991 show how Stockhausendecided on the ordering of the couples’ entries. The final “synchronization plan” for the couples’entries can be found in the score to FREITAG-VERSUCHUNG (Stockhausen 1997), pp. vi-viii.

59 Toop 2005, p. 138.60 Stockhausen 1997, p. xxxvii.

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musicians is spread out in a geometric pattern around and within the seated audience.

During the main section of the work, two monophonic melodies are passed around from

player to player, each instrumentalist intoning a note or group of notes. The action of

these two melodies winding their way around and through the audience sets up a two-part

spatial counterpoint. As we will see in chapter 4, the speeds, shapes, directions, and

densities of movement can be measured because of the detailed box diagrams in the score

and the precise tempo indications.

1.2.4. Final works. Stockhausen’s last, incomplete work-cycle KLANG (2005-07),

contains some remarkable spatialized music. The unusual work HIMMELS-TÜR (2005),

which is performed on a special door by a percussionist, uses spatial location as part of

the program. After beating on “Heaven’s Door” for a certain amount of time, the

percussionist finally walks through it and plays on a battery of cymbals which cannot be

seen by the audience. But one of Stockhausen’s very last works, COSMIC PULSES

(2007), takes the composer’s lifelong interest in rotating sounds to the extreme.

In COSMIC PULSES, “24 melodic loops...rotate in 24 tempi and 24 registers...each

section of each of the 24 layers has its own spatial motion between 8 loudspeakers, which

means that I had to compose 241 different trajectories in space.”61 The effect of hearing a

superdense spatial composition such as COSMIC PULSES surely causes some “entropy”

to set in, of the kind Toop mentioned earlier with regard to the Friday Tonszenen. Still,

the work is a kind of bookend, summing up a lifetime’s experience with spatial music in

a thrilling synthetic electronic space.

1.2.5. Conclusion. We can break down Stockhausen’s use of space into two categories.

First, space can be used to clarify complex contrapuntal textures. Second, it can be used

for sheer effect, either by rotation or by creating characteristic shapes for particular

sounds. For Stockhausen, the performing space, or the space which his music

61 Stockhausen 2007a (CD 91), pp. 6-7.

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“illuminates”, is a kind of “filter”. Whether one of his works is performed outdoors, in a

concert hall, or in a cave far beneath the earth’s surface,62 Stockhausen’s music changes

our perception of space, and space changes it.

1.3. Spatialization in Stockhausen’s Essays, Interviews and Sketches

The evidence presented in the previous section overwhelmingly indicates that

Stockhausen organized space as a compositional element. Both the quantity of

spatialized works and the variety of ways that the space is organized in Stockhausen’s

music support this claim. By surveying his essays, interviews and sketches, we can gain

a deeper insight into how he regarded space and what it meant for the structure of his

musical compositions.

1.3.1. Essays. Two essays published at critical junctures in Stockhausen’s career help to

crystallize his orientation towards spatial music. The 1958 essay “Musik im Raum”

(“Music in Space”) was based on a lecture by the same name.63 Here, Stockhausen asked

why one should spatialize music in the first place. He writes that there was a

compositional imperative in early serial music to treat all musical aspects equally. But, in

such “pointillistic” music, the individual characteristics of different musical elements

change so rapidly that one often finds oneself in a “state of suspended animation”. The

music essentially “stands still”.64 Allowing one musical dimension to predominate could

bring a renewed sense of temporality to the music; but by doing this, the spirit of equality

among all parameters is lost. By spatializing compositional elements, the composer can

highlight musical processes without contradicting “the spirit which gave birth to the idea

of equal valuation for all sound-characteristics.”65 Thus, spatialization can clarify a dense

62 In November 1969, Stockhausen’s music was performed in the Jeita caves of Lebanon. See Texte 3, p.377.

63 This lecture is now available on Stockhausen’s Text-CD 7. The English translation is in die Reihe 5,pp. 67-82 while the original is in Texte 1, pp. 152-175.

64 Stockhausen 1961, p. 69.65 Ibid. Cage’s idea is similar: “Rehearsals have shown that this new music...is more clearly heard when

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serial texture; or, put another way, it can allow the composer to write a denser texture

than is ordinarily possible.

But how can spatialization be accomplished in a way that allows movements in space to

relate to other musical elements, while, at the same time, allowing the spatial structure to

maintain some sort of internal coherence? Stockhausen suggested three ways of

quantizing space around the listener: the first is to set up sound sources in a line; the

second in a triangle, and the third in a circle. The last solution is most favorable because

of the greater possibility for continuous motion and the lack of associations with

traditional musical space relations.66 Stockhausen proposed two different ways of

quantizing the circle around a listener; the first method relies on a left-right symmetry (1

at the front, 0 at the back) while the other involves measuring proportional distances

around the circle (such as 3:2, 1:2) which can be brought into correspondence with

equivalent proportions in the rhythmic or pitch domains.

In “Music in Space”, Stockhausen offered a practical guide both for composers wishing

to organize the space around them, and for listeners trying to make sense of

Stockhausen’s own musical works. The ideas in Stockhausen’s article explain both his

own personal approach to spatialization in the 1950s, and also leave open avenues of

exploration to other composers by inviting the reader through clear graphical diagrams

and relatively simple language. Although he briefly considered how music might move

in three dimensions, Stockhausen made no definitive statements about it in this early

article. These questions are addressed in his next major article about spatial music.the several loud-speakers or performers are separated in space rather than grouped closely together.”Cage 1961, p. 12. Another sentiment is echoed by Henry Brant: “The spatial procedure, however,permits a greatly expanded overall complexity, since separated and contrasting textures may besuperimposed freely over the same octave range...with no loss of clarity.” Brandt 1978, p. 224.Heikinheimo puts it differently: “...the use of multiple channels is a kind of emergency measure whichhelps the composer to create clarity in a compositional weave that is either too monotonous or toocompact.” Heikinheimo 1972, p. 86.

66 The association of the left side with low sounds and the right side with high sounds – which comes fromexperience with keyboard instruments – could make spatial “lines” problematic, according toStockhausen, Texte 1, p. 169.

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In an extemporized lecture from 1972, Stockhausen spoke at length about “Vier Kriterien

der Elektronischen Musik” (“Four Criteria of Electronic Music”). This lecture was

transcribed and printed a year later.67 Between “Music and Space” and the “Four

Criteria” articles, Stockhausen had the intense experience of three-dimensional spatial

auditory “immersion” during the 1970 World’s Fair, as mentioned above in §1.2.3.

Regarding spatialization, Stockhausen’s third criterion (“Composition of multilayered

space”/“Die Komposition mehrschichtiger Raumlichkeit”) is the most important. Unlike

his earlier “Music in Space”, there is no technical description and there are no diagrams.

Instead of serving as a practical guide or proposing another system of spatial

organization, Stockhausen concentrated his discussion on experiences, sensations, and the

potential of aural imagination. Stockhausen focused on the possibilities of shapes that

sounds create in the air, and their qualitative effect on the human organism. Through the

integration of space in music, Stockhausen claimed one can be “emancipated from the

physicality of the body”;68 thus, spatial music “forms new people”.69 Through

spatialization, Stockhausen claimed he could “form spatial constellations like chords, and

bring them into relation with other constellations”, “compose spatial melodies”, and

“describe all possible geometric configurations”.70 According to Stockhausen,

In the next few decades there will be no limits placed on the imagination to createnew resources for music so that it progresses, emancipates itself, and becomes

67 The original publication was in Beuys and Herzogenrath 1973. The reprint is in Texte 4, pp. 360-424.The lecture is on Stockhausen 2007c (Text-CD 13).

68 Texte 4, p. 39069 Ibid., p. 384. Other writers have echoed this sentiment: “...for as long as there is little encouragement or

opportunity to acquire, let alone to develop, auditory spatial awareness, our society will surely have animpoverished aural architecture.” Blesser and Salter 2007, pp. 331-332.

70 Ibid., p. 385. “Ich könnte im Verlauf einer Komposition genauso räumliche Konstellationen etablierenwie Akkorde, und darauf andere Konstellationen beziehen; ebenso Raummelodien komponieren, diealso durch das Hoch- und Tiefgehen von Klängen, durch das An-mir-vorbei-kommen in bestimmtenHöhen oder Tiefen oder geraden oder gekrümmten Linien, alle möglichen geometrischenKonfigurationen beschreiben, wenn sie vom Komponisten strukturell benutzt werden.”

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free of the physical limitations that we have, up to now, accepted as natural.71

Many questions arise from statements in the “Vier Kriterien” essay. What are the spatial

constellations Stockhausen used? What might a spatial “melody” sound like? What

might constitute “geometric configurations” of space in music? While he never answers

these questions directly, Stockhausen’s statements are strong guideposts that suggest

ways of approaching his spatial music after 1970.

Finally, Stockhausen recognized the ability to use microphones to amplify various sounds

produced by musicians, and then project them into space using loudspeakers. In this

respect, he wrote:

The use of microphones is something quite new. It indicates another discovery,that every interpreter can have a wireless microphone, so that one can hear eventhe quietest consonants projected into the hall, and so that the sound of theperformance can be projected throughout an auditorium by a sound projectionistwho is responsible for the entire acoustic.72

This quote verifies that even works that are not specifically spatialized could be

“clarified” by some spatialization during performance. As previously stated in §1.2.1, the

sound projectionist is then responsible for the spatial design.

In a third article, called “Die Zukunft der elektroakustischen Apparaturen in der Musik”

[“The future of electroacoustic apparatus in music”]73, Stockhausen described two

improvements that could be made to spatialization in music: the use of “sound

wanderers” [Klangwändler] and better rotation tables. The proposals are further evidence

71 Ibid., p. 392. “Da sind in den nächsten Jahrzehnten der Phantasie keine Grenzen gesetzt, um der Musikneue Hilfsmittel zu schaffen, damit sie weiterkommt, sich emanzipiert und frei wird von denphysikalischen Grenzen, die wir bisher als selbstverständlich hingenommen haben.”

72 Texte 9, p. 594. “Die Mikrophonierung ist etwas ganz Neues. Es bedeudet auch eine neueEntdeckung, daß jeder Interpret einen Sender haben kann, so daß man auch die leisesten Konsonantenin den Saal projiziert hört, und daß der Klang der Aufführung rundum projiziert werden kann durcheinen Klangregisseur, der die Verantwortung für die gesamte Akustik hat”.

73 Texte 4, p. 425-436. The article began as a radio lecture in 1972 and was first published in 1974 in‘Musik und Bildung’, vol. 7/8.

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of the crucial role rotation and movement played in Stockhausen’s concept of musical

spatialization.

1.3.2. Interviews. In addition to the numerous comments about spatialization quoted

throughout §1.2, Stockhausen made a large number of important statements during

interviews. In his interviews, Stockhausen often underlines the importance of space to

his compositional projects. Referring to the loudspeaker arrangement in GESANG DER

JÜNGLINGE, Stockhausen said:

The speed of the sound, by which one sound jumps from one speaker to another,now became as important as pitch once was. And I began to think in intervals ofspace, just as I think in intervals of pitch or durations. I think in chords of space.74

Stockhausen lamented that

...we’re used to sounds which have been fixed to objects that have producedsounds, like instruments or human voices. And we have lost the ability to fly likebirds...75

These comments pertain most directly to his works TELEMUSIK and HYMNEN, where

the overt musical program includes travel at superhuman speeds. Stockhausen remarked

on the experience of “projecting” music in the spherical auditorium during the 1970

Osaka fair:

This polyphony of spatial movements and the speed of the sound become asimportant as the pitch of the sound, the duration of the sound, or the timbre of thesound.76

In addition to creating a kind of “spatial polyphony”, Stockhausen was also interested in

quantifying the sources, distances, and speeds of sound, to bring them under control

74 Cott 1973, p. 92.75 Ibid., p. 44.76 Ibid., p. 46.

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through his serial working methods.77 Although we encountered part of this quote earlier,

it is now presented in more context.

...any point in space should be precisely defined with respect to where the soundoccurs and how it travels from one point to any other...what I have in mind is asituation where you’d sit and project your sound in space with any given speedand draw the musical configuration of the sound in the air.78

In later sets of interviews, Stockhausen continued to emphasize his hope that listeners

would perceive movement in his space-music. Referring to MICHAELS REISE (1978),

Stockhausen revised his compositional priority somewhat, saying

...in my method of composition, the movement and the direction of the sounds areof paramount importance. They count as much as the volume and the timbre, andlittle less than the sound-frequencies.79

In a 1997 interview, which was conducted while he was working on the Sunday opera,

Stockhausen remarked that

...I now use a series of constellations of sounds moving in space or standing in acertain constellation. I work with series of space constellations.80

Finally, Stockhausen had some hope that “highly perceptive persons” would, in the

future, learn to listen to spatial music and find meaning in it just as they found meaning in

music of the past.

Listening to music which has been spatially and collectively perfected by modernacoustic equipment will assume a social weight of great importance, the weight of

77 Although Coenen 1994 defined several serial “working methods” in Stockhausen’s “compositionalparadigm”, he did not address the issue of space. Comments such as those quoted in this sectionsuggest that space should be elevated to a “working method” on nearly an equal footing as the othermethods Coenen described.

78 Cott 1973, p. 202-203.79 Tannenbaum 1987, p. 38.80 Paul 1997.

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a ritual.81

Stockhausen’s statements about the role of space in music give an idea of just how

important it was for him to organize this aspect of experience. What evidence do we

have for spatialization in his sketches?

1.3.3. Sketches. Countless sketches indicate that Stockhausen thought in terms of

spatial forms and shapes while composing. One of the first indications of spatialization

can be found in Stockhausen’s sketches for GESANG DER JÜNGLINGE. Appearing

either as a schematic diagram of the speaker setup,82 or as a drawing with lines

connecting different speakers from a perspective seen from above,83 these sketches show

a certain tentativeness in dealing with the spatial issues.

Heikinheimo remarked that “composing for several channels...requires the creation of a

very detailed work-plan made to meet the demands of a systematic working technique.”84

We can observe a more elaborate spatial graphology than GESANG in the sketches for

KONTAKTE (1959-60). Here, a repertoire of symbols denote a particular sequence of

movements – not individual movements on their own.85 The “shapes” are then deployed

serially, along with other musical parameters. This system of spatial diagrams is found

throughout the sketches for KONTAKTE.86 Once the motion was determined serially,

Stockhausen’s sketches show detailed instructions – or, “performance plans” – for the

rotation table. These plans are arranged precisely in seconds, and include annotations81 Tannenbaum 1987, p. 41.82 Stockhausen 2001a, p. 31.83 Ibid., p. 23.84 Heikinheimo 1972, p. 83.85 A key page, which lends great insight to the spatial movements in KONTAKTE, is reproduced in

Heikinheimo 1972, p. 131. Unfortunately, parts of this sketch are not entirely clear even afterexamining it at the Stockhausen archive.

86 The sketch-book for KONTAKTE is arranged in ten sections, each of which is designated by a Romannumeral. Each section has a number of pages ranging from 8 to 132. Some of these are subdivided(“31.1”, “31.2”, etc.) These subdivisions are always given in Arabic numbers. The relevant sketches inwhich Stockhausen shows detailed plans for the rotation table [Rotationstisch] include sketch pagesVI/67.1, VI/67.2, VI/68.1, VI/68.2, and VI/69. They deal primarily with formal sections XIV A-D ofKONTAKTE.

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such as “accell – rit” and “schnell”.

While the perspective of “looking down” from above the audience was briefly put to rest

in KONTAKTE, Stockhausen returned to it in the sketches for CARRÉ. These box

diagrams – which became standard in almost all of Stockhausen’s subsequent

spatialization sketches – are crucial for negotiating the spatial design of CARRÉ. But in

addition to the box diagrams, Stockhausen often made numerous “Sitzpläne” or “seating

plans”, both for the musicians and audience. Like the box diagrams, these seating plans

are always from the perspective of “looking down” on the performance space.

In sketches of the LICHT operas, spatial information is plentiful. For each of the works

mentioned in §1.2.4, many sketches with spatial information are available for study.

Most often, Stockhausen’s sketches contain box diagrams. In COSMIC PULSES, spatial

movement was such a central concern that Stockhausen devoted at least seven sketch

pages to planning the movement of 241 sounds in 24 layers around the audience.87 If

anything, the attention given to space in Stockhausen’s music increased towards the end

of his life, culminating in the hyperdense texture of COSMIC PULSES.

We can see that Stockhausen put great emphasis on spatialization in his writings,

interviews and sketches. Yet, most literature about Stockhausen’s compositional

methods tends to focus on his techniques of organizing pitch, rhythm, and form. What

literature is there that can help us to understand the spatial aspects of

Stockhausen’s music?

87 These pages were on display at the 2007 Stockhausen Courses in Kürten; the diagrams are reproducedin Stockhausen 2007a (CD 91). Three pages contain the 241 box diagrams while four contain relatednumerical sequences. We will investigate this work in more detail in §5.2.4.

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1.4. Literature on spatialization and its role in Stockhausen’s music

1.4.1. Spatial Cognition, Perception and Psychoacoustic Studies. A large body of

literature seeks to explain and discover the cognitive systems that humans use to perceive

their acoustic environment, and how we make sense of it. These studies are useful

because they can help guide us in interpreting the measurements we will make of spatial

activity in music. Three authors who have surveyed a broad swath of literature in this

field while also making contributions of their own, are Albert Bregman, Jens Blauert,

and Blesser/Salter.

Reporting on evidence that “space is considered as a continuum by the auditory

system”,88 Bregman concluded that spatial location is in fact a grouping mechanism in

auditory cognition. However, it is not always the primary one: Deutsch’s celebrated

“scale illusion” studies showed that we sometimes prefer to group auditory stimuli by

frequency range rather than by spatial location.89 Bregman writes that “spatial

location...seems to lose out when it is placed into conflict with other bases for

grouping”.90 Although his studies support the idea that we can analyze sounds moving

from one place to another continuously (even if they are not actually moving themselves),

they also suggest that we should keep pitch as well as spatial location in mind when

grouping sounds together to form larger structures.

While Bregman investigated the way in which auditory streams are segregated, Blauert’s

research focused more on the way in which we actually localize sound. His interest lies

in the realm of the psychophysics of sound localization. Blauert summarized many

studies that measured the precision of human sound localization for various frequency

bands. He concluded that the “localization blur” is much greater in the median plane

88 Bregman 1990, p. 74.89 Deutsch 1973, 1974; Bregman 1990, p. 76.90 Bregman 1990, p. 82.

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(above the listener’s head, from front to back) than in the horizontal plane (around the

listener in a circle).91 The implications for spatial music in three dimensions have been

largely borne out by electroacoustic music practice. The relative sparseness of spatial

music that moves above the listener can probably be attributed to the fact that we cannot

establish the location of sounds above us with as much precision as those around us in the

horizontal plane.

Part of Blauert’s goal is to establish “head-related transfer functions”, or mathematical

functions that can be applied to auditory inputs in order to mimic the spectral distortions

introduced by the ear structure. One use of transfer functions is to create simulations of

room acoustics for architects. Composers can also use HRTFs to create “synthetic”

spaces in their electronic music. The connection between sound and architecture is one

critical aspect of negotiating the connection between sound and space – one that Barry

Blesser and Linda-Ruth Salter researched for their study.

Blesser and Salter’s research is an attempt to synthesize many different disciplines that

pertain to the relationship between sound and space. Their term “aural architecture

“refers to the properties of a space that can be experienced by listening.”92 Through the

centuries, humans have created various spaces with different acoustical properties in

order so that the quality of sound is altered. These range from ancient Greek

amphitheaters to “artificial” spaces created by computers and loudspeaker arrays (such as

the BEAST system).93 In crafting an interdisciplinary approach to the idea of “selecting,

designing, and experiencing spaces by listening”,94 Blesser and Salter look to the fields of

musicology, archeology, neurobiology, mathematics, anthropology, psychophysics,

cultural evolution, religious ceremonies, theatrical sound, and virtual space simulation, to

91 Blauert 1997, pp. 41-44.92 Blesser and Salter 2007, p. 5.93 Ibid., p. 17494 Ibid., p. 275.

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name just a few.95 While they lose some focus by casting their net so wide, Blesser and

Salter use aural architecture as a way to learn more about the sound qualities that a given

culture values. Their study is valuable because it helps us to understand what people at a

given time and in a given culture listen for in the sound world around them.

1.4.2. Category/Classification Studies. In her 1994 dissertation, Maria Harley/Maya

Trochimczyk96 completed a thorough study which locates the concept of space in the

Western philosophical tradition, while also covering a broad range of historical issues

surrounding the use of space in music. Her analyses, which center around music of

Henry Brant, Iannis Xenakis, and R. Murray Schafer, help to support her classification of

spatial designs. This classification scheme consists of:

1. Acoustic environments, such as enclosed space, concert hall, open space, orheadphones;2. Sound-space types, such as vocal-instrumental or electroacoustic soundsources, or a mixture of the two;3. Categories of static or mobile performers and/or audience.97

Harley also proposed a categorization of spatial theories. These categories are

1. An extension of polyphony (essentially, a way of heightened “streamsegregation”);2. Music built from sound objects projected into space;3. Spatialization as a new “parameter” manipulated compositionally;4. Spatialization as conceptual experimentation with performance rituals and theircontexts.98

In addition, Trochimczyk proposed categories of different spatial arrangements of sound

95 Ibid., p. 367, note 1.96 As a result of the author’s 2001 name change, there is the potential for some confusion. Articles by

Maria Harley and Maja Trochimczyk are by the same person. Citations are to the name under which agiven work was published.

97 Harley 1994a, pp. 185-186; also Trochimczyk 2001, p. 40.98 Harley 1994a, pp. 99-160; summarized in Harley 1994a, p. 341.

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sources, ranging from point to sphere. Although the dimensions can vary from zero to

three there is also the possibility of fractional or “fractal” dimensions. Simple geometric

patterns of performers could be a point, line segment, triangle, square, circle, cube

or sphere.99

“Net-based designs”, on the other hand, “underlie the chaotic happenings of Cage and

experiments with performance formats in which the traditional form of the concert has

disappeared.” In these “centerless spatial designs, musical performances can take place

outdoors or in unusual spaces...Here, the division between performer and audience

disappears, standard musical notation gives way to written instructions, and the ‘supra-

temporal’ musical work is replaced with a process or an action that cannot be repeated

twice in the same way”.100 Examples of such spatial designs include works by Boulez,

Cage, Brant, Hölszky, Bryars, and others.

Harley/Trochimczyk’s research into spatial music is primarily concerned with making

categories or classification schemes that can be applied to a large number of musical

pieces or sound designs. Her work is valuable because it helps us to see similarities

between a variety of works, and differentiate spatial arrangements according to the

general attitude towards space and the purpose of spatialization.

1.4.3. Analytical Studies. There are not many sustained or detailed analytical studies of

spatial motion in the published music theory literature. Composers (such as Xenakis

1970 and Brant 1978) have explained some of their techniques when composing spatial

movements, but do not provide much insight into how one might build an analytical

methodology. The music of composers such as Varèse has attracted a substantial amount

of scholarly work, but little information remains of Varèse’s actual spatialization

schemes, making it difficult to know exactly what spatial effects his music might

99 Trochimczyk 2001, p. 41.100 Trochimczyk 2001, p. 51.

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communicate to audiences.101 Another class of writers, such as Imke Misch, recognize

the importance that space plays for Stockhausen, but offer only general summaries of the

spatial techniques that are used in his music.102 Three authors have made significant

contributions to the field, however: Gisela Nauck, Boris Hofmann and Sara Overholt.

Nauck’s study, Musik im Raum – Raum in der Musik [Music in space – space in music],

is her 1997 dissertation from the Technical University of Berlin. She traces the history of

modern theories of space composition from Beyer’s “music of the future” article103

through Webern and the Darmstadt school of the 1950s. According to Nauck, notable

composers that contributed to this line of development include Varèse, Boulez, Cage,

Schnebel, and Stockhausen. Although she analyses several pieces of spatialized music,

the most interesting work for our purposes is her analysis of the spatial structure in

Stockhausen’s GRUPPEN.

Nauck designed a Hörpartitur or “listening score”, which essentially is a graph showing

when each of the three orchestras plays.104 In the tradition of visual analyses by von

Burow and Silberhorn for STUDIE II,105 Nauk’s Hörpartitur is a valuable visual aid in

forming an idea of the way the work engages the spatial domain.106 Part of the graph is

reproduced in Example 1.4.3a. In addition to this graph, Nauck makes a “typology of

musical spatialization” [Typologie musikalischer Räumlichkeit] in which she divides the

various types of spatial movement into three categories. This is shown and translated in

Example 1.4.3b. She further divides each of her three broad categories into sub-

categories. The categories are a quantization of spatial movement in and of itself, and

range from very active motion (category A1a) to static (category C3c). Nauck links

timbre with spatial motion in her concept of the “sound field” [Klangfeld], and considers

101 Meyer 2006, p. 342-343.102 Misch 1999a, p. 149.103 Beyer 1928.104 Nauck 1997, pp. 215-218.105 Von Burow 1973, Silberhorn 1980.106 For more on Nauck’s analysis, see §2.1.3.

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sounds with the same timbre less active than those whose timbres change.

Thanks to her categories, Nauck can propose a hypothetical level of spatial activity and

function for each of the 144 groups in GRUPPEN. This is all to show that there is a

dramatic process going on in space. We reach a “high point” in this process during the

last insertion (groups 114-122) by combining four different types of spatial motion, and

by the simultaneous presence of a considerable amount of “active” motion.107 By

tabulating the amount and type of motion through the twelve main sections (and four

insertions), Nauck’s analysis of GRUPPEN shows how the spatial motion contributes to

the overall musical drama in a complex serial work.

Hofmann’s work uses Nauck’s as a springboard to formulate some basic analytical

approaches towards spatial music. Hofmann is primarily concerned with analyzing the

relationship between tone-space and real-space [“Tonraum” vs. “Realraum”] in several

Xenakis works.108 Although Hofmann suggests that in many of Xenakis’ works, sounds

can be assigned an x- and y- coordinate in polar coordinate space, he never follows up on

this very suggestive idea.109 Instead, Hofmann diagrams some general aspects of

Xenakis’s music, especially in the orchestra work Terretektorh, using a kind of “tin can”

diagram. The tin can diagram allows Hofmann to relate pitch height to spatial location.110

Overholt’s 2006 dissertation attempts to apply a theory of “shapes” to two Stockhausen

works, GRUPPEN and OKTOPHONIE. Her reading of Wörner 1973 and Maconie 1976

leads her to propose four categories of shapes, based on what she terms “aural

salience”.111 She divides her shapes into those with “motion” and those that are “static” –

107 Nauck 1997, p. 227.108 Hofmann 2008, p. 79.109 Ibid., p. 87.110 Ibid., p. 85.111 The passages Overholt cites in Wörner 1973 (p. 93, 169) are particularly laden with spatial imagery:

“Stockhausen makes a distinction between a motive (or theme) and shape. He only speaks of a musicalmotive when it actually motivates something—when it leads to something, whether to variations, totransformations, to disintegrations, or anything else of the sort. When this does not happen, there will be

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reminiscent both of Nauck and Harley/Trochimczky.112 Overholt terms her static shapes

either “points” or “blocks”, while shapes with motion can be “lines” or “wedges”.113

Wedges can be further subdivided into different shapes whose boundary points involve

various locations in the performance space.114

After proposing a reading of the entire shape structure of GRUPPEN using her

classification, Overholt investigates the different spatial characteristics of seven larger

sections, concluding that certain shapes are conspicuously absent from some sections.115

By removing the static shapes from her entire analysis, Overholt finds that there is a kind

of alternation between sections in which wedges predominate versus those where lines

constitute the main spatial language. Through the further application of this

methodology, which has its roots in reductive techniques in tonal music, Overholt

concludes that the spatial “background” of GRUPPEN is essentially wedge-line-wedge-

line-wedge-line-wedge.116

Overholt advances her analysis of the spatial movement in GRUPPEN by proposing that

there are certain “magic spots” in the space. “Magic spots simply take a sound (usually a

well-defined timbre) and move that sound linearly to a second location.”117 According to

Overholt, there are three techniques that Stockhausen uses to achieve this effect: (1)

change of volume; (2) “Doppler” effect (change of pitch); and (3) “sensory overload”.118

In this last technique, Stockhausen “overloads” our senses with data so that we lose track

no “motive” but rather a unique shape, isolated and without any motivic character.” (Wörner 1973, p.93)

112 Unfortunately, neither of these two authors are cited in Overholt’s bibliography. Although Overholtdoes not mention the sketches for KONTAKTE, her idea of finding “shapes” to categorize spatialmovement has precedent there, as mentioned above in §1.3.3.

113 Overholt 2006, p. 51.114 Ibid., p. 54.115 Ibid., p. 86.116 Ibid., p. 91.117 Ibid., p. 101.118 Ibid.

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of where a sound is, only to have it appear in another place shortly thereafter.119

Essentially, Overholt identifies and outlines a set of compositional devices that interlock

with spatial processes to facilitate the effect of spatial movement.

Finally, Overholt applies her shape theory to a later work of Stockhausen’s,

OKTOPHONIE. This electronic music, mentioned above in §1.2.3, is composed in eight

channels and serves as the “background music” to the second act of Stockhausen’s

TUESDAY opera. During OKTOPHONIE, several shorter pieces are performed at the

same time as the electronic music, including “1st Invasion”, “2nd Invasion”, “Pietà”, “3rd

Invasion”, “Explosion”, “Jenseits”, “Synthi-Fou”, “Abschied” and “Spiegelwelt”.

Overholt parses the piece into four sections:

Section 1: 1st Invasion, 2nd InvasionSection 2: PietàSection 3: 3rd Invasion, Jenseits, Synthi-FouSection 4: Abschied

By applying her theory of “shapes” to this three-dimensional spatial work,120 Overholt

concludes that there is “four-part shape counterpoint in the first ivasion [sic]; two-part

shape counterpoint in the second invasion; 3-D two-part counterpoint in the third

invasion; and a single shape for all the materials in the last section.”121 After Overholt’s

analysis, we are left wondering how exactly this counterpoint might operate, or what

rules govern the movements in it. Do the spatial “voices” interact at all like they do in

the pitch world?

119 Ibid., p. 112.120 According to Overholt, the “aurally salient” shapes in OKTOPHONIE consist not of the drones (which

slowly rotate around the listener in various ways) but the “shots”, “crashes” and “sound bombs” thatpunctuate the electronic texture.

121 Ibid., p. 159. Stockhausen suggested “spatial polyphony” in his interviews with Cott (see §1.3.2) butlater analyzed OKTOPHONIE as a work with eight-part spatial counterpoint (Stockhausen 2000b, p.74).

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1.5. Argument for more analytical rigor

1.5.1. Lack of knowledge of spatial musical constructs. The writers mentioned in §1.4

engage the spatial element of music in various different ways, reflecting the diverse

approaches that have been taken by scholars. Although there have been many important

contributions to the study of space in music, research has not yet engaged the idea

Stockhausen had of measuring spatial motion. We return one last time to Stockhausen’s

pregnant remark to Jonathan Cott, cited twice earlier:

...any point in space should be precisely defined with respect to where the soundoccurs and how it travels from one point to any other. (Cott 1973, pp. 202-203)

Were we to have the kind of stunted understanding of pitch that we have of space, it

would be as if we were only able to make categories of “clouds” of pitches, and to say

whether the general direction of a pitch complex is “up” or “down”. We would not have

a very highly developed notion of a solitary “pitch” (in space: a “point”), or where it

could or might move to. There would be no way to measure an interval (in space: a

“distance”), or know much about melodies (in space: “shapes”).

Other aspects of spatial music interact with traditional musical elements in more complex

ways. The concept of velocity – or, the rate of change in the location of sounds – could

apply to rhythm. However, the velocity of sounds in the space around us involves

another aspect of experience which is difficult to capture in any way other than on its

own terms. Although melodic contour has a “shape”, spatial melodies can have complex,

multidimensional contours if they move around in three dimensions. These and other

specific issues relating to spatial music have not yet been explored in a sufficiently

rigorous way.

1.5.2. Objections to measuring elements of spatial composition. Is it even feasible to

measure elements of music in space? Perhaps the most vocal opponents might come

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from the music cognition camp. Experiments have revealed that reverberation has a great

effect on our ability to localize sound.122 Thus, important structures in spatial music may

be perceived wholly differently – or not at all – in different spaces. Yet, if spatial

location is an integral element of the music, performers should and must seek out

performance venues with low reverberation, allowing space to be perceived clearly. If a

hall is like an instrument, a hall with a large reverberation time is like an out-of-tune

instrument: it is simple not suited for performance.

Another objection to measuring the distances, speeds and shapes in spatial music is that

sound sources may be set up differently than what is specified in the score. This too may

have a significant effect on the perception of the spatial design. However, the pieces we

are studying by Stockhausen usually have very specific spatial designs which are clearly

laid out in the score. Stockhausen often specifies even the distances between sound

sources in his scores. To further the analogy with more traditional music: if the

instruments in a symphony orchestra are slightly different from those that the composer

would have had when the music was composed, the timbral profile of the work is still

similar and recognizable. It is likely that even if sound sources are set up slightly

differently from the composer’s specifications, much of the intended spatial design will

remain intact.

A third question relates to the issue of the location of the listener relative to the sound

sources. If the spatial setup is such that sound sources are located in a ring around the

audience, listeners near the edge of that ring will probably perceive spatial motion much

differently than those seated near the center. Different listeners may have wholly

different perceptions of spatial movement, owing to complex psychoacoustic phenomena.

However, crafting an analytical methodology that would resonate with every listener

122 In a situation with a lot of echo, “the largely incoherent ear input signals resulting from thereverberation generate a diffusely located auditory event whose components more or less fill thesubject’s entire auditory space”. Blauert 1997, p. 279.

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would be an impossible task, due to the many variables involved. What we propose here

is to measure the spatial movement as it is composed in the score, not as it is necessarily

perceived by any one listener. By analyzing and characterizing the structure that the

composer intended to create, we may then follow up by speculating further about how –

or even if – this structure is perceived by an engaged listener.

1.5.3. Weaknesses in published literature. Although Harley/Trochimczyk provides a

valuable classification for spatial designs, her method does not engage spatialized works

on a detailed level – she only analyzes arrangements of sound sources and audiences. As

a result, she rarely points to any specific techniques that operate within a spatial composition.

Overholt, while courageously trying to develop techniques and vocabulary for analyzing

spatial movement where there are few in existence, makes a error in assuming that the

shapes which are primarily a tactile or visual sensation can be applied to those heard in

spatial music. She creates her categories first, without first identifying the precise shapes

that a category might contain. After making a shaky case for deciding on the form of her

shapes, she applies reductive techniques – developed for the analysis of pitch – to her

discussion, without thoroughly justifying them in the spatial domain. Her conclusion that

the spatial motion in GRUPPEN reduces to an alteration of two shapes seems overly

speculative, and it is not clear how this helps the listener to gain a greater appreciation of

the spatial design. Moreover, to discard the rotating background sounds in

OKTOPHONY is to ignore this development in one of Stockhausen’s main spatial

techniques: the “rotation table” tradition.

Nauck’s analytical work will be the most successful if we wish to create analytical

techniques for spatial music. In her study, the goal is to make an interpretation of motion

within a dramatic context in GRUPPEN. But her equation of timbre with spatial location

is questionable, since a change in timbre may not necessarily imply a change of perceived

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location. However, Nauck’s approach is stimulating because her methodology applies

the same ideas of serial composition – namely, quantization of the domain of spatial

movement – that were used to compose the piece in the first place. Still, she does not

make precise measurements of the spatial motion, leaving her continuous arrangement of

categories of spatial activity on a continuum from static to active open to some debate.

Hofmann’s starting point is Nauck’s classification schemes. He identifies many different

general characteristics of spatial motion in Xenakis’s music. However, Hofmann comes

tantalizingly close to suggesting the quantitative approach used in this dissertation when

he writes that “every sound can theoretically be assigned an x- or y- coordinate in space”.

However he never follows up on this idea and only offers a descriptive analysis –

although this analysis is very useful in identifying the myriad possibilities of spatial

motion in a piece. Hofmann’s work is stimulating because he has close contact with

spatialization techniques: his doctorate is in sound engineering. While his work on

Xenakis’s spatialization is extensively researched, offering great insight into its

production methods and the kinds of events that happen in space, there is still much that

is left unknown.

1.5.4. Statement of goals. In the following chapters, I intend to engage two specific

spatial works by Stockhausen. To gather data, I rely primarily on a statistical approach,

which is more neutral than methods dependent on inventing categories or reducing

foreground layers of activity to their background progenitors. By collecting a large set of

baseline measurements, we will have a foundation on which to build more complex

analyses involving set theory, graph theory, transformation theory, and scale theory. I

will propose some criteria by which we can determine a “set” of locations in the spatial

domain, thereby laying the foundation for relating these spatial sets by mathematical

transformations. Finally, I will use the mathematical structures of graph theory to

characterize the various connections between different parts of the space in spatial music,

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thereby unveiling the variety and subtlety of structure in the spatial domain.

The goal of the study is not only to elucidate an important aspect of Stockhausen’s

musical language, but also to point the way towards understanding other spatialized

music. After applying analytical tools to two major Stockhausen works, I will briefly

examine three other Stockhausen works in the final chapter, along with more selected

works by Xenakis and Tallis. By applying my analytical methods to spatial music, we

will better appreciate an already rewarding musical experience.

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Chapter 2.Analytical Methodologies

Introduction

The impetus for measuring space is inspired by a statement Stockhausen made suggesting

that it is an important factor in understanding his compositional practice.

...any point in space should be precisely defined with respect to where the soundoccurs and how it travels from one point to any other.123

In response to Stockhausen’s idea, our goal is define the location of sounds in space, and

then measure the movement of those sounds through space. This may be simple to do if

the composer specifies exactly where a sound is located, but it becomes more challenging

if we have to pinpoint the sound’s coordinates between two or more loudspeakers.

First, we will explore techniques of published analysis that have been applied to

Stockhausen’s music. This will provide some basic orientation as we build our methods.

Next, we will develop a method for locating sounds in space. Then, we will consider the

shape of the space itself, and whether sounds must “jump” around or whether they move

continuously from one place to another. Related to this is the question of whether

modularity holds in spatial music in a similar way that it does in pitch. By exploring

some of the basic properties of mathematical graph theory, we will find ways of

expressing the structure of spatial music in a useful, abstract way. We then explore the

idea that shapes traced out in physical space may be related to one another; for this we

apply the notion of mathematical transformations. In certain spaces, these

transformations may have symmetries which can be described by mathematical groups.

Finally, we will find that the data we collect can be analyzed statistically. These

analytical methods, when used together judiciously, can uncover a great deal of

information about spatial music which otherwise would go unnoticed.

123 Cott 1973, pp. 202-203.

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As we stated in the previous chapter, Stockhausen mainly uses space in two ways: first,

to clarify dense musical texture; and second, as a compositional premise in and of itself.

For the first case, we have to know what the musical texture is and why it might need to

be clarified. If we were to analyze works that seem to use space in this way, we would

have to go through a fair amount of analysis of the structure of pitch, rhythm, and form

prior to tackling the spatial question. Works that treat space in the second – as a

compositional premise – provide provide a more direct way of engaging the spatial

question.

2.1. Analytical Orientation

2.1.1. Composition analysis. The method used in many published analyses of

Stockhausen’s music is one that I call “composition analysis”. It is also the method that

Stockhausen himself used most frequently. In a composition analysis, the writer typically

begins by presenting the fundamental or generative “idea” for a given piece. This can be

as abstract as a numerical series, or as prosaic as a vision of a squadron of helicopters

containing the four musicians of a string quartet. Stockhausen’s numerous “form

schemes”, which are common for works after GRUPPEN, often provide the key to the

work’s basic structure, and are a convenient starting point for composition analysis.124 In

the next stage of this type of analysis, sketch material is often consulted in chronological

order, focusing first on the decisions that build the basic framework of the piece, and then

progressively filling in more compositional detail. This stage is greatly facilitated by the

voluminous sketch material that Stockhausen generously made available to scholars. The

final analytical stage engages surface details such as embellishments, insertions, and

foreground elements.

The goal of composition analysis is to unveil the compositional process, and to elucidate

124 Toop 2005, pp. 165-207.

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details that were relevant and important to Stockhausen himself. A successful analysis

requires that the analyst has at his or her disposal a fair number of sketches, and a notion

of how Stockhausen might have proceeded according to his typical working methods.

Composition analysis has the distinct advantage of allowing us metaphorically to sit next

to the composer’s work bench and get a glimpse of his sequence of decisions and

problem-solving.

2.1.2. Score analysis. Composition analysis contrasts with another analytical method:

“score” analysis. In score analysis, the printed score is the starting point. Here, the focus

is not on the process of composition, but rather on the outcome of that process: namely,

the score. One such analysis of Stockhausen’s music that has attained a kind of canonical

status in the music theory literature is David Lewin’s essay on KLAVIERSTÜCK III.125

Lewin parses the musical surface of this brief work into pentachords. He then relates

these musical objects through a sophisticated transformational network. The end result –

an abstract transformational network – is far removed from what one obtains in a

composition analysis.126 By mapping out a logically consistent way of hearing the work

from the perspective of a well-informed composer and theorist, Lewin finds a persuasive

way of structuring the piece that the composer himself might not have imagined.

Gisela Nauck made a different kind of score analysis of the three orchestras in

GRUPPEN.127 She created a visual transcription of the spatial information in the score.

Although she calls her analysis a Hörpartitur, she approaches the task from a similar

perspective, since her transcription clarifies aspects of the score that are difficult to

discern even after a fair amount of study. Boris Hofmann’s analyses – which are rooted

125 Lewin 1993, pp. 16-67.126 Blumröder’s analysis of the same piece (Blumröder 1993, pp. 109-154) follows the “composition

analysis” method. The contrast in approach between Blumröder and Lewin is quite great. WhereasLewin parses the surface into pentachords, Blumröder sees chromatic tetrachords. Lewin’s approachfollows Harvey’s lead (Harvey 1975, p. 24). Other notable analyses of this brief piece are by Cook(Cook 1987, pp. 354-371), Maconie (Maconie 2005, pp. 118-120) and Stephan (Stephan 1958, pp. 60-67).

127 Nauck 1997, pp. 215-218. The first page of her analysis is shown in Example 1.4.3a.

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in Nauck’s methodology – also suggest that one can relate spatial motion with pitch

graphically.128

2.1.3. Listening analysis. Lewin’s work, while primarily falling under the rubric of

score analysis, also flirts with a third kind of analytical methodology that has been

applied to Stockhausen’s music: “listening” analysis, or Höranalyse. Listening analysis

involves making a transcription of a performance, or reorganizing the elements of the

composition in a way that is performable. As a method of aiding the reader in hearing his

transformational networks, Lewin wrote a kind of “aural skills” exercise for the piano

which rearranges the elements of Stockhausen’s music so that the transformations among

them are easier to discern.

For works where the score is not very specific, listening analysis is particularly useful.

Hopp’s work on KURZWELLEN is an example.129 Since the musical material in this

composition is not entirely determined, Hopp makes tables and charts of the events in

various recordings. His listening analyses are essentially transcriptions of various unique

and unrepeatable performances of a work. This is an effective way of helping one grasp

what is actually happening during a realization/performance of this unusual piece.

2.1.4. Suitability of various analytical methods in spatial music. All three of these

analytical methodologies have shed light on the principles at work in Stockhausen’s

music. But since Stockhausen often left spatialization to his intuition – in contrast to

other aspects of the music – there is often very little sketch material available for study.

What does exist shows that Stockhausen was usually less concerned with exerting strict

control of distances and speeds, and more apt to engage in a kind of playful discovery of

the different possibilities within his chosen loudspeaker setup or performance

128 Although he does not present any graphic analyses himself, Hofmann 2008 does draw severalsuggestive diagrams which could visually relate space to pitch in novel ways. See especially Hofmann2008, pp. 99, 125.

129 Hopp 1998.

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arrangement.130 Form schemes, though crucial in providing a starting point for

composition analysis, rarely if ever contain any specific information relating to

spatialization. Therefore, this kind of analysis is not particularly suited for our goals.

A serious drawback in listening analysis is that the ways we organize information

visually in a Höranalyse may not be the same ways that we organize information

acoustically. Making a visual transcription of an acoustic phenomenon has the potential

to influence one mode of perception – namely, hearing – through another – sight. In

addition, listening analyses can have the side-effect of confining us to the current cultural

limitations of hearing.131 While the listening analysis method can prove useful in

clarifying analytical outcomes, such as with Lewin’s work, we must apply it with caution.

Although Nauck’s graphical score analysis of GRUPPEN helps us to gain a better

understanding of what is going on in the space around us in that work, we want to do

more than simply transcribe what we are able to see in the score. Part of the goal of the

present project is to stimulate people to listen in ways they might not otherwise be aware

of; in other words, to develop the ability to hear spatial movement as “music”. Score

analysis seems to be the most fruitful method for the present study, although tempered by

the inclusion of any relevant source material and commentary by the composer. Similar

to Lewin’s analysis, ours should result in an understanding of a work that might not be

evident either by recreating the generative process from the ground up, as in

compositional analysis, or by transcribing a performance. Our analysis should reveal

information about the work that the composer himself did not necessarily anticipate. The

first challenge in building such a methodology for spatial music involves pinpointing the

130 Günter Peters argues that this underlying “playfulness” is tempered by an idiosyncratic and “earnest”element of spiritual devotion. Peters 2003, pp. 235-264. Stockhausen’s improvisational practice canalso be observed in numerous “insertions” he added to many of his works, most famously perhaps inGRUPPEN. Misch 1999b tracks the various compositional vs. improvisational techniques inGRUPPEN. See Misch 1999b, pp. 142-153, 171-188, and 202-218.

131 There are, of course, limits to what our perception and cognition systems can process. It is importantnot to confuse cultural evolution with biological evolution. Cultural evolution can take place over thespan of a decade; biological evolution takes millennia.

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location of sounds in space, a task we will discuss in the next section.

2.2. Locating a sound in space

2.2.1. Methods of spatializing sounds. There are many different commercial methods

of spatializing sounds. Stereo recording and playback is the earliest and one of the

simplest ways of delivering a realistic spatial environment electronically. As stereo

broadcasts and recordings gained commercial success in the 1960s, sound engineers

began to search for ways of more accurately reproducing the sonic environments in real

performance spaces.132 Stockhausen’s techniques of spatialization vary considerably, but

132 Various attempts to develop commercially viable multichannel sound systems existed before the two-channel compromise of stereo was reached. One notable early experiment, called “Fantasound”, wasused in Walt Disney’s film Fantasia (1940), which was originally released in three-channel sound forprojection over approximately 80 loudspeakers (Garity and Hawkins 1941).

Somewhat later, Dolby labs (which is highly invested in the theater industry) developed several systemsfor creating the effect of surround sound in the theater and home. Although Dolby has developed abewildering number of standards, the two most commercially successful systems are called 5.1 and 7.1surround sound. In both of these systems, the number before the decimal point refers to the number offull-frequency range loudspeakers set up around the listener, whereas the number after the decimalpoint refers to the number of required subwoofer speakers.

In Dolby 5.1 surround sound, a speaker situated in front of the listener carries most of the spokendialog, whereas the left and right speakers, placed at an angle of 22° – 30° from the front center, areused for most of the movie soundtrack. Speakers set up in back of the listener, at an angle of 90° –110° from front center, provide ambient sound which help to create a better sense of space. Thelocation of the subwoofer is less important, since the directionality of low-frequency sounds is morediffuse. A Dolby 5.1 speaker setup is shown in Example 2.2.1a. The system in Dolby 7.1 surroundsound is much like that of 5.1. An additional pair of speakers is added further behind the listener atangles of 135° – 150° from front center. These speakers further enhance the ambiance of the soundenvironment. However, the main movie soundtrack is still heard through the pair of front speakers at22° – 30° from the front center point. An ideal Dolby 7.1 setup is shown in Example 2.2.1b.

While Dolby’s surround sound system is primarily geared towards reproducing movie soundtracks,another system of spatialization is lumped under the term “ambisonics”. Ambisonics is a technique ofspatialization that has a small, devoted following but has not had much commercial success. Many ofthe fundamental techniques of ambisonics were developed by the British engineer Michael Gerzon andin the 1970s and 80s. The standard format for ambisonic sound is called “B-Format” and includes fourdifferent channels: mono, front-back, left-right, and up-down. What is special about the ambisonicapproach to spatialization is that the recorded signal is the same, regardless of the number of speakersin use. The signal is encoded in such a way that an ambisonic decoder can split the signal into differentchannels. The goal is to have a continuous sound-field where the precise location of the loudspeakers isnot even noticeable. Further distinguishing this method from Dolby’s is the fact that in ambisonics, it

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he usually spatialized his electronic music by hand.133 He never adhered to any of the

commercially available methods of spatialization, and despite the suggestions of many

participants at the annual Stockhausen courses, he consistently resisted the idea of

producing DVD versions of his spatialized works in Dolby surround sound. It is

probably in the best interests of Stockhausen’s music that it does not get issued on these

commercial formats, since they are primarily designed for home theater systems and not

for the complex and finely-tuned sonic environments Stockhausen sought to create.

In live performances, Stockhausen often arranged musicians in different shapes and

patterns throughout space. In some works, such as TRUMPETENT (1995), musicians’

movements are less precisely defined, and vary according to the shape of the hall they are

performing in. In other works, such as LICHT-BILDER (2002), the musicians’

movements on stage are precisely formulated according to symbols in the score. Seldom

do musicians move rapidly from one place to another; movement is almost always at a

slow or moderate pace.

2.2.2. Locating sounds in space: box diagrams. The locations of sounds in space are

the basic building blocks of spatial music. Without the locations, we will have nothing to

analyze. Fortunately, Stockhausen usually indicated the locations of the sounds using a

method that I call the “box diagram”. Box diagrams, such as those in Examples 1.2.1d

and 1.2.3d, show where sounds should ideally be located in the space. Often they

indicate the actual physical moves Stockhausen made during the spatialization process,

e.g., movements of a joystick. Box diagrams can be found in scores of both electronic

doesn’t matter very much where the speakers are located. Rather, the crucial element is the directionthat the speakers are pointing.

Articles by Fellgett and Gerzon (Fellgett 1975, Gerzon 1975) describe the basic technology and earlyimplementation of ambisonic systems. Like many niche technologies, ambisonics has a web site ofenthusiasts that can be found at http://www.ambisonia.net.

133 Even when utilizing sophisticated electronic methods for spatializing sounds, Stockhausen still reliedon motions of his hands to guide joysticks, rotation mills, rotation tables, or potentiometers.Stockhausen’s methods of spatializing electronic music are described in more detail in Chapter 1.

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and instrumental music. Although the diagrams indicate the location of the sounds in the

“sound field”, they do not generally show the sounds’ relationships to the listeners, and

they never factor in cognitive perception issues. Stockhausen usually indicated the ideal

location of listeners on other sketches or diagrams.

By fixing the size of the performance space, we can overlay a Cartesian coordinate

system onto box diagrams. Then we can find the coordinates of a sound source with

reasonable degree of precision. Coordinates are essential to expressing the location of a

sound source in space; they are like the frequencies of the fundamentals in pitch analysis.

2.2.3. Locating sounds in space: calculating the location. In other scores,

Stockhausen notated information about where he imagined sounds are centered, but not

in as precise a way as in his box diagrams. In the score to OKTOPHONIE, Stockhausen

made diagrams which show how he controlled the decibel levels of speakers in order to

“nudge” certain sounds around in space. Stockhausen wrote that “Changes in loudness

are the premise of (one) method to move sounds in space.”134 With an efficient

algorithm, it would be possible to calculate the coordinates of these sounds too.

Example 2.2.3a shows one of Stockhausen’s decibel diagrams in a stereo panorama,

along with a hypothetical space – 14 meters in width – that defines the performance

space.135 In each case, the same sound is emitted by both loudspeakers, but at different

volume levels. Examination of Stockhausen’s spatial diagrams for OKTOPHONIE

shows that the greatest positive adjustment in speaker volume is +8 dB while the greatest

negative modification (aside from -� dB – “silence”) is -25 dB. Using this information,

134 Stockhausen 2000b, p. 68. Other aspects of sound that Stockhausen says are important to theperception of spatial motion are phase difference and the Doppler effect. See Cott 1973, pp. 97ff.Stockhausen might have used the slight tendency of most instruments’ pitch to go higher when playingloudly to his advantage in GRUPPEN. The “searchlight” chords, which achieve their spatial effectthrough crescendo and diminuendo, are susceptible to slight variations in pitch, which through theDoppler effect could potentially heighten the perception of spatial movement.

135 The size of 14 meters is from Stockhausen’s diagram of the spatial setup. Stockhausen 1994, p. OXIX. This was the height of the hall in the Leipzig opera house (Stockhausen 2000b, p. 63).

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we can weigh the dB settings so that a positive dB value will draw the hypothetical sound

center towards the loudspeaker, while a negative value will push it away. Even though

the same sound is technically coming from two loudspeakers, the dynamic balance should

create the illusion that the sound is centered somewhere between them. Since the decibel

scale is logarithmic, we can use a simple exponential function to make the calculation.

An optimal exponential base can be determined for positive and negative decibel settings

through trial and error. The difference of these “attraction” or “repulsion” factors can

then used to estimate the location of the sound. The analysis of a typical stereo sound in

OKTOPHONIE is shown in Example 2.2.3b.

By extending this technique, we can localize sounds in a quadrophonic panorama as in

Example 2.2.3c. We first determine the hypothetical center points for the dB values on

each edge of the square. Then by connecting the points on opposite edges, we form two

intersecting lines in the space. The point where these lines intersect (which can easily be

determined algebraically,) is the theoretical center of the sound. This analytical

technique is shown in detail in Example 2.2.2c. By combining the results of our

calculations in Examples 2.2.3b and 2.2.3c, we can say with some confidence that the

sound Stockhausen composed begins and ends on the path indicated in Example 2.2.3d.

This method hinges on the illusion that a sound produced from several loudspeakers is

localized at a certain point between them. In this type of spatial technique, the composer

can – in theory – move a sound anywhere within the performance space. But in a

performance situation, many listeners report that they perceive sounds to be moving more

around the periphery of the space, and only to a limited degree towards or away from

them.136 These perception issues are extremely important and worth investigating more

rigorously from a cognitive viewpoint. Still, weighing Stockhausen’s decibel settings

takes into account elements of the score that were painstakingly notated, but have been

left unanalyzed by other writers. Furthermore, by analyzing the raw spatial data in the

136Thanks to Jerome Kohl for this important observation.

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score, we may learn much about the composers’ basic intent, which can help us to craft

more effective cognitive studies later on. Ultimately, our analytical methodology lies

squarely within the tradition of “score analysis”, not “listening analysis”. The results of

our analysis should give us a “best guess” result which takes advantage of these hitherto

uninterpreted elements of the score.

2.2.4. Uncertainty. Many factors may cause listeners to perceive the location of a sound

differently. Even so, we can say that the coordinates we determined through analysis are

the most probable center of the sound, within a cloud of uncertainty of radius r. Factors

involved in determining the value of r might include the amount of reverberation in the

space, the position of the listener relative to the sound, and the physical qualities of the

sound including frequency, timbre and duration. We have reason to believe that for a

performance of a spatialized work, the sound projectionist has every incentive to limit the

value of r to its lowest possible value for a given space.

2.3. The Shape of Space in Spatial Music

2.3.1. Continuity in space. Knowing the properties of the medium we are studying –

physical space – is important. The shape of space will help to suggest analytical tools. If

the composer uses space in certain ways, we may be able to relate places in it through

mathematical transformations. The idea of the transformation, which we will develop

later in §2.5, has been used to gain deep insight into many different phenomena in music,

including the serial music that Stockhausen composed.137 Depending on the aspects of

continuity and discontinuity a given performance space has, transformations applied to

things in the space may form a mathematical group. If we are able to say that a given

space and its transformations form a group, we can draw connections between the

137 Transformational analysis is particularly useful in works such as MANTRA, since the entire piece isessentially a series of transformations of the opening “formula”. See Stockhausen 2003, Conen 1996pp. 59-100, and Toop 2005 pp. 73-98. Of course, Lewin’s 1993 essay is also an excellent analysisbased on transformational principles.

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structure of spatial music and other musical domains, such as pitch.

2.3.2. Dimension. The notion of dimension in space underwent significant changes in

mathematics during the nineteenth and twentieth centuries. For our purposes, however,

dimension may be defined simply as the Euclidean space common in experience. We can

say that the dimension of the space is equal to the number of coordinates, or parameters,

needed to describe the location of a point.138 We define a one-dimensional performance

space as one in which sound sources are arranged in a line, a two-dimensional space as

one where sources are in a plane, and a three-dimensional space as one where sources are

located in more than one plane around the listener. The dimension should be defined for

every piece of spatial music.

2.3.3. Modularity of space. Modularity is an equivalence relation that defines elements

of a set as the same under some operation (usually addition) and a modular interval.139

Why would we want to define space as modular? Often, spatial music is performed in

places that have physical boundaries. Many of Stockhausen’s performance spaces have

such “horizons” – i.e., four walls, a floor and a ceiling.140 What if a continuous sound in

spatial music moves outside the physical performance boundaries? If modularity in

spatial music holds, points outside the boundary will “wrap around” to other points which

are in the original space, but on the other side of the horizon. This would be analogous to

the way pitch is perceived in much Western music.

Similar to the way pitches are arranged on a one-dimensional continuum, we can place

musicians along a straight line. If a sound were to move from one musician to another

down to the end of the line, it could continue if we wrapped around to the other end of the

138 This is a somewhat naive definition, especially in light of the discoveries of Cantor and Peano in theearly 19th century. Nevertheless, it corresponds well to everyday experience and is thereforeappropriate for our purposes.

139 More precisely, a is equivalent to b modulo M if aMb. 140 Some performance spaces such as those outdoors may not exhibit such strong boundary limits.

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space.141 The space would then have a modular interval that was equal to the length of

the line of musicians. An analogous type of modularity can exist in two-dimensional

spaces. In such spaces, sounds that moved off the edge of the plane would simply re-

enter at the other end. In three dimensions, the situation would be similar. In all three

kinds of spaces – line, plane, and volume – space wraps around at the edges.

While this may seem like it is a counterintuitive notion, there are precedents for

modularity in spaces other than the pitch continuum in music theory. The Tonnetz,

originally conceived of by Leonhard Euler in 1726,142 represents relationships of keys in a

two-dimensional plane. Though their representations of it are slightly different, the

Tonnetze of Euler, Oettingen and Riemann extend infinitely in all directions of the plane

because all three utilize a just intonation.143 Richard Cohn, in generalizing the properties

of an equal-tempered Tonnetz, remarked that such a structure should be generalized as a

torus (or donut).144 Tempering the musical intervals results in modularity in the structure

of the Tonnetz.145

Another instance of modularity in music theory plays an important role in Wayne

Slawson’s theory of sound color.146 Some of Slawson’s transformations on sound color

contours operate in a modular space. According to Slawson, the characteristics of sound

colors essentially involve two variables, so the primary space he deals with is a two-

dimensional plane.147 In sound-color “transposition”, colors must sometimes move off

141 Although we are not as accustomed to this notion as we are to the more intuitive concept of the octavein the pitch domain, some composers have utilized a similar kind of wrap-around in their pitched music,especially Ligeti.

142 Euler’s original “Tonnetz” is reproduced in Cohn 1997, p. 7.143 Cohn 1997, p. 11. Hyer 1995, p. 102 reproduces Riemann’s Tonnetz from his “Ideen zu einer ‘Lehrevon den Tonvorstellungen’” (1914-15).144 Cohn 1997, p. 18.145 In the same way, the equal-tempered scale allows the circle of fifths to close by tempering the cycle of

justly-tuned fifths (Schechter 1980, p. 41).146 Slawson 1985.147 The variables are the frequencies of the first two formants – what Slawson refers to as F1 and F2.

Ibid., p. 55.

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the horizon of his continuum. A “wrap around” applies best to the sound colors he terms

Acuteness and Openness.148 When transpositions (or translations) are applied to these

sound colors, causing them to move off the edge of the plane on which they are defined,

they simply wrap around to the other side. For the properties of Smallness and Laxness,

this procedure does not work as well, so Slawson does not require complete closure.149

Although it may seem at first glance appealing to use modularity in analyzing spatial

music, it is questionable whether applying it will be particularly useful. Consider a

spatial shape that is wrapped around space so that part of it lies on one edge of the space

and the rest has moved to the other side. It is likely that without considerable mitigating

circumstances, a listener will perceive this shape as disconnected.150 For this reason,

Stockhausen rarely – if ever – crosses over the edge of space in a way that is analogous to

the pitch continuum. Rather, Stockhausen will “wrap” sound around to form a continuity

that is more amenable to human perception. For example, a “line” is curved into a circle,

and the audience sits in the middle; a “plane” is curved into a sphere. Thus, even though

modularity often plays an important role in pitch analysis, we will find that it is usually

not a particularly useful concept in spatial analysis except in the circumstances just

described.

2.3.4. Internal structure of space. Does space contain a finite number of points from

which sounds may originate, or can sounds move freely in an infinite continuum?151

Evidence from Stockhausen’s scores indicates an ambiguous attitude towards this

question. In the composer’s box diagrams, discrete points in the performance space are

usually clearly defined; the location of sound sources is typically indicated with a dot.

148 Ibid., pp. 72-76.149 Ibid., p. 193.150 Bregman understands space as an important grouping mechanism. It seems likely therefore that two

sounds coming from opposite sides of the space would be perceived as discontinuous. See §1.4.1.151 This question has, of course, been debated since nearly the dawn of philosophy. Zeno’s paradox is one

early formulation of the question as to whether there are an infinite or finite number of points betweenany two positions in space.

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But as we saw in Chapter 1, Stockhausen often intended for sound to have the illusion of

moving continuously around the audience. The rotation table paradigm is a prime

example.152 Stockhausen’s great spatial experiment at the spherical auditorium in Osaka

supports the idea that even when there are a finite number of sound sources, they are only

a means of circulating continuous spatial motion. The notation of motion in

OKTOPHONIE, as we saw in §2.2.2, also suggests that the environment may best be

thought of as continuous, even though only eight loudspeaker groups surround the

audience.

We can conclude from Stockhausen’s scores and writings that a continuous effect is

intended even when a finite number of sources are available.153 However, we may still

shed light on how the music might actually be perceived by focusing on points along the

arc of movement. Any analysis that deals with these points will come close to modeling

the shape of continuous motion anyway.

2.4. Sets and Graphs in Spatial Music

We have discussed techniques of delivering spatialized music to listeners, determined

how to find a point in spatial music, how to label the point with coordinates, and

considered some properties of the medium of space, such as modularity. Grouping

spatial events together into sets allows us to consider spatial shapes and spatial motives.

152 Even though Stockhausen took care to make the rotation effects audible in works like KONTAKTE,some listeners report a kind of “flapping” phenomenon – especially when the rotation occurs at slowspeeds. Cardew mentions this in a passage cited in §1.2.1. The discontinuous flapping effect seems tobe mitigated by three factors: (1) sitting near the middle of the space; (2) increasing the number ofindependent channels and speakers placed around the horizon; (3) increasing the speed of the rotationsaround the listener.

153 Stockhausen 2000b, p. 62. “Natürlich wollen wir nicht nur vier, acht oder acht mal zwei Lautsprecheroder 32 in einem Raum verteilen, sondern wir wollen, daß eines Tages jeder Punkt im Raumlokalisierbar wird und daß man dafür spezielle Auditorien baut..” Die werden kommen, wenn dieRaum- Musik wieder wichtiger wird.” [Naturally, we don’t want to divide only four, eight or 8 � 2speakers or 32 speakers in a space; instead, we would like one day to make every point in spacelocalizable, and for this we need to build special auditoriums. This will happen, when spatial music isagain important.]

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This allows us to apply mathematical ideas of set theory and graph theory to develop a

greater variety of analytical tools. In some spatial music, concepts from graph theory will

help us to define the shape and quantity of sets there are for a particular spatial structure.

Exploring these questions will not only benefit analysis but also composers. If we are

able to grasp some of these aspects of the structure of a given compositional space, we

will better know what kinds of motion are feasible. Then, we can understand more

completely how a composer has or has not taken advantage of the available possibilities.

2.4.1. Dimension and spatial sets. In spatial music, a set can take different forms

depending on the dimension of the space we are working in, and the cardinality of the set.

The various possibilities are shown in Example 2.4.1. A spatial set can define a point, a

line, an area, or a volume. We may add a “fourth dimension” – time – to differentiate

between sets whose elements are heard successively or simultaneously. A set heard

successively is comparable to a spatial “melody” while one heard simultaneously is

analogous to a spatial “harmony”. This corresponds to the idea of ordered and unordered

sets: a melody is an ordered set while a harmony isn’t, with respect to time. This is the

reason why determining the dimension of the space is an important step in understanding

what kinds of sets are possible.

2.4.2. Measures of distance/interval. In spatial music, the distance between sound

events is important because it is analogous to interval in pitched music. If we assign

coordinates to points in space using the methods we outlined above, it is easy to

determine the distance that is traversed, or the interval between two sounds by using the

distance formula. For spatial music in 2 dimensions, the distance d between any two

points (x1, y1) and (x2, y2) on a Euclidean plane is simply the distance formula

d = �[(x2 – x1)2 + (y2 – y1)2 ]

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In some cases it may be useful to use a distance vector, which tells us distance and

direction. Here, it is necessary to specify exactly the direction of motion. We can define

directions as “north” or “south” even in works where Stockhausen conceived of the space

as not having a definite “front” or “back”.154

For our analysis, it is most convenient to assign the top edge of a box diagram as the

“north” end, and the bottom as the “south” end, since Stockhausen usually oriented

listeners facing the top edge. This distinction takes into consideration the orientation of

the audience in determining where “north” is. These questions will be explored in more

detail later in §2.5.2.

2.4.3. Closeness and connectedness. While it is useful to have an absolute measure of

distance, it is also helpful to introduce the idea of “connectedness”. A set of connected

points could look like Example 2.4.3a. Here, the distance from r to s is less than the

distance from r to t. But, sometimes it is advantageous to define s and t as somehow

adjacent to r, since the difference in distance to point r is slight.155 Of course, points do

not need to have the same, or nearly the same distances from each other to be considered

connected; in fact, we may consider any two points in space connected as long as music

passes from one to another.

We can formalize the property of closeness that is indicated in Example 2.4.3a by

154 In works such as CARRÉ or LICHTER-WASSER, the audience is seated facing the middle, andtherefore the space is possibly conceived more in terms of polar coordinates from the listener’sperspective. In works such as GRUPPEN, HYMNEN and OKTOPHONIE, the audience is seatedfacing the same direction (“front”), and the idea of the space having a “front”, “back” and “sides” isarguably more apt. However, we are most concerned here with analyzing elements of the score,without focusing greatly on the audience perception. Since Stockhausen’s box diagrams are verysimilar for all the works mentioned above, we may – for the time being – safely adopt the convenienceof thinking in terms of “front”, “back”, and “sides” even in works where polar coordinates mayultimately seem more appropriate. In addition, adopting a consistent Cartesian coordinate systemfacilitates comparison among different works.

155 The same happens when we make an identity between fifths in a mean-tone system. Some perfectfifths in this tuning are different sizes than others, but they are typically all treated as equivalent eventhough some may be very dissonant.

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representing the musical situation as a mathematical graph. Although mathematicians

differ slightly in their definitions of a mathematical graph, in general a graph is a set of

vertices (in our case, sound sources) and a set of edges that connect those vertices.156

Some works of spatial music can be represented abstractly by graph structures.

A composer who (like Stockhausen) often tries to keep an illusion of continuous motion

even when music is passing discontinuously from musician to musician – will probably

tend to make sounds move to closely-positioned instruments, because the movement will

seem to be more connected to the listener – as remarked in §2.3.5. Whereas small moves

in the space will probably be more common, large moves are more likely to be perceived

as disconnected events. Representing the situation in spatial music as a mathematical

graph is an elegant mathematical abstraction that is useful for modeling the possible

structures in this perceptual domain.157

2.4.4. Basic mathematical concepts of graph theory. In the previous section, we

defined what a graph is and showed how it is a useful mathematical abstraction in some

spatial music. Since graph theory is not commonly used in music theory, we will now

build a vocabulary for talking about graphs by defining several basic terms.158

2.4.4.1. Fundamental definitions and terminology. Two vertices connected by an

156 More precisely, a graph is an ordered pair (V(G), E(G)) consisting of a set V(G) of vertices and a setE(G), disjoint from V(G), of edges, together with an incidence function �G that associates with eachedge of G an unordered pair of (not necessarily distinct) vertices of G (Bondy and Murty 2008, p. 2).Diestel 2005 defines a graph using different terms: “A graph is a pair G = (V, E) of sets such that E �[V]2; thus, the elements of E are 2-element subsets of V.” (Diestel, p. 2).

157 The idea that spatial shapes will be be more well defined (and more memorable) if the intervalsbetween their constituent elements are small is a fundamental difference between sets in pitch and setsin space. A widely-spaced voicing of the elements of a pitch-class does not necessarily impede theconnection among them that causes us to regard them as a unit. Spatial sets whose elements are definedsuccessively through time will probably be easier to perceive together as a unit if the intervals betweentheir elements are small. Spatial sets whose elements are defined simultaneously – as a chord or“constellation” – may include internal intervals that are greater.

158 All of the definitions in this section (except the definition of a spatial set) are taken from Bondy andMurty 2008, and Diestel 2005.

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edge are called neighbors; the term adjacent is used interchangeably. The number of

vertices in a graph is called its order. There is no particular “correct” way of

representing a graph on paper; however, in such a representation, edges should always

connect the proper vertices. An edge is incident to a vertex if it connects that vertex to

another vertex. If all the vertices in a graph have the same degree, the graph is regular.

A complete graph is one in which any two vertices are adjacent, while an empty graph is

one in which no two vertices are adjacent (thus, there are no edges). The degree of a

vertex is the number of edges that are incident to it; thus, all vertices in an empty graph

have degree 0, while the degree of all vertices in a complete graph are equal to the order

of the graph itself.

A loop is an edge that links a vertex with itself, whereas an edge that links two distinct

vertices is a link. If a vertex has an edge that is a loop, that vertex is incident to itself.

Two or more links that connect the same pair of vertices are parallel edges. A simple

graph has no loops or parallel edges.159

The union of two graphs G and H is the union not only of their vertices, but also of their

edges, and written as G � H. In the same way, the intersection of G and H is the

intersection of their vertices and edges, and written as G � H. If G � H = �, then the

graphs G and H are disjoint. The difference of graphs G and H is obtained by deleting all

the vertices and edges in H from G. Since it is impossible to have an edge that does not

connect two vertices, some edges in G that are not in H may be removed under the

difference operation. If V(H) � V(G) and E(H) � E(G), then we can write H � G and

we may say that G is a subgraph of H. The complement H of a graph G contains the

same number of vertices in G, but it has no edge that is in H; rather, the complement of H

has edges between all vertices that H does not. Graphs illustrating all the definitions in

159 While the graphs we will create to model Stockhausen’s spatial structures rarely have loops, they oftenhave parallel edges, so they are not simple.

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this section may be seen in Example 2.4.4.1.160

2.4.4.2. Isomorphic graphs and automorphisms. In general terms, two graphs are

called isomorphic if they have the same structure. Thus, isomorphic graphs G and H

must have the same number of edges and vertices, but in addition there must be a

bijection161 from each vertex in G to each in H. If the graphs G and H are isomorphic, we

write G � H. In more precise mathematical language, two graphs G and H are

isomorphic if there are bijections (or “one-to-one correspondences”) �: V(G) V(H) and

�: E(G) E(H) such that �G(e) = uv if and only if �H(�(e)) = �(u)�(v).

Consider Example 2.4.4.2a. These two graphs are isomorphic because they have the

same structure. Only their labels are different. In Example 2.4.4.2b, the two graphs have

the same number of vertices and edges, but are clearly not isomorphic because there are

some edges in G that are not in H. Unfortunately there is no simple general method,

aside from brute force, to check to see if two graphs are isomorphic.162

A automorphism is an isomorphism of a graph to itself. In a simple graph, an

automorphism is simply a permutation of its vertices which retains adjacency. Thus, the

set of automorphisms of a graph are its symmetries. It can be shown that the set of

automorphisms on a graph forms a mathematical group.163

2.4.4.3. Paths, cycles and connectivity. A path is a simple graph whose vertices can be

arranged in a linear sequence in such a way that two vertices are adjacent if they are

consecutive in the sequence, and are nonadjacent otherwise. Paths can be abbreviated Px,

where x is the number of edges involved. In formal terms, a path is a non-empty graph P160 These definitions are taken from Bondy and Murty 2008, p. 3, with several of my own explanations.161 A bijection is a one-to-one mapping from one set to another. More formally, a bijection is a function f

from a set S to a set T with the property that for every x in S, there is exactly one y in T, such that f(x) =y.

162 Bondy and Murty 2008, pp. 12-15.163 Ibid., p. 15.

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= (V, E) of the form

V = {x0, x1, ..., xk} and E = {x0x1, x1x2, ..., xk-1xk}.164

The ends of a path are the vertices x0 and xk, while the number of edges in a path is its

length. A cycle is a path whose last vertex is the same as its first vertex. Like paths,

cycles can be referred to as Cx, where x is the number of edges involved. A cycle

containing only one vertex is a loop, whereas a cycle with two vertices has parallel edges.

A 3-cycle is often referred to as a triangle. A cycle need not include all of the vertices in

the graph. Example 2.4.4.3 shows a path and a cycle on a graph.

Two other terms relating to connections among vertices will be of use in describing and

analyzing the graphs in spatial music. A set of vertices X is called a separator in a graph

V if V - X is disconnected. More specifically, a single vertex which, if deleted, would

cause the graph to be disconnected is called a cutvertex, and an edge separating the ends

of two such cutvertices is a bridge. Examples of all of these properties are shown in

Example 2.4.4.3b. The location of separators, cutvertices and bridges can be a factor in

determining the potential density of musical activity in one or another area of space, since

one may hypothesize that there will be greater activity around these structures due to the

property of linking subgraphs.

2.4.4.4. Digraph (or, directed graph) and weighted graphs. In some graphs, we may

limit the “flow” from one vertex to another so that it may be only in one direction. This

structure can be useful in spatial music because the trajectory of a single melody from

one point to another point may be unidirectional. In other cases, the flow in one direction

may be significantly greater than in the other. In mathematical terms, a directed graph D

is an ordered pair (V(D), A(D)) consisting of a set V := V(D) of vertices and a set A :=

164 The informal definition is from Bondy and Murty 2008, p. 4, while the formal definition is fromDiestel 2005, p. 6.

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A(D), disjoint from V(D)) of arcs, together with an incidence function �D that associates

with each arc of D an ordered pair of (not necessarily distinct) vertices of D. The

addition of the incidence function is critical because it allows us to say precisely that if a

is an arc and �D(a) = (u, v) then a must join vertices u and v. The number of arcs in a

directed graph D is referred to as a(D).

Digraphs may conveniently be represented as graphs with arrows on their edges. A

digraph is shown in Example 2.4.4.4. While in many cases it will be sufficient simply to

refer to the connections in a graph in spatial music as edges, we will find that the concept

of the digraph with its arcs allows us to gain a significant amount of precision in defining

the properties and structures of spatial music.165

Weighted graphs are graphs that have some sort of weighting assigned to their arcs. In

traversing a weighted graph, we incur more or less of a “penalty” by moving from one

vertex to another. This kind of graph is part of what allows us to build models of the

well-known “traveling salesman problem” in which the salesmen tries to determine the

shortest path through a given town in order to reach the maximum number of customers,

while minimizing the distance of his trip. Formally, each arc a is associated with some

real number w(e) in a weighted graph. Although weightings can represent distances, we

will also use them in a slightly different way in our analysis – namely, to model the

likelihood that a spatial melody moves from one point in space to another.

2.4.4.5. Trees/Forests. The general term forest is applied to a graph that has no cycles.

A specific forest that is connected is a tree. Every nontrivial tree must have at least two

vertices of degree 1. These vertices, which are at the most “outlying” parts of the tree,

are called its leaves.166 Lying at the “other end” is the root of the tree, out of which its

branches stem. There are many theorems and corollaries that arise through study of

165 The definitions in this section are related to Bondy and Murty 2008, p. 31.166 Bondy and Murty 2008, p. 100.

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trees, and later we will see one useful application of this structure in Stockhausen’s

spatial music.167 The important thing about a tree is that it contains no cycle; in fact, a

complementary way of defining a tree is that if we were to link any two non-adjacent

vertices in a tree, that tree would contain a cycle.168 A graph demonstrating all these

properties is shown in Example 2.4.4.5.

2.4.5. Adjacency and Incidence Matrices. How are graphs represented so that

calculations can be done on them efficiently? Any graph can be represented by an

adjacency matrix and an incidence matrix.169 An adjacency matrix of a graph G is the n ×

m matrix AG := (auv), where auv is the number of edges joining vertices u and v. A loop

counts as two edges in such a matrix. On the other hand, an incidence matrix of a graph

H is the n × m matrix MG := (mve), where mve is the number of times (0, 1 or 2) that vertex

v and edge e are incident.170 We will find that representing graphs using adjacency

matrices can yield significant analytical payoff in spatial music, since they allow us to

calculate the number of possible paths from one vertex to another for a path of a given

length.

Having defined some of the basic concepts in mathematical graph theory, as well as a

way of analyzing some of their properties by using matrices, we now see how they might

apply to some structures in spatial music.

2.4.6. Musically interesting graphs and their structures. In the next section, we take a

look at four different graphs which are musically interesting, showing how the definitions

and properties from the previous section can tell us more about their structure and

compositional possibilities. It is not easy to generalize about what characteristics might

167 See §4.4.11.168 This property of a tree is defined in Diestel 2005, p. 14.169 Chartrand 1977, p. 217.170 Definitions from Bondy and Murty 2008, p. 6.

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make up a musically interesting graph.171 However, we might hypothesize that graphs

with only a few vertices probably do not provide enough structural possibilities to make

them musically interesting. Furthermore, the amount of musical interest peaks when the

degree of the vertex reaches a certain optimal range. Vertices with only one adjacent

edge will not allow a great variety of spatial motion through them, whereas those with too

many adjacent edges will allow only unpredictable, diffuse motion.

2.4.6.1. Three contrasting graphs. Consider Example 2.4.6.1a. Since we shall be

making frequent reference to this graph in the following sections, we will call it Graph

1.172 Since all 25 vertices, including vertex r have eight edges, this is a regular graph of

order 25 (but note that according to our definitions above, it is not complete). The graphs

shown in Examples 2.4.6.1b (Graph 2) and 2.4.6.1c (Graph 3) are not regular. Graph 2

is similar to Graph 1 but it has a “gapped” structure, while Graph 3 resembles a kind of

“spider-web”. In these graphs, one region is more or less connected to the graph than

another. In Graph 2, s has the fewest edges of any vertex in the graph; as such it has the

minimum degree of Graph 2. The situation is the reverse in Graph 3; here, t has the most

edges of any vertex and consequently it has the maximal degree.

2.4.6.2. Definition of a spatial set in a graph. Although we have explored the shape

and characteristics of a spatial set in relation to the dimension of the performing space in

§2.4.1, no formal definition was attempted. The terminology we have developed and the

graph structures we have examined in the previous section now make such a definition

possible. We therefore define a spatial set of cardinality c in a graph G as a subgraph of

order c-1. Whether this graph is a path depends on whether the set is heard melodically

171 Of course, the composer is responsible for finding musically interesting spatial structures in much thesame way as he or she might also be expected to chose musically interesting scale structures in pitch.

172 It may seem like we are using a needlessly large graph – in the case of Graph 1, consisting of 25interlocked vertices and 200 edges – to demonstrate some relatively simple principles. However, thearrangement of nodes in Graph 1 might be thought of as a musically interesting and practicalarrangement because it allows for a fair number of musicians in a space that could be contained in agymnasium or a rectangular black-box theater. Also, the work we analyze in Chapter 4 – LICHTER-WASSER – has 29 musicians. We are setting the stage for the analysis of that piece.

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or not. Since this subgraph will include information regarding not only its nodes, but also

its edges, a spatial set contains considerably more information than a pitch or pitch-class

set does.

2.4.6.3. Structures and sets in Graph 1. The adjacency matrix A for Graph 1 is shown

in Example 2.4.6.3a. Multiplying A with itself creates another matrix called A2. This A2

matrix tells us how many paths of length 2 (cardinality 3) there are through the graph

from va to vb.173 To generate the A3 matrix (not shown), we would multiply A × A2. The

A3 matrix would, of course, tell us how many paths there are through the graph of length

3 (cardinality 4).

The sum of all the entries in Graph 1’s A matrix equals 200, which – by definition –

equals the number of edges in the graph. The A2 matrix for Graph 1 is shown in Example

2.4.6.3b. The sum of the entries in A2 equals 1600. This means that there are 1600

different paths of cardinality 2 in Graph 1. Since there are no values of 0 in the A2 matrix

for Graph 1, we know that there is at least one path from any given point to any other

given point in the graph for cardinality 3. We interpret one row of Graph 1’s A2 matrix in

Example 2.4.6.3c, which shows all the paths of length 2 for vertex v13. Here, the number

of paths from v13 to every other vertex in the graph is indicated inside each box and can

be confirmed by tracing the paths through the graph. The 64 paths of cardinality 3 that

begin at v13 in Graph 1 are shown in Example 2.4.6.3d.

2.4.6.4. Structures and sets in Graph 2. Example 2.4.6.4a shows the adjacency matrix

A for Graph 2, while Example 2.4.6.4b shows the graph’s A2 matrix. We can discover

more subtle structure by studying the entries of the A2 matrix. The paths of length 2 for

the least well-connected point of the set – v11 – are found in Example 2.4.6.4c.

Investigation shows that there is no way to get from v11 to the four edge vertices in this

173 Of course, path need not move from one vertex to another if there is a loop at a vertex. However, inGraphs 1, 2, and 3, there are no loops.

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graph structure in two moves. However, there are some vertices – such as v16, shown in

Example 2.4.6.4d – that have paths of length 2 that connect it to every other vertex.

Thus, different parts of the space have different ability to reach other parts.

2.4.6.5. Structures and sets in Graph 3. Example 2.4.6.5a shows the adjacency matrix

A for Graph 2, while the A2 matrix for the same graph is given in Example 2.4.6.5b. This

matrix confirms our intuition that the center vertex, v9, has the most connections when

two moves are involved. As shown in Example 2.4.6.5c, there are a total of 32 paths of

length 2 that begin at v9.

The A2 matrix reveals further shades of difference between vertices that seemed to have

an equal number of connections in the A matrix. This is confirmed by comparing the

data for v14 and v17. Both of these vertices had 4 edges in the A matrix. Because v14 is

adjacent to v9, we hypothesize that it has more paths length 2 than v17, which lies more

distant from v9. This intuition is confirmed by the A2 matrix: v14 has 20 different paths of

cardinality 3, while v17 only has 16. We conclude that in Graph 3, the closer a vertex is to

the “hub” (in this case, v9), the more places it can get to in two moves. Thus, in a spatial

composition, it is likely any location in space that functions as a “hub” will have a higher

density of spatial movement than other parts of the space.174

2.4.6.6. Hamiltonian Cycles in Graphs 1, 2 and 3. Suppose a composer wanted to

move a pitched melody through each vertex of a graph, without ever touching a vertex

twice, and end up at the starting point. This particular problem is classic in graph theory

and is analogous to constructing a 12-tone cycle in pitch. A cycle that traverses all the

174 An analogy between space and scale can be made here. A complete graph is like a chromatic space,where each pitch – in theory at least – has the same “value” in non-tonal music. Graph 3, with its“hub”, is more akin to the structure of a a tonal space, where the tonic typically gets used much morethan any other scale degree. The central vertex in Graph 3 could be considered analogous to the “tonic”in a tonal system. These speculations naturally lead us to wonder whether a graph structure could beisomorphic to a scale structure. Temperley’s work suggests that it is extremely likely that a tonalpractice could in fact be modeled by a weighted graph. See Temperley 2001, pp. 167-201.

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vertices of a graph such that no vertex is touched twice is called a Hamiltonian cycle, or a

Hamiltonian path. Example 2.4.6.6a shows a Hamiltonian cycle through each of our

three graphs. Determining Hamiltonian cycles gets easier when more edges connect

vertices in a graph; but in general, the task of finding these cycles is very difficult. In

fact, the general task of finding Hamiltonian cycles in a graph has been shown to be NP-

complete,175 meaning that in many cases the only way to determine such cycles is to go

through an exhaustive “brute-force” search. The complexity of this search increases

exponentially with the addition of new vertices.

2.4.7. Other ways of determining spatial structure and sets. So far, we have explored

graphs in which all the vertices have the same, or nearly the same degree. These graphs

do not contain any bridges. We conclude this section by considering the ramifications in

spatial music of a graph structure with several bridges.

A variation on Graph 3, Graph 4 is shown in Example 2.4.7a. Here, we use five

different instruments, representing three different instrumental families: strings (violin,

viola), woodwind (clarinet), and brass (trumpet, trombone). If we draw edges between

vertices according to their timbre, three disconnected subgraphs result. However, we

might utilize one instrument in each subgraph as a bridge to another. These instruments

will then have particular functions – we might say “spatial pivots”.

In Example 2.4.7b, edges connecting instruments with similar timbres are represented

with solid lines, while bridges that function as spatial pivots are indicated with dotted

lines. Ordering space in this way introduces variety of function into the edge structure.

The instruments located at cutvertices (those that are incident to bridges) might

experience a high density of spatial activity because of their function as links to other

subgraphs. In other words, a spatial set would have to include one of the cutvertices in

order to “modulate” from one subgraph to another. Of course, there are many other

175 Gary and Johnson 1983.

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possibilities for connecting spatial structures such as these, limited only by the

composer’s imagination.

2.5. Transformations in Space

In the previous section, we investigated some properties of three different arrangements

of sound sources. Through matrix multiplication, we were able to generate adjacency

matrices, which tell us the number of sets available in these spaces for various walks

through the graph. The possibilities are summarized in Example 2.5a. In all cases, we

find that there are a large number of sets, particularly when the cardinality increases to 4.

While this allows a vast quantity of raw material for composers, the variety could

become unwieldy. Are there similarities among the shapes that would allow us to reduce

the number in order to make the spatial possibilities more managable? In this section, we

will develop the idea of transformations in space. This will have the by-product of

allowing us to create classes of objects that are equivalent in some way.

The question of relating motives also applies to listeners and the way they might hear the

same shape. Certain shapes may sound similar for a listener depending on the spatial

relationship of that listener and the musical shape. Consider a spatial set such as the one

shown in Example 2.5b. Each listener faces towards the center of the space, and hears

the same spatial set. Is it possible to relate the shapes that each listener hears? The

answer to this question is yes. By developing the idea of spatial transformations, we will

show that these four motives are indeed related.

2.5.1. Transformation in pitch. A transformation is a function that maps one thing to

another thing. Examples of transformations are translations, rotations, and reflections. In

pitch, transformations are usually referred to as transpositions or inversions.

Multiplication (scaling) is also occasionally used. Transformations can be represented as

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matrices. The three common types of transformations in pitch are defined in matrix form

in Example 2.5.1. By multiplying a pitch-class by the matrix, we effect the

transformation.

2.5.2. Transformations in a plane defined by matrices. It is also possible to define

transformations by matrices in spatial music. Applying matrices in this way makes it

unnecessary to specify a set of points in space where sound sources may be localized,

though of course the matrix operates on sets of points. This kind of transformation

models situations where the composer has exercised complete freedom to place a sound

anywhere in the space. It is probably most practical for works such as OKTOPHONIE

where sounds may be technically localized anywhere in a continuous space.

The matrices of four basic transformations in a plane are shown in Example 2.5.2a. The

transformation matrices shown in Examples 2.5.2a-b are 3 × 3 – one order larger than the

2 × 2 matrices we used in pitch transformation. Transformation matrices for spatial

music in three dimensions would be 4 × 4. Adding another spatial dimension simply

makes the transformation matrix one order larger.

The analytic and compositional payback for defining transformations in this way

immediately becomes apparent when we try to relate the four different shapes shown in

Example 2.5b. All four motives are related under rotation. Applying rotation

equivalence to the 64 sets of cardinality 3 from Example 2.4.6.3d – all of which involved

pathways of length 2 – yields only six distinct set-classes. These six set-classes are

shown in Example 2.5.2b. Of course, the diagonal set – whose edges are of length �2 –

does not exactly relate to the set going straight up and down, or to the right or left, where

the distance between vertices is exactly 1. We must overlook these slight differences if

we want the equivalence classes to hold.176 If we wish to further reduce the number of

176 We do this quite regularly when relating pitch structures in tuning systems other than equaltemperament.

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spatial set-classes, we may note that two of the six classes in Example 2.5.2b are

equivalent under reflection. Under rotation and reflection, the 64 sets form only five set-

classes, as shown in Example 2.5.2c. Knowing the basic shapes that are possible in a

given space can help a composer to better grasp the spatial materials he or she has to

work with.

2.5.3. Transformations in a plane defined by mappings. Instead of defining

transformations as matrices, we may specify mappings from one element of a set to

another. Suppose we have four sound sources set up around a listener, as shown in

Example 2.5.3a. This structure can again be represented by a graph. Example 2.5.3b

shows a set of transformations on the graph, consisting of four rotations r and four

reflections f. These transformations preserve the symmetry of this graph since all of the

eight rotation and reflection operations are automorphisms – that is, isomorphisms of the

graph onto itself.

Examples 2.5.3a-b modeled the kinds of motion a composer might want to use on an

arrangement of sound sources that is essentially a regular polygon.177 However, we can

also define transformations in a space that is not a polygon. In Example 2.5.3c, we

define a graph which is a subgraph of Graph 1. We may define “transposition” or

translation for this graph. There are four ways of moving through the graph, which we

will call east (transformation a), north (b), northwest (c) and northeast (d). The four

directions are indicated in Example 2.5.3d. Nine transformations are defined on the

space in Example 2.5.3e. Each transformation is a different path in the same direction

through the graph.178

A third kind of transformation is defined on Graph 3 in Example 2.5.3f. The

177 A regular polygon is a polygon of n-sides in which the length of each side is the same. Examplesinclude the equilateral triangle and the square.

178 Except, of course, for the identity transformation, whose path moves back to the same vertex where itbegins.

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transformation m1 defines movement originating in the center vertex, and expanding

outwards through eight spokes to the middle ring. Transformation m2 continues moving

outwards along a second set of spokes to the outer ring. Finally, m3 maps each vertex in

the outer ring back to the center of the graph. The transformations are cycles through the

graph; though each cycle starts with the same place (vertex 1). While the transformation

m3 is a function (it is many-to-one), m1 is not (since it is one-to-many). Although these

transformations express a useful technique in spatial music, their structure is unlike those

defined above. We will formalize the structure of these different transformations in §2.6.

2.5.4. Application of spatial transformation to spatial music. The idea that “shapes”

in spatial music are somehow transformed is an appealing concept. First, we are

accustomed to transformations in music that apply to elements of the pitch domain.

Second, we are used to making spatial transformations in everyday life. We mentally

“flip” the image of something in a mirror, or we rotate the name of a book on its spine. It

is not impossible to imagine that we can perform the same mental transformations to

shapes heard in space.

In this section, we have developed two ways of understanding transformations in spatial

music. Defining transformations as matrices is probably most useful when the space is

continuous and sounds may be localized anywhere. Defining them as mappings is

probably more useful when there are a finite number of sound sources. Applying

transformations to different spatial sets can show us relationships among sets, grouping

them together into set-classes. Although we may occasionally have to overlook slight

differences in the absolute distance between elements of the spatial set – e. g., equating

points with edges of length 1 with those of length �2 – we do the same thing when

relating sets of pitches in a tuning system that is not equally tempered. While further

defining the structure of space, transformations in space aid us in appreciating the

many structural possibilities of spatial music, while also suggesting ways of analyzing it.

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2.6. Groups

2.6.1. Application of group theory. Groups, useful for studying symmetry, appear

prominently in post-tonal theory.179 For instance, the set of twelve transposition

operations is a group, as is the set of Tn and TnI operations. We shall explore the

similarities between groups of pitch structures and groups of spatial structures below.

2.6.2. Dihedral groups. Each of the eight transformations of rotation and reflection on

the graph in Example 2.5.3a are automorphisms.180 Together, they form an

automorphism group.181 The four groups of these cycles form a Dihedral group,

abbreviated D4. Their group table is shown in Example 2.6.2a. Many structures in music

theory form Dihedral groups, and comparing them to our spatial group is useful for

understanding the potential similarities between the structure of pitch and space.

Consider the set of twenty-four Tn and TnI operations in pitch-class space, modulo 12.

This set is a Dihedral group of order 24, abbreviated D24.182 Now imagine that we have

twelve (instead of four) sound sources equally spaced around the audience in a circle.

Clockwise spatial movement from one sound-source to an adjacent one is naturally

179 A group is a nonempty set S together with some binary operation � on the set which satisfies thefollowing properties:

1. Closure. For all a, b, in S, a � b is also in S (� is the group operation).2. Associativity. For all a, b, and c in S, (a � b) � c = a � (b � c).3. Identity element. There exists some element e in S such that for all a in S, e � a = a � e = a.4. Inverse element. For each a in S, there exists some element a-1 in G such that a � a-1 = a-1 � a = e,

where e is the identity element. (Rosen 1995, pp. 6-10).180 Chartrand 1985, p. 231.181 Properties of automorphism groups, including inner and outer automorphisms, are described in Morris

1987, p. 167 and Morris 2001, volume 1, pp. 129-131. They also play a role in the structure ofKlumpenhouwer networks (Lewin 1990) and Carey and Clampitt’s definition of a “well-formed scale”(Carey and Clampitt 1989, p. 196).

182 Morris 2001, p. 17.

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analogous to transposing a pitch-class n semitones. Reflections across the circle are the

same thing as inversions in pitch-class space. Thus, the structure of this physical space,

taken along with the rotation and reflection operations also form a Dihedral group of

order 24. In this particular spatial configuration, spatial motion is isomorphic to the

familiar transposition and inversion operations of pitch.

2.6.3. Cyclic groups. A related group structure is the cyclic group. The set of

transposition operations defined on the space of pitch classes mod 12 is a cyclic group of

order 12, abbreviated Z12.183 We could represent a similar group in the circular

arrangement of twelve sound sources we previously imagined in §2.6.2. If we define

only the operation of rotation in this space, we eventually arrive back where we started by

completing one cycle around the circle. The transformation of rotation on this space

forms a cyclic group Z12 which is isomorphic to the group of transposition on the twelve

pitch-classes.

The transformations we defined in Example 2.5.3c are somewhat more complicated.

These translation operations form a rectangular point lattice which is a subset of the

Euclidean plane. The group of these translations is a translation group. Their group table

is shown in Example 2.6.3a.

2.6.4. Other structures. The three multiplication transformations defined on Graph 3 in

Example 2.5.3f do not form a group because some of the transformations are not

functions.184 Even so, these m-transformations – and this kind of spatial arrangement of

sound sources – might be very useful for a composer. Even though Graph 3 and the m-

transformations on it do not form a group, we can still analyze its structure and make

predictions about its use in spatial music by generalizing its structure as a graph.

183 Ibid., p. 15.184 This structure is strikingly similar to Slawson’s sound-color contour of Laxness. Slawson 1985, p. 55.

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If the space and the transformations do not form a group, we can also use statistical

analysis to obtain a great deal of information about the work. Analyzing the density of

spatial motion in the performance space is one way of proceeding along these lines. It

may be possible to find “hot spots” in the space where a disproportionate amount of

spatial activity occurs. If there is a weak or ill-defined graph structure, statistical

measurements allow us to make measurements which are useful in determining the

spatial structure of the piece.

2.7. Statistics

2.7.1. Use of statistics as an analytical method. The methods developed in the

previous sections are geared towards analyzing a presumed motivic/transformational

language in spatial music. But in some spatialized music, such a language may not be

present. Even in music that does have such a language, there are relevant elements of a

spatial composition that cannot be captured by these methods alone. By employing some

very simple statistical techniques, we can fill in some of these gaps.

Knowing how often a particular sound source, or – in a graph, a vertex – intones musical

events is an important measure in a spatial composition. Listeners may have a vague

impression that one part of the space may be used more frequently than another. Such

determinations are impossible to verify unless we measure the number of times each

musician is used. Above, we remarked that in spatial music that can be represented by a

graph structure, some parts of the graph may be used more than others (such as

cutvertices or bridges). These statistical analyses can tell us whether the composer has

simply followed the latent structures in a graph, or worked against them.

The amount of time each musician is used through the course of a composition may differ

from the number of times he or she is used. Simply measuring the number of times each

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musician is used does not tell us how long that musician is the focus of attention.

Statistically calculating the amount of time music dwells in a particular section of space

can give further nuance to our understanding of the overall spatial profile of a piece. We

may also consider how certain locations in space function relative to other points in

space. We might characterize a “collector” as a point in space that functions as the

destination of a great variety of different spatial moves. A “reflector” directs spatial

motion principally to one or two different places, whereas a “diffuser” passes motion on

to a wide variety of different places. Statistical analysis of spatial data can tell us if

different places in spatial music have different functions.

2.7.2. Relationship of speed and spatial motion. We have already described how the

speed of a spatial motion can be calculated. But, the average speed of music over larger

spans of a piece can yield useful information about the total speed profile of the work.

One hypothesis that can then be tested pertains to the relationship of the speed of spatial

motion and its complexity. Is there a greater variety of motion when the speed is slow?

This seems probable, since listeners are perhaps more likely to hear subtle spatial motions

when the speed is slow. At high speeds, motion may become more repetitive, making the

overall spatial pattern easier to perceive. Relating statistical data for spatial speed with

spatial motion can tell us more about perceptual strategies in spatial music.

2.8. Conclusions

The sections in this chapter suggest the following method of approaching analysis in

spatial music, along with questions that can be answered along the way.

1. Look principally to the score for things to analyze. (§2.1)

2. Calculate the location (coordinates) of sound sources. (§2.2)

3. Define the kind of space that is used. Is it continuous or discontinuous space?

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Is there modularity? What is the dimension of the space? (§2.3)

4a. If the space is discontinuous, catalog the moves that are used from one sound

source to another, and determine the edges between vertices. Then,

determine the number and shape of sets that are possible. It is possible also

to calculate the probable density of spatial motion in a particular area of

space by determining how connected a given vertex is to the graph as a

whole. Does the composition reinforce the graph structure or work against

it? Are there sound sources that function as “hubs”, “spokes”, “portals” or

“pivots” to other parts of the graph? (§2.4)

4b. If the space is continuous, determine the shape of sets by calculating their

locations. (§2.4)

5. By observing the kinds of motion around the space, define transformations,

either through matrices or through mappings. Are there patterns or

symmetries? (§2.5)

6. If so, define a group structure of the transformations on the space. A group

structure may possibly operate on a subset of the space or a subset of the

transformations. Is the structure of space analogous to the structure of

other musical dimensions, such as pitch? (§2.6)

7. Use statistical data to obtain more information about the way space is used in

the work. (§2.7)

Using these methods, we can learn a great deal about spatial motion in a composition.

They also open many avenues for further exploration. One possible direction for future

research is to make further use of graph theory. There has been a significant amount of

mathematical research in this field ever since Euler made his first studies in the 1730s.

Composers may wish to develop graph structures in their spatial music more thoroughly.

While we have only scratched the surface of graph theory in this chapter, a more

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thoroughgoing approach could open up many new avenues of musical possibility and

analytical insight.

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Chapter 3.Stockhausen’s OKTOPHONIE

3.1. OKTOPHONIE and LICHT.

3.1.1. Stockhausen’s LICHT project and the TUESDAY opera. Beginning in 1974,

Stockhausen embarked on a trio of works reflecting cycles in the natural world. The first

work, SIRIUS (1975-77) is a four-part work structured on the seasons of the year.

Second, Stockhausen tackled the days of the week with his seven-opera cycle, LICHT

(1977-2003).185 Finally, Stockhausen’s KLANG project – left unfinished at the time of

his passing – was to include one piece for each hour of the day. Of these works, LICHT

contains a section from the Tuesday opera called OKTOPHONIE that features

spatialization among its many compositional elements. This chapter is concerned with

the elements of spatialization in OKTOPHONIE. After an overview of LICHT and its

compositional basis in a “Superformula” we will turn to these spatial elements.

Virtually the entire structure of the LICHT cycle is derived from a one-minute

composition called the Superformula [Superformel]. The basic building material for the

Superformula consists of three “nuclear formulas” [Kernformeln]: one has 13 pitch

classes, the second 12, and the last 11. These nuclear formulas, which contain the basic

“kernel” pitches, are shown in Example 3.1.1a. Each of the three nuclear formulas

corresponds to a protagonist in the opera: Michael, Eve and Lucifer. Each also has

characteristic intervals – a perfect fourth for Michael, a rising major third for Eve, and an

185 Though SIRIUS has been performed in its entirety many times, LICHT has not yet been heard as aseven-day opera cycle. While Thursday, Saturday, Monday, Tuesday and Friday have been performedas entire operas, Wednesday and Sunday have not yet been performed as a whole work; only sectionshave been heard. The same is true of KLANG: only individual works from this cycle have beenperformed as of this writing.

Although the comparison with Wagner is almost inevitable, some scholars have speculated thatStockhausen’s motives for composing the enormous LICHT cycle were influenced by the prevailingeconomic climate facing many composers today. Claus-Steffan Mahnkopf describes the “aggregate ofindividual works” in the LICHT cycle as “resembling a cunning mercantile strategy for introducingincoming commissions and performance conditions into a formally completely open, and thus far frombinding totality” (Mahnkopf 2002, p. 49f).

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ascending major seventh for Lucifer).186 Although these intervals occur often in the three

formulas, the general contour of the lines also sets them apart from each other. The

overall contour of Michael’s formula is falling. Eve’s rises, has a sudden fall in the

middle, then rises again. Lucifer’s formula twice leaps up and falls back again.

To create the Superformula from the nuclear formulas, Stockhausen added rhythm and

various “Akzidenzien”.187 The five types of Akzidenzien used by Stockhausen are

“variation”, “echo”, “scale”, “modulation”, and “wind” or “anteconsonants”. The

Akzidenzien provided general categories for embellishment and extension that

Stockhausen used to add musical subtlety to the kernel pitches. Although Stockhausen

also added other embellishments such as gefärbte Pausen (“colored pauses”) and

“yodels”, these are not, strictly speaking, Akzidenzien.188 Stockhausen further increased

the rhythmic subtlety in his Superformula by adding tempo changes to nearly every

measure. These tempi are derived from his well-known “chromatic” tempo scale.189 As

remarked before, there is no spatial information in the Superformula.

The finished Superformula, or “master plan” for LICHT, is shown in Example 3.1.1b. It

is essentially a combination of the three nuclear formulas, with added layers of

embellishment and compositional design just described. The seven segments are called

“limbs” or “Glieder”. A system of transpositions (both for pitch and tempo) derived from

186 “Characterizing” a simple group of pitches is a technique Stockhausen had already developed to aconsiderable degree of subtlety in works such as MANTRA. Pitches in MANTRA are associated withother musical elements such as form of attack and duration. See Stockhausen 2003a.

187 The German word Akzidenzien is the plural form of Akzidenz. While it can mean “accidental”, as in asharp, flat or natural, in Stockhausen’s case it also means “supplement” or “additional profit”. The listof Akzidenzien can be found in the Superformula sketches in Texte 5, pp. 154–55. Thanks to JeromeKohl for pointing out this subtlety.

188 For a much more comprehensive analysis of the Superformula design, see “Die Tonhöhen derSuperformel für LICHT”, Texte 9, pp. 12-28. Analyses of the Superformula can be found inSchwerdfeger 1999 and Bandur 1999a. The sketches for the Superformula can be found in Texte 5, pp.147-160. Jerome Kohl’s analysis of the Superformula’s structure (in Kohl 1990) provides anothercrucial angle on the design.

189 Stockhausen’s tempi are derived from an “equal tempered” division of a tempo “octave” (i.e., twelveequal logarithmic divisions of the tempo scale from quarter = 60 to quarter = 120). See “...wie die Zeitvergeht...” (“...how Time Passes...”), Stockhausen 1963, pp. 99-139, translated in Stockhausen 1959.

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elements of the Superformula, determine the center of the overall tempo and pitch scale

for each individual opera. In principle, one quarter note of the Superformula is expanded

to 16 minutes in each final composition of LICHT.190 When expanded, events in the

Superformula create formal boundaries in the structure of individual component works.

Each composition in LICHT theoretically shares some material with every other

composition, yet each is also derived (on the deepest level) from a brief section of the

Superformula.

Unique in the LICHT cycle, the TUESDAY opera is not entirely derived from the

Superformula. Stockhausen composed the first act – called JAHRESLAUF – in 1977,

before the Superformula was even composed. It is only Act 2, written from 1990-1991,

that stems from the Tuesday limb of the Superformula. The official name given to Act 2

is INVASION – EXPLOSION mit ABSCHIED (Invasion – Explosion with Farewell),

but this 74-minute-long piece of music (listed as No. 61 in the Stockhausen catalog) is

subdivided into many smaller works. OKTOPHONIE is the background electronic music

for the entire second act. Like several other lengthy, spatialized works of electronic

music by Stockhausen, OKTOPHONIE is meant to be coordinated with and played back

during live musical performance.191

3.1.2. Premise of OKTOPHONIE. The TUESDAY opera is the day of conflict

between the characters Lucifer and Michael. The action of the second act takes the form

190 Following this logic, one would expect the duration of all seven operas to equal approximately 960minutes or 16 hours. In practice this is not the case, not only because of the countless nuances in tempo(which Stockhausen often added only in rehearsals), but also because of the addition of the many“insertions”, or “Einschübe”. Because of this, the total duration of LICHT is approximately 29 hours(Stockhausen 2005, p. 4). Even so, by the time Stockhausen was working on the Monday opera,Jerome Kohl concluded that “on the whole...LICHT...exhibits an almost unprecedented strictness incarrying out the form-plan” (Kohl 1983-4b, p. 175).

191 Stockhausen first attempted synchronization of electronic and live music in KONTAKTE (1966). Thetechnique can be found throughout his LICHT operas. For example, UNSICHTBARE CHÖRE is meantto be played back during the first and third acts of the Thursday opera. Electronic and concrete music isalso intended to be heard during performance of ORCHESTER-FINALISTEN in the Wednesday opera.Similar procedures can also be found in the Monday and Friday operas.

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of open warfare between two “armies” – one representing Michael and the other Lucifer.

The Michael army “repels” two “invasions”, but a trumpeter – representing Michael – is

wounded. A lament called “Pietà” mourns the fallen soldier. Thereafter follows the third

invasion, culminating in three rather dramatic “explosions”. Here, Lucifer’s troops

successfully break through whatever defenses were erected, and reach the defenders.

Lucifer’s forces are represented by trombone-wielding characters who, in the Leipzig

premiere, gradually made their way from the back of the hall to the stage on raised

gangway. After the three explosions, there is a section called “JENSEITS/BEYOND”.

Following this is a piece for synthesizer called SYNTHI-FOU (also known as

KLAVIERSTÜCK XV). In the final ABSCHIED/FAREWELL section, the spatial

activity dies down and thirteen chords, symbolizing Lucifer’s “number”, close out the

opera.192

During OKTOPHONIE, the idea of warfare is audibly represented as a nighttime aerial

bombardment. “Shots”, “bombs”, and “crashes” punctuate the musical texture. The

sound-bombs actually correspond to an enormous augmentation of the rhythm in the

Lucifer formula, whereas the shots correspond to the augmented rhythm of the Michael

formula.193 These sounds are spatialized so as to heighten their dramatic effect.

Simultaneously, slow drones rotate around the audience. The drones suggest the idea of

squadrons of bombers. Stockhausen’s informal sketch “Studie zur Oktophonie”

(Example 3.1.2)194 gives some idea of the overall composite impression of all these

“drone” movements – without the added sound bombs, shots, and crashes. Wirtz

observed that the second act of TUESDAY is an example of the way in which “musical

192 A more detailed summary of the staged and concert version can be found in Texte 9, pp. 217-230 and231-236.

193 More specifically, the “deep middleground” structure in the first half of OKTOPHONIE – that is, up toPIETÀ – is based upon the augmentation of the Monday, Tuesday, Wednesday and Thursday limbs ofthe Superformula. From PIETÀ to JENSEITS, the Friday and Saturday limbs of the Superformula formthe background material. SYNTHI-FOU is based on an extension of material from the end ofJENSEITS, while the Sunday limb determines the structure for the final section of OKTOPHONIE,namely, ABSCHIED and SPIEGELWELT.

194 Frisius 1996, p. 333.

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forming” can be generated by scenic design; it shows an unusual variant of scenic music

in the LICHT cycle.195 According to Maconie, the sonic design reminds one of the kind

of warfare Stockhausen himself experienced in his childhood, and thus has

autobiographical components.196

3.1.3. Formal Structure of OKTOPHONIE. As stated, elements of the Tuesday limb

from the Superformula determine the length of the subsidiary sections in the second act

of the opera.197 Essentially, OKTOPHONIE consists of two halves which are roughly

equal in duration. The first one, which is about 36 minutes long consisting of the two

“air battles” and two “invasions”, includes the most dynamic spatialized elements. The

second half, which is nearly 32 minutes in duration, contains the subsections PIETÀ, the

3rd INVASION/EXPLOSION, SYNTHI-FOU and ABSCHIED. There are far fewer

dramatic spatial events in this section; the spatial language dwindles down to the slowly

rotating drone sounds mentioned earlier. The entire temporal structure of

OKTOPHONIE is shown in Examples 3.1.3a and 3.1.3b, which is a kind of Hörpartitur

for Act 2 of the Tuesday opera.

3.1.4. Analytical and Critical Literature on OKTOPHONIE. In a lecture he gave in

1998, Stockhausen spoke extensively about the actual techniques he employed while

spatializing the sounds in OKTOPHONIE.198 Other writers, such as Misch, describe the

use of space without making any significant attempt at analysis.199 The only independent

analytical work that appears to have been done on the shapes of the bombs, shots and

crashes is by Overholt.200

195 Wirtz 2000, p. 48.196 Maconie 2005, p. 484. Of course, the THURSDAY opera is perhaps the most clearly autobiographical

work Stockhausen ever wrote. In OKTOPHONIE, Maconie found the sounds “anything but terrifying,gigantic, or realistic...the synthesizers Stockhausen is relying on have failed to deliver”.

197 The sketches that show the development of smaller-scale structures are in Texte 9, pp. 177-184.Richard Toop’s lecture of 3 August 2004 at the Stockhausen Courses illuminated the structure and logicbehind one of the parts of OKTOPHONIE, PIETÀ.

198 Stockhausen 2000b.199 Misch 1999a, pp. 152-155.200 Overholt 2006, pp. 134-167.

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Overholt employs a system of “shapes” to categorize the bombs, shots and crashes in

OKTOPHONIE. “The same four main categories of shapes — points, blocks, lines, and

wedges — are found in OKTOPHONIE [as in GRUPPEN], but the individual shapes are

considerably more complex, and, through the use of electronic technologies, they are

much more clearly delineated and no longer left to illusion.”201 Overholt suggests that

different sections of OKTOPHONIE are characterized by different deployments of her

shape repertory.202 The first part along with Invasion 1 contains “four-part [spatial]

counterpoint”, involving “blocks”, points”, “lines” and “wedges” altogether. During the

final section, (“JENSEITS, SYNTHI-FOU, ABSCHIED”), spatial activity has dwindled

down to a single spatial motion: “blocks”, or drones.

Overholt’s contribution is twofold: first, she categorizes spatial events in OKTOPHONIE

according to their generic spatial shape. Then, she analyses the order of the “shapes” – or

events – given in the score. Therefore, her conclusion that “... unlike GRUPPEN, in

which the shapes outlined a mirror form in the overall plan for the work, in

OKTOPHONIE the shape deployment seems to start with great complexity and then

dwindle down to just the background block formations”203 is useful but evident from the

score.

To gain a more nuanced understanding of the spatial aspects in OKTOPHONIE, we will

apply some of the analytical techniques we defined in Chapter 2 to elements of the score.

We are interested in determining the level of precision Stockhausen had over the

spatialization of the sounds in OKTOPHONIE. We will also explore the internal spatial

structures in event groups like the distribution of shots in different parts of the space.

201 Ibid., p. 140. Compare the first four shots in Example 2.5.5c with her analysis in Overholt p. 145.Each “shot” or “bomb” appears to be a block; each “crash” a triangle.

202 Ibid., p, 159.203 Ibid., p. 166.

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3.2. Elements of Sketches and Score that Pertain to Spatialization

3.2.1. Basic approach to analysis. We begin from one particular detail from a sketch,

to orient our analytical questions.204 This lower-left corner of the sketch, boxed in

Example 3.2.1a, shows how Stockhausen conceived of the various shots (“Spur 3+4”)

[“Channel 3+4”] and bombs (“Spur 3”) [“Channel 3”] moving in the cubic space.205

Although the decibel levels which allowed him to create the movement shown in the

sketch are notated in the score, Stockhausen did not make any more sketch diagrams for

the spatialized sounds.206 A diagram for one spatialized sound from OKTOPHONIE first

shown in Examples 2.2.3a-d is reproduced here as Examples 3.2.1b-e to facilitate this

expanded discussion.

3.2.2. Bombs. After performing the calculations described in Example 3.2.1b-e on the

score data in OKTOPHONIE, we can determine the theoretical location of spatial events

in the cubic space. Examples 3.2.2a-c show the location of each of the total number of 65

bombs as it lands on the floor. Example 3.2.2d combines these data on one graph to show

where all of the bombs fall. Example 3.2.2e gives the decibel settings of each bomb and

the coordinates where the bomb falls, in a 14 × 14 square floor area. In this example, the

coordinate (0,0) is at the center of the space whereas (-7, -7) is in the back-left corner.

It is clear from a cursory examination of the data that the majority of bombs fall in the

back half of the space. Whereas 15 bombs fall in the front half, 49 fall in the back half.

Significantly, one bomb (the first) falls exactly in the middle of the space. This event

may serve to help orient listeners at the beginning of a complex sequence of spatial

204 Along with sketches pertaining to the temporal structure of OKTOPHONIE which were cited infootnote 11, Stockhausen made an additional series of sketches for the “Verräumlichung”. Texte 9, pp.521-524.

205 This example is from Texte 9, p. 521.206 Although he was clearly aware that other sonic elements such as phase difference and the doppler

effect contribute to the perception of spatial location, Stockhausen exclusively used changes in loudnessamong the speakers to spatialize sounds in OKTOPHONIE. “Lautstärkeveränderungen sind vorläufigdie einsige Methode, Klänge im Raum zu bewegen.” (Stockhausen 2000b, p. 68).

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events. Although a great many bombs fall close to the center of the space, there is a

noticeable concentration about two-thirds of the way back, along the center axis of the

space. Bombs 7, 23, 24, 29, 34, 37, 38, 43, 51, 56 and 64 all fall in an area of increased

spatial activity towards the bottom-back-center. This distribution already suggests ways

of interpreting the spatial activity in OKTOPHONIE which we will return to in our

analysis.

3.2.3. Shots. 48 shots occur in the first half of OKTOPHONIE, while 18 occur in the

second half. All of the shots end on the ceiling (loudspeakers V-VIII), and are meant to

be “anti-aircraft fire”. In the first half of the piece, shots start at the front-floor pair of

speakers (II/III). During the second half, shots are subdivided into two groups. The first

ten shots begin from the floor array (I-IV), while the last eight begin from the front array

(II, VI, VII, III). In order to make this more clear, the shapes of these three groups of

shots is shown in Example 3.2.3a.

The spatial location for each of the 48 shots is shown in Examples 3.2.3b-d. In Example

3.2.3e, the individual data are combined on one graph. Example 3.2.3f shows the decibel

settings and the calculated coordinates of these first 48 shots in a table. An analysis of

the starting points of the shots again shows several “clumps” where a disproportionate

number of shots begin along the bottom-front edge of the cubic space. The endpoints of

the shots are strongly skewed towards the back of the hall. Only six shots make it to the

front half of the ceiling, whereas 39 end at the back half. Three shots end exactly in the

middle of the ceiling, directly above the sound-projectionist. Surprisingly, there is also a

fairly dense area of shot activity in exactly the same place where a disproportionate

number of bombs fell. The endpoints of shots 4, 7, 10, 13, 16, 21, 32 and 34 are located

towards the bottom-back-center. There seems to be a relationship between shots and

bombs in this area of the space. We will further refine our measurements in the

following sections to determine exactly what this relationship is.

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Of the eighteen shots in the second half of OKTOPHONIE, the endpoints are once again

highly skewed towards the back half of the space. Not one of the eighteen shots in the

second half end at the front half of the ceiling. The starting and ending points of the first

ten shots are shown in Example 3.2.3g; the last eight are shown in Example 3.2.3h. Their

combined distribution can be seen in Example 3.2.3i, and the coordinates in space are

given in Example 3.2.3j-k.

3.2.4. Crashes and Explosions. A crash occurs when a “shot” hits an imaginary aircraft

dropping the bombs. The beginning and end points of the eleven crashes are shown in

Example 3.2.4a. The aggregate spatial distribution is graphed in Example 3.2.4b, and the

calculated coordinates can be found in Example 3.2.4c. Stockhausen added eight

“superimposed crashes” later in the first half of OKTOPHONIE, but the exact spatial data

for these crashes is not recorded in the score.207

The analysis of starting and ending points is probably the most speculative of the

analyses done so far from the perspective of perception. As the crash sound descends

from the ceiling to the floor, it moves around the space in a kind of “corkscrew” shape.

The corkscrew can be seen from Stockhausen’s diagram, shown in Example 3.2.4d.

Because of the rapid movement around space, it is likely that unless special precautions

are taken by the sound projectionist, the starting and ending points of the sound may be

difficult for listeners to localize.

With this caveat, however, the crashes present an important opportunity for our analysis.

In the dramatic context of OKTOPHONIE, a crash occurs only as the result of a shot

successfully “hitting” an imaginary aircraft. There is a trivial correlation between the

207 Stockhausen 1994, p. O XXII. “For the 8 superimposed crashes from 24:02.3 to 26:42.3, 8individual rotations were controlled manually.” Stockhausen clearly relished this part of the score,with its curving lines descending from one staff to another, since he made a color postcard of it. Thepostcard is called “8 Abstürze aus INVASION, 08. 10. 1990”.

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ending points of the shots and the starting points of the crashes. This can easily be

verified by comparing the top part of Example 3.2.3e (endpoints of the shots) to the

bottom portion of Example 3.2.4b (starting points of the crashes). Unlike other spatial

activity, the endpoints of the crashes tend to be distributed more towards the front half of

the floor, as can easily be seen in the bottom portion of Example 3.2.4b (“Ending

points”). Eight of the eleven crashes end in the front half of the floor. Perhaps the spatial

distribution of the crash endpoints was intended to give somewhat more balance to the

overall spatial design of the piece, since so many other events are skewed towards the

back half of the space.

Although our analyses of the location of spatial activity have been informal in this

section, they raise several suggestive questions. While there is a trivial cause-and-effect

relationship between the shots and the crashes, is there any such relationship among the

shots and the bombs? And, is there an internal structure to the pattern of bombs and shot

endpoints? In order to decide how best to approach the problem, we will define the shape

of the space in OKTOPHONIE.

3.3. Shape of the space in OKTOPHONIE

3.3.1. Arrangement of sound sources. An extended correspondence between Solf

Schaefer, director of the Darmstadt Ferienkurse für Neue Musik from 1995-2008, and

Stockhausen indicates how important the correct speaker arrangement was for the

composer of OKTOPHONIE.208 In the exchange, Stockhausen emphasizes how crucial

the correct height is for the top bank of four loudspeaker groups. Stockhausen later

claimed that he could follow movements even when sitting “outside” the space defined

by the cube.209 When performed in an opera house, some of the audience would almost

certainly be seated outside this cube. For “emergency situations”, Stockhausen allowed

208 Misch and Bandur 2001, pp. 571-615, especially p. 566.209 Texte 8, p. 583.

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an alternative speaker arrangement. Concerning this arrangement, Stockhausen wrote

If it is impossible to have a cubic octophonic arrangement of the loudspeakers,one can possibly choose a quadrophonic arrangement (whereby of course half ofthe composed spatial movements are lost). One then balances the eight channelsof the tape with help from a panorama-regulator in the following four loudspeakergroups... Even so, one should avoid this emergency solution.210

The alternative arrangement is shown in Example 3.3.1a. Stockhausen left this

suggestion out of his score, published in 1994 (the citation is from 1998, after the

correspondence with Schaefer). Stockhausen’s suggestion invites speculation about how

music such as OKTOPHONIE might be perceived in this loudspeaker arrangement.

Instead of sounds moving from the floor to the ceiling, they would spiral around arcs in

the space. Instead of moving in three dimensions, the spatial motion would be squashed

down to a plane. Needless to say, our analysis of the sounds would have to be

significantly altered if such an arrangement were in place. We do not speculate on the

results of such an analysis, first because, in the author’s experience, the use of this

alternative loudspeaker arrangement is exceedingly rare, and second, because

Stockhausen considered it only permissible in “emergency situations”. Therefore, the

shape of the space is a cube, and its dimension is 3.

3.3.2. Drones. In section 3.2, we located and informally analyzed the bombs, shots and

crashes in OKTOPHONIE. While these sounds start and end in particularized locations,

the drones cycle about in a continuous fashion.211 Clearly the idea of the drones is to give

the impression of continuous spiraling motion. In the score, Stockhausen indicates the

shape of these sounds with continuous closed shapes. There can be no doubt that despite

the discontinuous arrangement of speakers, Stockhausen wished to give the impression

that these sounds were in continuous movement around the listeners.

210 Texte 9, pp. 308, 319.211 A pitch analysis of one of the layers of drones can be found in Kohl 2004, pp. 127–28.

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3.3.3. Other sounds. On the other hand, there are discrete points where bombs, shots

and crashes start and stop. Although these sounds can be placed anywhere within the

boundary of the cube, their endpoints are specified. Since Stockhausen manipulated the

relative decibel settings of the speakers through balancing the amplitude of the sounds

assigned to them with a potentiometer, these starting and ending points can theoretically

be centered anywhere in space. Therefore, these sounds have elements of both

discontinuous motion (their starting and ending points) and continuous motion.

3.4. Spatial shapes in OKTOPHONIE

3.4.1. Spirals/Drones. Although the drones consist of slow rotating sound, their shape

actually varies considerably: some are like corkscrews while others are like “figure 8s”.

In the first half of OKTOPHONIE, Stockhausen composed three different layers of

spirals or drones. The first layer, identified as layer 5 in the score, moves from the front

plane of the space (speakers II-VI-VII-III) to the rear (speakers I-V-VIII-IV). Later in

this level, drones spiral from the left plane to the right, and then back again. In layer 6,

two looping movements traverse loudspeakers V-III-VIII-II and I-VII-IV-VI. Finally, in

layer 7 there is very slow clockwise rotation around the floor in speakers I-II-III-IV.

The second half of OKTOPHONIE has considerably more complex drone activity. In

PIETÀ, there are four layers of drone motion (identified in the score as layers 1-4).

These include very slow spiral rotation at a 20 second rotation rate; slow counter-

clockwise rotation; slow looping movement and very slow cross rotation.

Through the remainder of the second half of OKTOPHONIE, the complexity of drone

movement increases. There is double-rotation, lateral counter-movement, alternating

loop rotations, and more conventional rotation movements as well. These movements are

always quite slow in relation to the shots, bombs and crashes. For example, the drone in

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layer 11.4 takes 10.9 seconds to traverse loudspeaker groups V-VIII-VII-VI; layer 12.1

takes 10.7 seconds to complete one circuit of alternating loop rotation, and in layer 12.2

a complete cycle takes 13 seconds. Since the looping movements occur at so many

different asynchronous rates, and involve not only circular patterns but also “figure-

eights”, the compositional technique is strongly reminiscent of the asynchronous talea

employed by Messiaen in the first movement of his Quartet for the End of Time. It could

certainly be said that the affect Stockhausen tried to evoke in this layer of spatialization –

one of a certain timelessness – is somewhat analogous to Messiaen’s.

3.4.2. Bombs. The spatial locations of the bombs are the same at the level of the ceiling

and the floor; in other words, they fall straight down.212 In order to show the shapes that

the successive bombs create on the floor, the 65 bombs are divided into four groups (for

greater legibility) and shown in Example 3.4.2a. We may begin to identify any patterns

by calculating the angle between each set of three bombs. This is easily accomplished

using simple trigonometry. At the same time we can calculate the distances, in meters, of

points between consecutive bombs. The result of these calculations is shown in the table

in Example 3.4.2b.

Analysis of the simplest forms – those with three points and one angle – shows that there

is a great variety of shapes that do not readily lend themselves to simple transformational

relationships. Example 3.4.2c shows the three shapes with the most acute angles: all

three successive points create paths whose central angles are near 2°, yet, the distances

between the points are all quite different. Analysis of the most obtuse angles, shown in

Example 3.4.2d, yields similar results.

How does the angle and distance measurement change over time? Plotting the distances

212 Stockhausen 2000b, p. 72. “Klang-Bomben fallen von der Decke und sind Einschläge 1 bis 24 in demQuadrat, in dem Sie sitzen.” This is surprising, since if the bombs were dropped from moving aircraft,their descent pattern would be parabolic, not straight down. Perhaps Lucifer is dropping the bombsfrom stationary Zeppelins, balloons or helicopters!

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and angles between each of the 65 bombs yields the graphs shown in Example 3.4.2e.

We might hypothesize that bombers concentrate their runs on one particular target. If

this is the case, we might expect to see several dense clusters of bombs with large

distances separating clusters, similar to the way aerial bombardment was conducted in

various historical air campaigns. While there most certainly are clusters of bombs in the

space, they occur only over the entire course of the composition, and do not appear to

occur in any obvious temporal sequence.

3.4.3. Shots. As we have seen, the first group of 48 shots begins in the front pair of

speakers and lands on the top surface of the cube of speakers. However, we will now

analyze the sequence of shot ends. This will determine whether there is an internal

structure to the end locations of the shots. The first sequence of 48 shot endpoints is

divided into three groups and shown in Example 3.4.3a. The results of calculating the

angles between each set of three shots is shown in Example 3.4.3b.

Analysis of the angle and distance measurements over time yields somewhat different

results from the previous bomb analysis. The plots of distance and angle for the first 48

shots are shown in Example 3.4.3e. From examining these graphs, we could conclude

that there is perhaps some logic in the change of distance between each shot. Whereas

the first seven shots are very widely spaced, the average distance between shots decreases

over time.213 We could interpret the data to suggest metaphorically that those who are

directing the counterattack gradually focus their antiaircraft fire on specific areas on the

ceiling. The distance between the end of one shot and the end of the next becomes

somewhat smaller as targeting becomes more effective. But, by the time shot 40 has

occurred and the distances between shot endpoints have really noticeably decreased, the

bombs have long since stopped falling. Even though there is an overall trend in the shot

213 The angles of the first seven shots also are fairly small. We may hypothesize that large distances resultin small angles, since a large distance will tend to move the sound off to a corner of the space thusincrease the likelihood of the next sound “doubling back” on itself. While this is a logical assumption,the correlation between distance and angle is anything but clear in Example 3.4.3e.

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plot towards smaller distances between end-points, it seems that ultimately the data are

inconclusive in this respect; there is nothing to fire at by the end.214

Like the first group of 48 shots, we will only analyze the ending points of the second

group of 18 here. As in the first group, these shots also end on the ceiling. The sequence

of shot endpoints is shown in Example 3.4.3f. It is immediately apparent that these shots

are even more densely clustered around the top-back-center part of the space than the

previous group. The angles formed by each set of three shots are all acute except for one,

as the analysis in Example 3.4.3g shows.215 The predominance of acute angles gives the

impression that the shots are constantly circling around, doubling back on themselves.

There is little here to indicate a strategy of shot targeting.

One clear relationship between the shapes of the shot ends in the first and second groups

is that the first four shots in each group trace out approximately the same pattern. The

four shots in the second group are slightly contracted in comparison to those in the first.

Dilating them by about 140% – without any significant translation, rotation or flip –

would cause the first three shots to approach almost exactly the same shape as the shots

in the first group. This could be thought of as analogous to starting out two sections of a

musical work with a similar pitch-motive. This close relationship – without showing

dilation – is highlighted in Example 3.4.3h.

Our attempt to relate shapes yielded few useful results when we examined the internal

structure of the shapes created by the bomb ends. This also seems to be the case with the

shot ends. It was either too difficult for Stockhausen to organize these internal

relationships given the way he spatialized the sounds, or he chose not to. However, we

214 In this case, the structural necessity of correlating spatial activity with events from the expandedSuperformula has overridden the dramatic principle that ideally would associate shots with bombs intemporal proximity.

215 Since several shots occur in exactly the same place, the angle between several groups of three shots isundefined.

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can say with some certainty that for the most part, the shots do target areas of high bomb

density. Our analysis confirms that these aspects of spatial composition were, for the

most part, very likely left to chance by Stockhausen, who only managed the general

distribution within tolerances of several square meters.

3.4.4. Crashes. A crash occurs as a clear cause-and-effect relationship; the cause, of

course, is a shot which has “successfully” hit one of the imaginary bombers overhead.

As we have mentioned above, the the “corkscrew” shapes of the crashes are probably the

most difficult to perceive. Still, we can examine their shapes in some more detail to

answer one of our fundamental questions: is the targeting scheme of the shots at all

related to the location of the bombs?

If the targeting scheme of shots is related to the bombs, then it seems most logical to

hypothesize that the location of the start of the crash would be near the previous bomb.

As Example 3.4.4a shows, the distance from the top point of the crash to the immediately

preceding bomb (which, of course, lands on the floor) varies considerably in each case.

Although the first crash occurs exactly in the same place as the first bomb, we can

probably discount this as insignificant because all of the initial spatial events occur

directly in the center of the space. At the extreme, we find that the distance between

bomb 61, which occurs 5.1 seconds before crash 23, is 10.61 meters – a very large

distance considering the overall dimensions of the cubic space. The average distance

between a bomb and the crash that follows is 3.7 meters, which is far enough apart from

the crash in a 14 × 14 × 14 cube that it is difficult to support a spatial connection. We

can conclude from this that the location on the ceiling of the beginning of a crash has

little or no relationship to the location on the floor of the bomb immediately preceding it.

The antiaircraft fire is not, therefore, taking cues from the location of the bombs that land

on the floor.

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3.5. Other Spatial Relationships in OKTOPHONIE

3.5.1. Atemporal Relationship Between Shots and Bombs. In the previous section,

we traced the pathway through space of bombs, shot endpoints and crashes. We found

that crashes which begin on the ceiling do not have a significant relationship to the

bombs that precede them. But taken out of time, we noticed that they do have a

relationship; the shots tend to hit the same general region as the bombs below. How

close is this out-of-time relationship? We can tally up the number of shots and bombs

that occur in each vertical 1 × 1 × 14 “slice” of the space and create a ball diagram. This

diagram is shown in Example 3.5.1. Crosses indicate the location of a crash.

This diagram underlines the most important conclusion from our data, which we

informally stated in section 3.2.4. The diagram brings to light several significant

characteristics of Stockhausen’s spatialization techniques at this time in his career. While

he could spatialize the general location of sounds in a three-dimensional space, he either

could not make strong connections between specific starting and ending points of sounds,

or chose not to.

3.5.2. Trajectories and routes through space. Because of the precision and quantity of

our data, we can subject it to one last analysis. In sections 3.4.2 and 3.4.3, we established

that there are few, if any patterns in the shapes formed by the endpoints of bombs and

shots. Are there certain common trajectories that the shots themselves traverse as they

move through the space from floor to ceiling? To answer this question, we may consult

one last pair of graphic aids of this chapter, beginning with Example 3.5.2a. Here, the 3-

dimensional paths of shots 1-48 are shown in a two-dimensional representation.

Examining Example 3.5.2a in more detail reveals that there appear to be three main

groups of shots, based on their starting points on the floor. Group 1 begins from between

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-5 and -1 meters left of center-front-bottom. All of these shots end in the upper-right-rear

octant of the ceiling, with only two exceptions (shots 35 and 39). Group 2 includes all

shots starting from the center-front of the space and extending 2 meters to the right.

These shots end in more unpredictable places on the ceiling, but their endpoints tend to

land along a diagonal extending roughly from the upper-right-front corner to the upper-

left-rear (from loudspeakers VII to V). The third group, comprising shots starting

between 3 and 5 meters right of center, all land – without exception – in the upper-left

octant. Since the difference between shot beginnings and ends of Groups 1 and 3 is the

most significant, the shots in Group 2 can be interpreted as serving a mediating role

between these two contrasting extremes.

Highlighted in Example 3.5.2b are some shot trajectories that are significant for other

reasons. Shots 5 and 6, whose endpoints are both very distant from the centers of spatial

activity in either direction, also begin nearest the corners of the space on the floor. To

transform shot 5 into 6, we would need to translate the sixth shot 12.6 meters to the left

(towards the left wall, in the x axis), and rotate it by 44 degrees in the plane that the two

lines have in common.216 Clearly, these shots are very distantly related – a translation of

12.6 meters and a rotation of 44 degrees is large for this space. Because shots 5 and 6

begin on completely opposite ends of the cubic space, they may be interpreted as an aid

to “framing” the subsequent spatial activity in OKTOPHONIE.

On the other hand, several shots which have close endpoints also have close starting

points. In addition to the distantly related shots 5 and 6, two pairs or sets of shots are

highlighted in Example 3.5.2b. Shots 30 and 38 are very closely related. In order to

transform shot 30 into 38, we need only nudge it 0.2 meters to the left, and then rotate it

by about .7 degrees in the plane that the two shots have in common. Similarly, shot 20

transforms into shot 24 by moving only 0.1 meters to the right and rotating 1.8 degrees in

216 The angle of rotation was computed simply by trigonometry and is always in the plane that both linesare in. A transformation matrix could be derived from this relationship as well.

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their common plane. Although there are several other examples of shots that are closely

related to each other, there are also examples to the contrary. For instance, the three

shots that end at the center of the ceiling (shots 1, 31, and 33) all originate from very

different places on the floor.

Transforming one shot into another like this raises important questions about how

relevant translation versus rotation is in our ability to relate spatial shapes aurally. Are

two shots that are closely related by rotation, but distantly related by translation more

aurally similar to each other than two shots that begin near to each other but are not

closely rotationally related? Answering such questions would involve significant

discussion of perception which are unfortunately beyond the scope of the current study.

However, by measuring and quantifying the many different relationships among shots,

we have not only learned much about the way space is composed in OKTOPHONIE, but

also laid the groundwork for a far deeper understanding of the way spatial music behaves

on its own terms.

3.6. Conclusions

Computing the location of spatial events in Stockhausen’s OKTOPHONIE has allowed

us to uncover a significant amount of information about the structure of that domain. The

most important discovery we have made as a result of our study of the spatial

composition in OKTOPHONIE is the statistical distribution of shots, bombs and crashes

throughout the cubic space. Although the shots and bombs are not precisely coordinated,

we have conclusively established that regions of spatial activity on the floor correlate

loosely with regions directly above, on the ceiling.

Through analysis of the distances, angles and distances between shot ends, and between

bomb targets, we found only weak internal structure. Yet we can still see that there is

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some ordering process that affects the general statistical distribution of shots in the space.

If the spatial motion in OKTOPHONIE can be said to be “unified” in some way

according to one overriding principle – as so many other compositional elements in

Stockhausen are – we can say that the shots, bombs and crashes are spatialized within the

performance space of the cube. Since the motion takes place inside the cube, we have a

boundary between what is the piece and what is not. But the principal benefit of our

analysis is that we have been able to learn exactly how far Stockhausen’s compositional

control of the spatial domain extended. By showing how Stockhausen ordered the

general distribution of shots, bombs and crashes and leaving the very specific endpoints

and starting points more to chance, we have learned a great deal about this composer’s

compositional techniques in the spatial domain.

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Chapter 4.LICHTER-WASSER

4.1. LICHTER-WASSER and LICHT

4.1.1. Place of SUNDAY and LICHTER-WASSER within the LICHT project. The

Sunday opera was, chronologically, the final opera that Stockhausen composed in the

LICHT cycle. SUNDAY is “the day of mystical unity between Eve and Michael, out of

which the new life of Monday proceeds.”217 The opera itself is made up of six parts. The

first, LICHTER-WASSER, is given the dual appellation “Scene 1” and “Sunday-

Greeting”, and with its extensive spatial aspects, forms the main topic of this chapter.

Composed for 29 instrumentalists, two vocal soloists and a synthesizer, LICHTER-

WASSER contains the most complex spatial motion in the opera. The second scene,

ENGEL-PROZESSIONEN, is for a capella choir. This work also includes a fair amount

of spatial motion. The choir, divided into seven smaller ensembles, processes around the

hall in groups during the piece while singing in seven different languages. The third

scene, LICHT-BILDER, is scored for basset-horn, flute, tenor, and trumpet, with

synthesizer. The flute and trumpet are ring-modulated, but the other instruments are not.

Here, abstract images (“Bilder”) are projected on large panels behind the soloists, while

the instrumentalists move around the stage in stylized patterns which are notated in the

score.218 Scene 4 – called DÜFTE-ZEICHEN – is a work for seven singers, boy’s voice,

and synthesizer. Solos, duets, or trios are sung while different types of incense are

burned. HOCH-ZEITEN, which forms Scene 5, calls for five-part choir and five-part217 Texte 6, p. 156. “Sonntag ist der Tag der mystischen Verinigung Evas und Michaels, aus der das neue

Leben des Montag hervorgeht.” Also ibid., p. 175: “Und Sonntag ist der Tag der mystischenVereinigung von EVA und MICHAEL, der die Voraussetzung schafft für die Neugeburt des Montag.Und so ist LICHT ein Zyklus, der weder Anfang noch Ende hat.”...“Sunday is the day of mystical unionbetween EVE and MICHAEL, which serves as the prerequesite for the rebirth in Monday. Thus,LICHT is a cycle, which has neither beginning nor end.”

218 An crucial aspect of LICHT-BILDER is the text, which Stockhausen described as “venerations ofEVA-Maria” (Stockhausen 2005, p. 5). The naming of varieties of “spheres of life” -- such as stones,fruits, human saints, and star constellations – echos Goethe’s maxim no. 539: “ ‘Ich glaube einen Gott!’Dies ist ein schönes, löbliches Wort; aber Gott anerkennen, wo und wie er sich offenbare, das isteigentlich die Seligkeit auf Erden.” [“I believe in a God!” This is a beautiful, praiseworthy phrase; butto recognize God in all of His manifestations, this is the true holiness on earth]. <http://www.wissen-im-netz.info/literatur/goethe/maximen/1-09.htm#539>, accessed 10 January 2008.

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orchestra. While the five parts of each texture are simultaneously in different tempi, the

choir and the orchestra are designated locations in two different halls. The sounds of

choir and orchestra are linked together electronically so that listeners in each hall

occasionally hear a mixture of both.219 The final scene, SONNTAGS-ABSCHIED, is

essentially a wordless transcription for five synthesizer players of the choir portion of the

music of HOCH-ZEITEN.

All of these scenes have the Sunday limb of the Superformula as their formal background

structure. This limb, along with the entire Superformula, is shown in Example 3.1.1b.

But unlike the other LICHT operas, Stockhausen decided to omit part of the

Superformula in realizing the fundamental structure of SUNDAY.220 Since SUNDAY is

the “mystical union” of Michael and Eve, the Lucifer formula is not used as the

background structure in any of its scenes.221 Although this Lucifer material is not present

in the background level, it does play a part on a more surface level in the Sunday opera –

most memorably, perhaps, in the comic elements of the vocal solo “Ud (Samstag-Duft)”

in DÜFTE-ZEICHEN, which is derived from it.

4.1.2. Premise of LICHTER-WASSER. In English, LICHTER-WASSER means

“Lights-Waters”. According to Stockhausen, “life depends completely on light and

219 Stockhausen 2003b and 2004.220 The special Sunday-Superformula sketch, which can be found in the LICHTER-WASSER analysis

booklet pp. 40-41 (hereafter referred to as Stockhausen 2001b), shows six polyphonic lines. The threeon top are the projection of only the Sunday limb of the Superformula over all of the Sunday opera,while the three lower lines are the projection of the entire Superformula over the Sunday opera.Although the Lucifer formula is present in this sketch, it is clearly intended to be omitted in the actualrealization of the Sunday works. Stockhausen’s notes indicate that the work which was to involve theLucifer formula – LUZIFERIUM – would have been performed in a prison, or underground.LUZIFERIUM would have been separated spatially from the other parts of the opera. This technique isreminiscent of HOCH-ZEITEN and the HELIKOPTER STREICHQUARTETT (of WEDNESDAY),where different parts of the ensemble are not in the sample physical location. This conscious separationis yet another facet of Stockhausen’s interest in spatial relationships over large distances.

221 This Lucifer limb remains the only part of the Superformula which was not elaborated upon inStockhausen’s LICHT project. Originally Stockhausen planned a work called LUZIFERIUM thatwould have been based on this material, but the composition never materialized. For more on theplanned composition LUZIFERIUM, see the LICHTER-WASSER lectures, day 1.

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water”;222 thus, the work is a kind of celebration of life on Earth and – presumably –

extraterrestrial life. Stockhausen wanted “to compose a work about our solar system”223

and spent a considerable amount of time learning about the different aspects of it in order

to prepare for composing LICHTER-WASSER.

In LICHTER-WASSER, “the rotations of the notes in space are related to the rotations of

the nine planets and 61 moons of our solar system, whose names, astronomical

characteristics and significances are sung”.224 Although Stockhausen later gave up on

precisely coordinating the rotations of heavenly bodies with his music, LICHTER-

WASSER is still “closely related to our solar system, the planets, moons with their

rotations, [and] timings”.225 In the piece, instrumentalists are spread around the hall in a

geometric pattern. During the central sections of LICHTER-WASSER, a note (or group

of notes) in a melody is played by one instrumentalist, who then “hands off” the melody

to another instrumentalist. At the same time, two vocal soloists move around the space at

a slower rate of motion. While the actual speeds of planets, moons and other elements of

the solar system as they cycle around the sun are not literally reflected in the music,

Stockhausen imagined that there was nevertheless an indirect relationship.226 Its premise

being a kind of “artificial” solar system of music, LICHTER-WASSER is a work of

almost ritualistic meditation in which very little “dramatic action” happens. The text of

the work reflects this meditative “inactivity”; the words mostly describe or state the

physical properties of different elements of the local solar system.

Stockhausen described LICHTER-WASSER as the “Sunday Greeting”. Stockhausen’s

tradition of starting an opera off with a “Greeting” dates back to the first opera in the

222 LICHTER-WASSER lectures, 2001, day 1.223 Ibid.224 Stockhausen 2001b, p. 5. Of course, at the time Stockhausen composed LICHTER-WASSER, Pluto

was still a planet. In a personal conversation that took place in December 2006, Stockhausenmentioned that he felt it was a mistake to demote Pluto: “it will only confuse people”.

225 Stockhausen 2004b, p. 4.226 Many works by Stockhausen reflect this connection between music and the cosmos, perhaps most

notably SIRIUS. For more on this, see Peters 2003, pp. 233ff.

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cycle, Thursday. “Greetings” often take place in a performance space other than the main

space for the hall (such as a foyer), and vary considerably in length.227 Their tonal

material is generally derived transparently from the Superformula. In this way, the

audience is reminded of the basic material which is then expanded (or “projected”) over

the course of the ensuing opera. Compared with the other Greetings, LICHTER-

WASSER is not very unusual in terms of length, but it is more complex musically than

most of the others. LICHTER-WASSER is most notable because it is the only Greeting

that is also designated “Scene 1”.228

4.1.3. Formal Structure of LICHTER-WASSER. The basic pitch material of

LICHTER-WASSER is derived from the Michael and Eve nuclear formulas, which are

shown in Example 3.1.1a. These nuclear formulas can form the pitch material for

Greetings in other operas. Over the course of LICHTER-WASSER, characteristic

intervals from the Eve formula gradually “migrate” to the Michael formula. Curiously,

Stockhausen did not indicate in his sketches a reverse process, where intervals from the

Michael formula migrate to the Eve formula, except in the sense that some characteristic

intervals from the Michael formula are used as auxiliary decorations in the Eve formulas.

This technique is a way of “blending” one formula into another, which musically

expresses the “mystical union” between the two characters. The complete sequence of

twelve pitch blendings is shown in Example 4.1.3a.229 During LICHTER-WASSER,

each of these pitches in the nuclear formula is itself treated as a pitch center, and is

embellished or expanded in various ways.

227 Although the Thursday greeting is relatively short, the Friday greeting is as long as the entire first actof the opera.

228 The very idea of a Greeting to begin with has a connection with Christian religious services, where thecelebrant often greets the congregation (an “introit”) before the ceremony proper begins.

229 In the case of the Eve formulas, Stockhausen transposed the formula in his sketches but the“blendings” are limited to embellishments around formula notes. The composer employed an electronicversion of this technique in DER KINDERFÄNGER, in the MONDAY opera. Stockhausen 2002, p.10. Other uses of the technique are in MONDEVA (THURSDAY opera, act 1 scene 2), and theMISSION UND HIMMELFAHRT section of MICHAELS REISE (Act 2 of the same opera).

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In contrast to OKTOPHONIE, which was made of two roughly equal parts – but then

subdivided into several unequal smaller sections – the formal structure of LICHTER-

WASSER is based on a framing structure which surrounds a lengthy central structural

complex. The tenor and soprano soloists begin LICHTER-WASSER with a brief

“Anfangs-Duett”. This duet is followed by the entrance (“Eingang”) of the 29

instrumentalists, who slowly move to preassigned positions either along the walls of the

performance space or in the aisles. Each instrumentalist belongs either to the Eve-

orchestra or to the Michael-orchestra. Orchestra membership is indicated by a lit candle

inside a green or blue glass bowl, which is placed near the musician. Near each musician

is also a glass of water. The Eve-Orchestra tends to include instruments of a low range

complementing the soprano singer, whereas the Michael-Orchestra contains instruments

of higher range, contrasting with the lower range of the tenor singer. There are seventeen

musicians in the Michael orchestra, each corresponding to a note of the full (embellished)

Michael nuclear formula. Twelve instrumentalists make up the Eve-orchestra,

corresponding exactly to the notes of the Eve nuclear formula. The arrangement of

musicians in the space is shown in Example 4.1.3b.230 In addition, Stockhausen

sequenced the instrumentalists in both orchestras; we will refer to these paths as “basic

cycles” in the analytical sections below. The audience sits in the triangular spaces

between aisles, all facing towards center.

Following the Anfangs-Duett and Eingang, which serve expository functions in the

opening “frame”, the central sequence of interlocking “waves” and “bridges” begins.

During these sections, sequences of notes and segments of motives from the embellished

nuclear formulas get passed around instrumentalists in various shapes and at varying

speeds. In the twelve waves, the “durations of the two formulas are gradually enlivened,

from undivided durations of the formula notes in the first wave to maximally subdivided

230 Following the terminology we defined in chapter 2, we can say that the intersection of the Eveorchestra and the Michael orchestra is the empty set.

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durations in the twelfth wave”.231 In the following discussion, these twelve Michael and

twelve Eve wave sections will be abbreviated as “M1” or “E7”, etc.

Corresponding Michael- and Eve-wave sections are not always of the same duration; they

only align six times. Where they do align, a bridge section gets inserted. The spatial

motion is less active during most of the bridge sections. The pitches used in the bridge

sections usually are the nuclear formulas that will be heard in the next block of waves;

thus, bridges serve a dual function both as a contrast to the more active spatial motion of

the wave sections, and as an exposition of the pitch material that will occur in the next

“block” of waves. One bridge – the fourth – is repeated and interleaved between a series

of three “announcements” in which the musical texture becomes very much like a vocal

recitative.

At the end of the twelfth and final wave, the instrumentalists ritually process out of the

hall in a section appropriately named “Ausgang”. Before leaving, the musicians take a

drink of water from the glass set next to them. Finally, the two vocal soloists sing a

modified reprise of the opening “Anfangs-Duett”, simply called “Schluss-Duett”. In this

way, the central structural complex – comprised of alternating sets of waves with bridges

– is neatly sandwiched between vocal and instrumental “entrances” and “exits”. The

entire structure of LICHTER-WASSER is shown in Example 4.1.3c. The example also

clarifies the different pitched nuclear formulas that are used in each section.

4.1.4. Analytical and critical literature on LICHTER-WASSER. There is very little

analytical or critical literature on LICHTER-WASSER. The most extensive analysis of

the piece is by Stockhausen himself, and consists of the analysis booklet published for the

2001 Stockhausen Courses in Kürten. This booklet reproduces several important

231 Stockhausen 2001b, p. 5. Stockhausen made elaborate calculations relating to the number of notes ineach wave section during the planning stages of LICHTER-WASSER. These predictions, which can beseen in a somewhat cryptic but important sketch, can be seen in Stockhausen 2001b, p. 25. In mostcases, Stockhausen’s predictions come fairly close to the ultimate realization in the score.

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sketches for the work, and explains most of the essentual structural elements from the

composer’s perspective. Stockhausen’s analysis booklet is an indispensable artifact in

that it contains information that is crucial to forming a basic understanding of the premise

and methods of the work. A second important body of source material is the series of

seven lectures Stockhausen gave in 2001. The present author made transcriptions of

these lectures, which also aided in developing a sense of the work’s scope and structure.

Robin Maconie’s commentary on the piece is useful in that it places LICHTER-

WASSER in the context of Stockhausen’s other works. Aside from noting the obvious

similarity of LICHTER-WASSER to Xenakis’s Terretektorh (which we will investigate

more closely in Chapter 5), Maconie notes that there are certain “tonal implications” and

passages of “undisguised tonality” that seem to “resolve the dramatic and philosophical

contradictions of movement and cadence”.232 The work is “infused with a sense of

discretion and tact” and “iridescent color”. Stockhausen “returns to a sense of spiritual

purity...in which the spatiality of Brant and the lyricism of Webern are reconciled”.233

LICHTER-WASSER was commissioned for performance by the Southwest German

Radio Orchestra (SWR) for the Donaueschingen festival. It required extensive rehearsal

time and preparation;234 as of this writing, the orchestral version has not been performed

at all after the premiere. Although an eight-channel tape exists, its effect is not as

interesting spatially as the live performance, since loudspeakers are set up around the

audience in a ring.235 Consequently, many of the spatial effects which rely on

instrumentalists stationed in the aisles of the performance space are lost. The two-

232 Maconie 2005, p. 532.233 Ibid., p. 531.234 For the premiere, Stockhausen rehearsed two weeks with the orchestra and two weeks with the singers.

Three combined rehearsals followed before the premiere. LICHTER-WASSER lectures, 2001, day 4.235 Stockhausen described how he mixed down the 29-track recording of the musicians to make an 8-

channel tape in Stockhausen 2001b, p. 10. Stockhausen wrote that “when performed in this way, it isabsolutely necessary to inform the audience in the program book that the movements of the notes frominstrument to instrument as compared to a performance with orchestra are greatly reduced, butsimulated to a certain extent” [bold in original].

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channel stereo recording, available on CD from the Stockhausen-Verlag, does even less

justice to the work’s complex spatial language. LICHTER-WASSER does not consist of

smaller, self-contained works that can be split up and performed individually with less

substantial musical forces. Ultimately, the very aspect of LICHTER-WASSER that

makes it a stimulating piece in performance – that is, its spatialization – is difficult if not

impossible to capture in reproduction. This lends the work an certain aspect of “musical

theater” which is common in many other Stockhausen pieces. All of these factors

contribute to the difficulty of mounting an effective performance.

4.2. What elements of the score of LICHTER-WASSER pertain to

spatial movement?

4.2.1. Sketches for LICHTER-WASSER. The sketches for LICHTER-WASSER

consist of 153 pages of material. The development of the work through many stages of

composition can be fairly well understood through studying the material available at the

Stockhausen archive.

Concerning the spatial arrangement of the musicians, Stockhausen imagined many

different possibilities before settling on the final configuration. These sketches show that

he arrived at the general spatial layout of the 29 musicians236 fairly early on, but changed

the instrumentation several times. In the early sketches, musicians were moved around to

different locations several times. One very early sketch shows that Stockhausen briefly

conceived of the idea of putting similar instruments on opposite walls of the hall.237

Another shows that he considered using a tape with sampled sounds of “tennis shots”,

“baseball hits”, “billiard shots”, and different kinds of motion of spheres.238

236 This prime number is also the number of piano chords in the color of the first movement of Messiaen’sQuartet for the End of Time.

237 Lichter-Wasser sketch book, sketch 50 (dated 27 June 1996), middle-top sketch.238 Lichter-Wasser sketch book, sketch 51 (dated 18 July 1996).

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Once Stockhausen finally settled on the arrangement of musicians and the orchestration,

he began to draw shapes coursing through the space. Many of these shapes are

symmetrical, such as stars, circles, polygons, and lines. This sketch is shown in Example

4.2.1a.239 Initially, Stockhausen grouped his shapes according to the number of

musicians, or (in graph-theoretical language) vertices, that they involved. He then

created a series of decisive sketches in colored pencil showing the planned motion

through each of the twelve waves and bridges. These sketches, too numerous to

reproduce here, are in the 2001 analysis booklet. Movements for the vocalists appear as

well, on separate sketches. These motions were probably finalized after Stockhausen

determined the pitch and rhythmic detail, since he needed to know how many moves

there would be before expanding the nuclear formulas in each wave.

The compositional process Stockhausen went through, from deciding on the arrangement

of musicians, to composing the shapes, to integrating the shapes into the music, is

unusually well-documented and highly interesting in and of itself. Further study of this

compositional process could reveal important principles relating to Stockhausen’s spatial

preferences, as well as other practical necessities which spatial music must engage.

4.2.2. Revisions of spatial motion during rehearsals. During the rehearsals of the

work, Stockhausen made a considerable number of revisions and changes to the spatial

motion of melodies in LICHTER-WASSER. Most of these seem to be practical in

nature. Concerning an instrumental substitution made to the very first note in E2,

Stockhausen observed that the note in question was originally intended to be played by

the second trombone. In rehearsal, the second trombone could not play the note, so he

asked the first trombone to take over.240 These revisions and many other changes are

239 Reproduced from Stockhausen 2001b, p. 21.240 Stockhausen explained this on day 4 of his 2001 LICHTER-WASSER lectures. This change occurs in

bar 152. The original sketch, with the second trombone part indicated, is in Stockhausen 2001b, p. 28,top-right. Stockhausen did not explain why the trombonist could not play the note.

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detailed in a series of six pages of notes in the sketch-book.241

4.2.3. The final score to LICHTER-WASSER. Considering the complex process

Stockhausen went through in developing the spatial arrangement and distribution of

instruments, and the large quantity of revisions he made during the rehearsals, there are

often contradictions between the sketches and the final score. While considering all the

different implications of Stockhausen’s alterations would certainly shed much light on

the composer’s idiosyncratic method of composing, we must settle on one version of the

score to analyze here.

The present analysis is based on the final published score.242 It is drawn carefully in

Stockhausen’s own hand, and is generally very clear and unambiguous. The instrumental

movements are indicated above each note with an abbreviation for each instrument. In

addition, box diagrams for both the instrumentalists and vocalists litter the score. While

the box diagrams are an important aid for the conductor, they are also extremely helpful

for analysis because they clearly show the direction and shape of the movement. Since

Stockhausen incorporated all his revisions from the lengthy rehearsals and first

performances into this score, we can consider it to be a highly accurate representation of

the composer’s intentions.

4.3. Shape of the space in LICHTER-WASSER

4.3.1. Normal spatial conditions - Instrumentalists. The instrumentalists begin to play

during the Eingang section. As they process into the performance hall, each takes an

assigned place. Once reaching their places, each plays a single note of either the first

Michael nuclear formula or the first Eve nuclear formula, depending on the orchestra they

are in. The notes are irregularly repeated in a manner that Stockhausen commonly used,

241 These sketches are numbered pages 147-152.242Stockhausen 1998-1999.

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which he described in the score as somewhat akin to “morse code”. This continues until

all 29 instrumentalists are in their places.

Throughout the wave sections, instrumentalists do not move from their places. Under

normal conditions, two melodies – each based on their progressively “cross-influenced”

nuclear formulas – continually cycle around the space. Usually there is one single “trail”

through the space for each orchestra. Since there are two orchestras, the prevalent

musical texture is like a two-part spatial counterpoint – in other words, an “invention” in

two parts. The entire work LICHTER-WASSER could be thought of as a sequence of

“twelve two-part spatial inventions” – with six bridges sandwiched in between.

During the bridge sections, the normal spatial activity among instrumentalists is quite

different and generally more diverse than the wave sections. In the first three bridges,

instrumental solos are fairly prominent, even if there is still something of a spatial

“hangover” of rapid motion from the previous waves. During the fourth bridge, twelve

woodwind and brass instrumentalists move to the balcony and the entire ensemble of 29

instrumentalists is divided into two “quartets”. The eight polyphonic lines in these two

quartets contain instruments situated both on the floor and in the balcony, allowing for a

complex spatial interplay. In addition, the fourth bridge includes several brief

instrumental solos and ensembles. The fifth and sixth bridges feature more short

instrumental solos counterpointed against a fairly stable homophonic texture. In general,

the spatial activity in all six bridge sections is considerably less complex than in the

waves. The bridges are like islands of calm within the more frenetic spatial activity in

the surrounding waves.

In the Ausgang section, the instrumentalists resume their “morse-code” style. At a

determined point, given in the score, each pauses for a moment to take a drink of water.

The instrumentalists then move, one by one, towards the exit. As they exit, they continue

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playing until they are all outside the hall. Since each instrumentalist starts at a different

location and proceeds to the same door, the spatial movement can be represented as a tree

graph. Throughout the wave sections and most of the bridge sections, music moves

disjunctly from one instrumentalists to another. But given Stockhausen’s idea that the

spatial motion should represent the planets, moons and other objects in the solar system,

we can conclude that even though the shape of the space in the instrumental music is

normally discontinuous, it is meant to be perceived as continuous. The one time when

the motion is actually continuous occurs in the Ausgang section, where musicians move

towards the exit door while playing.

4.3.2. Normal spatial conditions – Vocalists. There are several significant differences

between the normal spatial motion of the two vocal soloists and the typical “implied”

continuous motion of the instrumentalists. First, the vocalists’ motion is altogether much

slower than the instrumentalists, since they must actually walk around the space instead

of “handing off” a melody to a nearby musician. Moreover, the vocalists cannot “move

through” the audience; they can only walk around the edges of the hall and through the

aisles. The vocalists’ spatial motion can be represented as a graph with considerably

fewer edges than the instrumental graph.

During the Anfangs-Duett, the tenor and soprano sing on opposite sides of the hall, facing

each other along the horizontal aisle. They begin to move around the hall during the

Eingang section, where each shows the members of the other orchestra to their places and

either “personally lights the musician’s... lamp...or gives a sign to the musician to light

the lamp himself”.243 In addition, the vocalists sing musical fragments as they gradually

“usher” in the instrumentalists.

The two vocalists sing from a different spatial position during each wave and bridge

section. In the course of a wave or bridge section, they remain stationary until near the

243 Stockhausen 2001b, p. 7.

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end. The precise moment where they should begin to move to the position for the next

wave or bridge is indicated in the score. Stockhausen timed how long it took each singer

to move from place to place. Based on these timings he suggested when movement

should begin in preparation for the next wave or bridge.244

During the Ausgang section, the singers perform more of a ritualistic role than a musical

one. They walk around the space, bidding each instrumentalist to drink from their glass

of water. After the musicians have left the hall, the vocalists move into position for the

Schluss-Duett. They perform the closing duet facing each other along the same

horizontal aisle as the Anfangs-Duett. However, the singers are considerably closer to

each other than they were in the beginning. The way in which the vocalists are closer to

each other than in the beginning is yet another subtle way of expressing the idea

“mystical unity” of Michael and Eve through the spatial design of LICHTER-WASSER.

4.3.3. Exceptions in spatial movement. In LICHTER-WASSER, there are several

exceptions to the usual type of spatial motion outlined above. Detailing the irregularities

is important because we should construct an analytical methodology that does its best to

take as many events as possible into account, but at the same time does not lose sight of

the most important elements of spatial design.

During the wave sections, there are occasionally instrumental solos at the same time as

one of the two melodies continues to cycle around the room. One of these solos occurs

towards the end of M1. The first violist plays several notes of the pitched melody while

the principal melody continues to cycle around the space. This passage, beginning at

measure 141, is shown in Example 4.3.3a. Another type of solo occurs when a single

instrument plays notes that are not doubled by the instruments that continue the main

spatial path. An example is the E-flat clarinet solo in M7. Beginning in measure 398, the

244 The fact that these timings are so precisely indicated again suggests constraints on the size of the hallin which LICHTER-WASSER is to be performed.

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E-flat clarinet articulates pitches that are more related to the tenor soloist than

instrumental material. This passage is shown in Example 4.3.3b.

A group of instruments may momentarily play the same melodic line in a kind of spatial

canon. An example of this can be found at bar 252, and is shown in Example 4.3.3c. In

this case, the structure of the spatial movement really is a three-part canon, as there is

only one path traversed through the space. Another example of a three-part spatial canon

begins in measure 337, and is shown in Example 4.3.3d.

While the two melodies cycle around, groups of musicians occasionally perform musical

material in unison, especially after the third bridge section. Stockhausen imagined some

of these passages, which take the form of rapid chromatic scales, as “comets” in his solar

system analogy. A “comet” occurs in measure 325, and is shown in Example 4.3.3e.

Most of the comets are brief, whirlwind occurrences which may be played with irregular

rhythm or pitch (as in mm. 351-352, mm. 386-387).

The most consistent exception in the general style of spatial movement occurs in the

twelfth (final) wave section. This wave is punctuated by seventy-two chords of between

three and seven pitches. However, there are often doublings in the chords, so that even

though there are only seven pitches, they may be played by up to twelve instrumentalists.

Often, the string instruments play the inner notes of the chords pizzicato, while wind

instruments play the lowest and highest notes. Although the spatial melodies continue

throughout the twelfth wave in both orchestras, the addition of the chords adds an

element of spatial complexity towards the end of this already highly intricate work.

4.3.4. Analytical methodology in LICHTER-WASSER. Although we have explored

several exceptions to the normal spatial conditions, they are not so many nor so great that

we cannot develop a simple, coherent and systematic approach to analysis. The most

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useful way to measure spatial motion in LICHTER-WASSER is simply to filter out the

most unusual or exceptional events, and focus on the two continually moving spatial

paths. There are several reasons to believe that the most important spatial structure is

expressed in these two counterpointed lines, heard during the twelve wave sections.

First, Stockhausen’s sketches clearly show that these two spatial paths are the most basic

motions in the work.245 Second, two spatial paths are consistently notated throughout the

score, and the greatest number of box diagrams relate to them.246 Third, Stockhausen

decided on the number of moves in each wave section early on in the composition

process, indicating that consecutive movement is part of the more fundamental

compositional structure.247 Exceptional events, such as instrumental solos and canons,

were added later to enliven the basic two-part spatial texture.

In sections where one instrument plays a short solo while the instrumental motion

continues, we will simply continue analyzing the more active spatial motion. The solo

instrument will be considered irrelevant after the principal melody moves to a different

location because it usually either doubles the music cycling around, or plays along with a

singer. In passages that are in “spatial canon”, we will simply follow the leading canonic

line (“dux”). When groups of musicians play “comets”, we will ignore them and

continue to focus on the unbroken spatial motion among other instrumentalists.

The greatest difficulty in crafting an analytical methodology in LICHTER-WASSER is

encountered in the twelfth wave section. The large number of chords which interrupt the

spatial motion breaks up the continuity in the previous eleven wave sections. However,

Stockhausen circled one particular instrument that is either the lowest- or highest-

245 The diagrams Stockhausen made, reproduced in Stockhausen 2001b, pp. 27-33, show the greatimportance he attached to these two principal spatial pathways.

246 There are very few box diagrams for instrumental motion during the bridge sections, none for the“comets”, and very few for the other exceptional spatial elements.

247 The crucial structural sketch of LICHTER-WASSER that shows this can be found in the Stockhausen2001b, p. 25. My own analysis has shown that Stockhausen’s estimates of the number of movements hewould need for each wave structure come quite close to the number present in the final score.

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sounding voice in most of the chords. These are the instruments we include in our spatial

trail. Even so, we will analyze the chords on their own terms in this exceptional section.

The result of our methodology is to find a single strand or trail of instrumental motion

through each wave section. The sequence of 3856 moves in the twelve wave sections

used for analysis is shown in Examples 4.3.4a-e. In conjunction with sequencing the

moves during the wave sections, calculations were made which associate the exact

number of seconds that the spatial melody lasts at each instrument. This meant that the

duration of every note or group of notes played by each instrument was determined,

taking into account the current tempo and rhythmic value. Because of their less

quantifiable quality, fermatas, ritards and accelerandos were not taken into account.

First, durations in seconds were computed for all the different rhythmic values in

LICHTER-WASSER, in each of the 22 tempos used from quarter = 35.5 to quarter =

120.0. Then, these durations were applied to the score. If an instrument played several

notes, or had a note followed by rest, all of the individual durations were summed to

determine the total amount of time that the spatial melody lingered at each instrumental

location.

The bridge sections present some aspects of spatial motion that complement or contrast

with the wave sections. However, there are few box diagrams in the bridges, indicating

that the motion that takes place in them is probably less important than in the waves.248

Even so, we can look for the most active spatial lines in some of the bridge sections to

analyze. In comparison to the motion in the wave sections, there is very little spatial

motion in the bridges that behaves as single paths. The sequence of instruments in the

fifth and sixth bridges, where the motion is most consistent, is shown in Example 4.3.4f.

Finally, the coordinates of each instrument in a 30 × 27 meter space were estimated. This

was the size of the space used in the premiere of LICHTER-WASSER and probably

248 The box diagram in the first bridge at measure 148 is an exception.

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cannot vary too much.249 The coordinates used for analysis are shown in Example 4.3.4g.

This example also shows an important structure in the arrangement of instruments. If one

compares the location of Eve- and Michael instruments, it is obvious that there is an axis

of symmetry in the space along the horizontal aisle, which is shown with an arrow. The

repercussions of this axis will be explored further in the following analysis.

Having determined the sequence of events in LICHTER-WASSER, the amount of time

each of those events takes, and each event’s location, we can feed the data into a database

and create computer programs that calculate information about the spatial motion.

4.4. Instrumental Motion

4.4.1. Speed of Spatial Motion. By relating the sequence of events, their duration, and

their location, it is possible to create the detailed graph shown in Examples 4.4.1a-b.

This graph shows the changing speed of the two melodies which move around the space

throughout the course of the entire composition. Average values for the speed in each

section (wave, bridge) are shown in the table in Example 4.4.1c. The graph and chart

contain a great amount of information which we shall refer to in the sections below.

4.4.2. Anfangs-Duett and Eingang. Apart from the synthesizer, whose sound comes

from each of the speakers in the four corners of the room and is meant to be stationary,

there is no instrumental music to accompany the singers’ Anfangs-Duett. Once the duet

is over, the instrumentalists enter the space from the door. The order of instrumental

entry is the same as the sequence of the basic M- and E-cycles shown earlier in Example

4.1.3b. Since the musicians enter silently, and only start to play their repeated note when

249 Stockhausen wrote that there was enough room for 728 seats in the auditorium (Stockhausen 2001b, p.6). A space much smaller would make it impossible to fit as large an audience, while a larger spacewould make it much more difficult for the musicians to synchronize their melodic hand-offs, whichoften occur quite rapidly. A larger space would also make it impractical for the vocal soloists to moveabout in a reasonable amount of time.

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they reach their place, there is no instrumental movement in this part of LICHTER-

WASSER either. However, it is significant that the order in which the instrumentalists

enter and take their places is the same as the two basic cycles because this helps to

reinforce the basic cycle sequence. Variants of the M- and E- basic cycle will be heard

several times in different variations during the following wave sections, so the entrance

of the instrumentalists functions as a silent spatial exposition.

4.4.3. First block: M1 and E1. The first wave sections in LICHTER-WASSER exhibit

spatial motion that is similar to the basic M-cycle and basic E-cycle shown in Example

4.1.3b. For reference, these two basic cycles are traced in Example 4.4.3a. The most

common motions through the orchestra during M1 and E1 are shown in Examples 4.4.3b.

In this and most of the diagrams of individual wave sections that follow, only motions

between instruments that happen two times or more are shown. Although they may be

significant on a local level, moves that happen once are not shown unless otherwise

stated, since these infrequent moves are less important within the large-scale structure of

spatial movement.

During M1 and E1, the general scheme is that melodies in each orchestra usually move

only among instruments in their own orchestra. The spatial motions, which closely

resemble the basic cycles, allow the membership of the two orchestras to be clearly

defined in a secondary spatial exposition.250 Consulting the part of Example 4.4.1a that

shows the fluctuations in speed of M1 and E1, we find that in general the speed of the

melodies as they cycle around the hall is relatively slow, but still quite lively. The M-

orchestra melody moves at an average rate of 9.4 m/sec while the Eve-orchestra melody

250 We may consider the importance Stockhausen attached to defining the two basic cycles of spatialmovement as similar to the importance that classical twelve-tone composers such as Schoenbergattached to defining the basic row. By specifying both that musicians enter according to the basic cyclesequence, and then reinforcing these cycles of movement in the M1 and E1 waves, Stockhausen createda basic set of expectations that may then later be transformed or modified in a way that engendersmore meaning for the listener.

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moves at 8.89 m/sec.251 The only wave with slower rate of movement is M4, where the

melody moves at about the same speed – 9.4 m/sec. According to one of Stockhausen’s

spatial principles, quoted at the very beginning of Chapter 1, we might hypothesize that a

slower rate of movement would allow for greater variety and complexity of spatial

movement. In the M1 and E1 sections, movement is slow and predictable, further

suggesting that Stockhausen thought of the function of these sections as expository.

In M1 and E1, the M-orchestra’s melody moves at a slightly faster speed than the E-

orchestra’s melody. However, the M-orchestra’s melody, if stretched out in one long

line, would be slightly shorter than the Eve-orchestra’s. We would expect that a melody

that moves further over the course of the same time would also move faster. How is it

that the M-orchestra’s melody moves faster? The reason has to do with the length of the

nuclear formulas. Because the M-nuclear formula used in LICHTER-WASSER has 17

notes instead of 12, Stockhausen must move the M-orchestra notes at a slightly faster rate

than the E-orchestra notes in order to get through the entire nuclear formula in the same

amount of time.

The overall shape of the motion in M1 tends to be around two concentric circles, while

the motions in E1 generally trace out two large circles that are adjacent to each other.

The shape of the movement in these waves, as well as the speed, are crafted so as to give

each orchestra differentiable spatial characteristics, even though the complex

arrangement of instruments causes the spatial shapes to interweave such that it can really

be said that a spatial counterpoint obtains.252

251 The speed of 9.4 m/sec corresponds to 21 miles per hour, while 8.89 m/sec corresponds to about 20miles per hour.

252 On day 3 of his 2001 LICHTER-WASSER lectures, Stockhausen said that in these two waves, thetendency of motion is to move from the edges to the center, and then back out to the edges. “It is like abreathing wave.” Our observations confirm this general observation. The idea of cyclically expandingand contracting is a trope in Stockhausen’s work which comes to the surface most notably in ATMENGIBT DAS LEBEN of 1974-77 (Maconie 2005, pp. 360-363). A very memorable and lengthy passageat the end of HYMNEN also reflects Stockhausen’s continuing interest in the sounds processes ofbreathing.

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4.4.4. 1st Bridge, Second Block M2-3 and E2. As discussed above, the spatial motion

in the bridge sections is generally less active than in the waves. The first bridge is no

exception. In relation to the preceding waves, the first bridge is quite short – only seven

measures long. In that time, a spatialized melody weaves its way around the M-orchestra

but it is partially obscured by the instrumental solos that predominate in this section. The

real function of the bridge sections, as stated above, is to introduce the nuclear formulas

for the next wave sections; however, only the M2NF (second Michael nuclear formula)

and E2NF are introduced here even though the next block of waves includes M3 as well.

Example 4.4.4a shows the most common spatial motions in the next block of waves,

which includes M2, M3, and E2. As we can see from the incomplete graph structure in

this example, the motions in these sections tend to be less frequently repeated as in M1

and E1. Many of the connections between parts of the space are made only once. Some

moves, such as f2–k, k–f2, and eh–f1 are explored that were seldom articulated in M1.253

The spatial movement in M3 is even more fragmentary than in previous wave sections.

As can be seen from his sketches, Stockhausen designed the movement in this wave to

resemble a kind of coiled-up shape similar to the letter “M”. Because of this, motion

tends to start near the conductor, moving away and then back again, traversing the space

along the vertical aisles. M3 is the first wave to include spatial movement that is

significantly different from the basic cycle that predominates in M1 and M2.

E2 includes a particular use of the instrumentation that seems unusual at first, but

becomes extremely important in LICHTER-WASSER. Although the spatial motions

kept mostly within their respective orchestras in M1 and E1, this is not entirely the case

in E2. Stockhausen designed the motion to cross over on itself through a relatively

frequent diagonal motion which traverses the instruments fa1–sax–va1–fa2–th–va4. Two

253 For an explanation of the instrumental abbreviations used here and elsewhere in the followingdiscussion, see Example 4.3.4g.

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of these instruments—va1 and th—are part of the M-orchestra, yet here they play a more

important role in the E-orchestra. These instruments are circled in Example 4.4.4a.

Stockhausen clearly felt that the importance of articulating a spatial shape (here, the

diagonal) justified the use of these instruments in the “wrong” orchestra.

We find that in these wave sections, the rate of spatial motion has sped up, especially in

M2 and M3. In particular there is a real burst of speed about two-thirds of the way

through M2. As the shape of motion begins to develop and less resembles the basic

cycles, the speed of the motion increases too, posing additional challenges to the

instrumentalists and the listeners. As we become more accustomed to the spatial motions

in LICHTER-WASSER, Stockhausen composed new and different structures to listen for

as the piece progresses.

4.4.5. 2nd Bridge, Third Block: M4-5 and E3-4. While the second bridge is

considerably longer than the first, the spatial activity is less active. The bridge is

dominated by instrumental solos accompanied by a homophonic three-part string texture.

We hear three nuclear formulas introduced here – M4NF, M5NF and E3NF.

The moves that occur more than twice in these waves are shown in Example 4.4.5a.

Stockhausen described the motion of M4 as a spiral that proceeded from the edges of the

space and in towards the middle.254 In fact, we can see the faint outlines of a spiral in our

diagram of M4, but most of the moves that define this shape do not occur frequently.

The average length of a move in M4 is the shortest of any wave. This would lead us to

predict that the average speed of motion is slower, since the melody must move through

so many instruments. In fact, the average speed of motion in M4 is considerably slower

than M2 or M3. This conflicts with Stockhausen’s plan that the speed of motion should

increase from wave to wave. Despite the “spiral” design, the most dense spatial activity

is clustered around the edge of the space in M4. Two instruments—va3 and va1—serve

254 See the 2001 LICHTER-WASSER lectures, day 4.

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as bridges to help pass the spiral motion from the edges towards the center of the space.

The instrument in the center of the space, fa2, plays an important role in this M-wave

even though it is in the E-orchestra. Since this instrument is used in the “wrong”

orchestra, it is circled. Although there is some basis for concluding then that the motion

in M4 does indeed outline a spiral, it seems to be only faintly defined.

The spatial structure of E3 is extremely tenuous since almost no moves occur more than

once. This is not only because E3 is a short wave, but also because it includes by far the

fewest number of moves of any wave. Because of this, moves on the graph in Example

4.4.5a are shown even if they occur only once. These infrequent moves are shown in

dotted lines. Stockhausen’s spatial strategy here seems to be to reinforce the basic cycle

but to concentrate spatial activity at certain instruments in remote parts of the hall.

Almost 50% of the spatial motion passes through six instruments h1, p1, fa2, h2, eu and

fa1, while the remaining twelve instruments involved in E3 are used more infrequently.

Like M4 and M5, the average speed of the motion in E3 is actually slower than previous

E-waves.

M5, like E3, is short in duration and includes the fewest movements of any M-wave.

Like M4, the average speed of motion in M5 is slower than M2 or M3. As in E3, we

show all spatial moves in M5, since most of them happen only once. The activity in M5

has a remarkable amount of variety for such a short wave: all instruments are used at least

once except for Eu and Tu. Here, orchestral membership is clearly blurred. Although the

average length of a move is quite short in M5 – 8.29 meters – it is by no means the wave

with the shortest moves among instrumentalists. Still, we find evidence to support the

conclusion that in M5, Stockhausen made an effort to compose a wide variety of non-

repetitive, short moves from instrument to instrument.

The E4 wave emphasizes motion in the two right-hand quadrants of the space. The

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instruments fa2–h2–eu–tu–p2–fa2 and fa2–va2–p2–sax–fa1–p1–fa2 are used frequently

in this wave. While these cycles are not literally traversed sequentially, the denser

motion that is centered around them is related by rotation around the center instrument

fa2. The M-instrument involved in these structures—va2—plays a vital role in E4

because of the amount of motion to and from it. In comparison with previous E-waves,

E4 has the fastest average speed – 11.60 m/sec.255 Still, many of the movements in E4,

especially those that traverse the central vertical aisle, are similar to E3.256

4.4.6. 3rd Bridge, Fourth block: M6-8 and E5-9. In this section we consider the

longest and most complex sequence of waves. However, this block of waves begins with

a bridge that is spatially quite inactive. The third bridge is the longest of the bridges up

until now, probably because it introduces all of the eight M- and E-nuclear formulas that

will subsequently be heard. Since it is almost entirely scored for instrumental and vocal

solos, the melodic motion through space – if it can even be said to be perceived as

“motion” at so slow a rate – slows to almost a complete stop in the third bridge.

Diagramming the more common spatial motions in this lengthy block of waves is

accomplished in Examples 4.4.6a. In these waves, the pitches in the M-orchestra and the

E-orchestra are transposed up and down an octave, respectively. Also, Stockhausen’s

aforementioned “comets” begin. Thus, we have an enlargement of the metaphorical tonal

space along with a new kind of spatial event. Spatial structures in these waves, which are

more distantly related to the two basic cycles, also correspond to the expanded

vocabularies. In M6, the central fa2 instrument is studiously avoided, in favor for

predominantly clockwise motion around the edges of the space. M7 introduces a

surprising shape – a star – which will be discussed in more detail later. A large

proportion of spatial motion in M7 flows through two instruments – f1 and t2. M8, like

255 The speed of 11.60 m/sec equals about 26 miles per hour.256 Stockhausen suggested that some movements might be similar across waves in his LICHTER-

WASSER lectures. 2001 LICHTER-WASSER lectures, day 4.

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M6, avoids the central area of the space while continuing to explore edge movements.

We notice in these three M-waves that although the motion through space resembles the

basic M-cycle less, it has also become more repetitive, with frequent moves – often ten or

more – strongly linking some instruments together. The motion has sped up considerably

too – M6 and M8 clock in with considerably faster spatial movement (25.64 and 27.06

m/sec respectively)257 than any previous M-wave.

While the M-orchestra tends to cycle around the edges of the space, the five E-waves in

this block also share some structural spatial characteristics. Many paths through E5-E9

include much longer “hand-offs” than before. One path, va5–b–f1–fa2, is extremely

common through all of these E-waves, and is a subgraph of the basic E-cycle. In E6 the

clockwise circular cycle h1–b–p1–sax–p2–eu–h2–va4–h1, which is a distinguishing

characteristic of this wave, comes to prominence. E5 and E9 have the longest average

move of any waves (11.09 and 12.21 meters, respectively). The fastest average speed of

any wave up until now – E9 – whizzes around the hall at a rate of 44.76 m/sec.258 Paths

though these E-waves often include the two corner instruments fa1 and tu, move to b, and

then through the area closest to the conductor. Surprisingly, th plays a significant role in

all five of these E-waves. Although th is an M-instrument, it may have a more prominent

role in these waves because of Stockhausen’s tendency to create large circles around the

center point. But th is not the only instrument to have a significant amount of

“crossover” here; in E8, three-quarters of the instruments in the inner circle – including

va3, th, v2, v1, va2, and va1 – figure prominently in the E-orchestra. The way that

instruments are “shared” from the M-orchestra in this block of E-waves again points to

the concept of “mystical unity” that Stockhausen tried to express in LICHTER-

WASSER.

In this block of waves, there is a surprising and unexpected relationship between the M-

257 These speeds correspond to 57 and 60 miles per hour, respectively.258 Slightly over 100 miles per hour.

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and E-orchestras in the speed of their spatial motion. Stockhausen took pains in his

sketches for spatial movement to give each wave its own special spatial forms and

shapes. Following Stockhausen’s plans, we would expect that each M- and E-wave has

its own patterns of change in speed. This is clearly not the case in E5. As we can see by

referring back to the long graph in Example 4.4.1b, the part of E5 that occurs during M6

shares the pattern of extremely rapid moves followed by slower ones. But as soon as M7

begins (about two-fifths of the way through E5,) the rate of change in speed abruptly

matches M7. This analysis shows how even though the shapes of the motion the M-

waves and E-waves can be considerably different, some of their characteristics – such as

the rate of change in speed – match quite closely and change in order to match each

other.259

4.4.7. 4th Bridge/Announcement block. The central block of “announcements”, which

alternates with the fourth bridge again brings the spatial motion to a virtual standstill.

During the first and third announcements, motion literally does come to a complete stop.

The soprano, accompanied by the synthesizer, sings self-referential texts containing a

certain witty semantic content. At the same time, twelve of the twenty-nine

instrumentalists leave their places on the floor, and move to the balcony. Thus, the fourth

bridge contains musical sources both on the floor and above the audience; thus, is has the

distinction of being the only spatial structure in LICHTER-WASSER that takes place in

three dimensions. The fourth bridge is quite densely scored and is made of groups of

instruments playing two homophonic parts together, punctuated by occasional brief solos.

After the second announcement and the repetition of the fourth bridge, the soprano sings

the third and final announcement. Here, she asks the instrumentalists to come back down

to the floor. Finally, she literally requests that the music continue (“Na, ihr Lieben, was

259 One might interpret this modification in the pattern of speed to be yet another expression ofStockhausen’s idea of “mystical unity” between Michael and Eve.

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sagen sie den nun?! / Stockhausen, wir können beginnen”).260 After a brief pause, we

launch immediately into the next wave sections, M9-10 and E10.

4.4.8. Fifth block: M9-10 and E10. Having introduced the three nuclear formulas M9,

M10 and E10 in the fourth bridge, we are ready to progress to the fifth block of waves.

Although these sections have the highest average speeds compared to any previous

waves, they also have the most repetitive moves through space, especially in the case of

E10. As can be seen in Example 4.4.8a, motions in M9 continue to outline the shape of a

“star” as in M8, with va1 serving as a kind of “hub”.261 Common motions in M10 can be

interpreted as defining two large “circles”: one on the left side of the space, including the

path fa2–f2–v5–k–ob–v4–fa2, and the other on the right side, comprising the instruments

fa2–kb–t2–f1–v3–fa2.

At the same time, E10 moves around the space in another circular pattern which recalls

the previous block of waves E5-E9. Again the path va5–b–f1–fa1 is traversed very often;

again th plays a role as an important vertex in the E-path th–h2–tu–eu–fa2–p2–fa1–p1–b–

h1–th.262

4.4.9. 5th bridge, Sixth block: M11 and E11. The frenetic activity in the previous

block of waves comes to an abrupt stop at the fifth bridge. This bridge is scored so that

some melodic material travels through the space while other instruments hold three-note

chords. This is perhaps a preparation or foreshadowing of the many three-note chords in

M12 and E12. Unlike most previous bridge sections, Stockhausen used box diagrams in

the fifth bridge to show spatial motion. Therefore, we can easily trace paths through the

space and calculate data telling us more about the motion. The average speed in the fifth

260 “Now, my dears, what do you say?! / Stockhausen, we can begin.”261 Stockhausen highlighted va1 on his sketch in M9. Analysis shows that a great variety of spatial moves

originate from va1 and point to va1. Its function as a “hub” is similar to fa2’s function as a hub in otherwaves.

262 This cycle is traversed in its entirety several times during E10.

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bridge is significantly slower than in the previous wave sections; in fact, it is comparable

to what it was in M1 and E1.

Example 4.4.9a shows the common spatial motions through M11 and E11. Here the

speed of motion through the two orchestras reaches its measurable peak, moving at the

rate of 46.01 and 48.47 m/sec respectively.263 At this rate of speed, spatial motion would

probably be impossible to distinguish if the paths traversed through the space were not

highly repetitive and predictable. Can the cycles through each orchestra during these

waves can be interpreted as variations of the basic cycles? The spatial motion in M11 is

almost exactly the same as the basic M-cycle, but reflected in the horizontal aisle.

Significantly, this motion abandons the instrument th, which though technically an M-

instrument, plays such an important role in the E-orchestra. The spatial motion in the E-

orchestra is less like its basic cycle, mostly because of the oft-repeated motion clear

across the performance hall, from h2 to p1. However, the E-orchestra picks up th,

incorporating it even more decisively into its movement pattern. Also, movement along

the diagonal va4–th–fa2–sax–fa1 is very similar to E2, but in the opposite direction. The

M- and E-cycles through the space in M11 and E11 are highlighted with a special gray

line in Example 4.4.9a.

4.4.10. 6th bridge, Seventh and final block: M12 and E12. Although the bridge

sections always articulated a contrasting musical texture from the waves around them,

this difference is greatest with the sixth bridge. This short bridge is almost entirely

homophonic and involves all of the instrumentalists and vocalists. The purpose of the

sixth bridge is to introduce an additional astronomical object called “MICHEVA”, which

has symbolic importance in the mystical universe of LICHT. Since the spatial motion in

this bridge is clearly defined (although there are no box diagrams, the instrumental

sequence is clearly written into the score), we can analyze the speed of motion through it,

as we did in the fifth bridge. Unlike the fifth bridge, the motion in the sixth bridge is

263 These speeds are 103 and 108 miles per hour.

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quite rapid and only slightly slower than the preceding block of M11 and E11 waves.

Therefore, the contrast it provides to the surrounding wave sections is principally through

texture, not spatial motion.

The spatial motion of the melodies in M12 and E12 is diagrammed in Example 4.4.10a.

Like the previous examples, we find that one principal path is traversed through both of

these waves. The average calculated speed of melodic motion through M12 and E12 is

fast, but not quite as fast as M11 and E11. This is because the way that chords interrupt

the texture. We find that the rate of spatial motion has a number of extreme “bursts”

followed by moments of calm.264 The path traversed through the space in M12 again

appears to be very similar to the basic M-cycle, but this time rotated 90 degrees

clockwise around the center point, represented by the instrument fa2. The cycle through

E12 is very similar to E11 but motion occurs in the opposite direction – in other words,

compared to E11, E12 is, almost instrument for instrument, a retrograde. The diagonal

fa1–sax–va1–f2–th–va4 again traverses a familiar path through space heard in E2 and

E11, and also in E12. However, the most commonly traversed path in E2 is the same as

in E12. There is a close relationship between these E-paths which helps frame the blocks

of wave sections.

The chords that interrupt spatial movement during M12 and E12 present an analytical

opportunity which unfortunately is not possible to explore in sufficient detail within the

bounds of the present work. However, the chords are such an important part of the

texture during these two exceptional waves that they are worth at least a cursory

examination. Of the 72 chords identified, the most common type – occurring some 30

times – includes seven notes played by 12 musicians. Ten of these musicians are always

the same and include the five violas and violins, playing pizzicato. The other instruments

264 The most rapid calculated motion in LICHTER-WASSER occurs in the second half of measure 654,where a 20-meter move from t2 to t1 during a sixteenth-note triplet at the tempo quarter = 71 takes .141seconds. This move clocks in at 141.9 m/sec, or 317 miles per hour.

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in each 12-part combination are pairings of two of the remaining nineteen wind and

brass. These pairings always involve different combinations of instruments except for

two chords, which combine f2–eu and kb–fa2 twice.265 These 30 chords, which we will

call “7/12” chords from now on, always come after a crescendo; they are always forte or

fortissimo; and, they are only of a quarter note duration. They function almost as

“clicks” or articulation marks at the end of a very rapid sequence of movements.

The next most common type of chord involves three notes played by three musicians.

There are 24 of these “3/3” chords. All of them occur among members either the M-

orchestra or the E-orchestra. Often the 3/3 chords are held for considerably longer than

the 7/12 chords and may even include glissandi. Because of these differences, the 3/3

chords clearly have a different function than the 7/12s and due to their longer duration

may play more of a timbral role than the 7/12s. The third group of chords involves six

instruments playing six notes. These “6/6” chords are related to the 3/3 chords because

they are always made up of an additive combination of two 3/3s. The 6/6 chords, like the

3/3s are usually held longer than the 7/12 chords. In addition to the 7/12, the 3/3, and the

6/6 chords, our data show that six chords are “irregular”. They have varying numbers of

notes ranging from 4 to 7. These irregular chords seem to function along the same lines

as the 3/3 and 6/6 chords – because of their longer duration and irregular instrumental

combination, they may have a timbral function as well. Analyzing the chords by the

number of pitches they contain and the number of instrumentalists they involve allows us

to see the different functions they serve in this section.

4.4.11. Ausgang and Schluss-Duett. In the Ausgang section, each instrument leaves his

or her position, one after the other, and walks out to the door while repeating a single

note.266 The structure of exits can be represented by a tree graph. It is quite likely that

265 The first f2–eu chord is in m. 650 and the second is in m. 691. The first kb–fa2 chord is in m. 640while the second is in m. 680.

266 “Each takes a drink and then walks out to the right (as seen by the conductor), playing .” Stockhausen2001b, p. 8.

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anyone wishing to perform LICHTER-WASSER would want to map out the path each

instrumentalist takes while exiting, in order to avoid any unfortunate collisions or

uncertainty among the instrumentalists themselves. Knowing something of the structure

of motion in the wave and bridge sections can aid a conductor or designer in crafting an

exit pattern that relates to the other structures in LICHTER-WASSER in a meaningful

way.

One proposed exit structure is shown in Example 4.4.11a. Here, a number of musicians

move first to the center of the performance space. Since the instrument at the center – fa2

– tends to receive a considerable amount of motion from a variety of places (something

we will explore further in §4.6), it makes sense that a number of instruments physically

move towards it at this point. In the proposed structure, there are 12 paths from a leaf to

the root; these paths are numbered 1 through 12 on the graph. As the paths traverse the

graph structure to the root, eight instrumentalists move through vertices of alternating M-

or E- membership. Paths 1, 8, 9 and 11 each move through similar E- or M- spaces only

once. The advantage to having the musicians exit in this way is that their movements are

symmetrical with respect to the horizontal axis along the k–p2 aisle.

An alternate arrangement, shown in Example 4.4.11b, allows each and every instrument

to traverse spaces which always alternate between M- and E- instruments. The structure

involves fewer paths from the leaves to the root – nine instead of twelve. However, this

exit structure does not always move each instrument in the most efficient manner towards

to door, nor is it symmetrical with respect to the horizontal axis. However, it could be

argued that this exit structure is desirable in keeping with the theme of “mystical unity”

between Michael and Eve. The pains Stockhausen took to mix Michael and Eve

structures together in LICHTER-WASSER are reflected in the way each instrument

always moves through spaces of alternating orchestra membership on the way to the

door. Whatever the conductor or director decides, he or she can make a more informed

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decision about how to manage this important section by analyzing the tree structure as we

have done here.

4.5. Vocalist Motion

4.5.1. Vocal movements in the Anfangs-Duett and Eingang. Since the singers stand

motionless during the Anfangs-Duett, there is no spatial motion to analyze. However,

their position in the space is important during this section. While the tenor stands at the

position k, the soprano sings from p2. The two singers are both oriented on the central

horizontal aisle – the horizontal axis of symmetry – where they will eventually return in

the Schluss-Duett.

During the Eingang section, the vocalists show instrumentalists to their positions. They

do this in the same sequence that the instrumentalists enter, corresponding to the two

basic cycles. While the vocalists show most of the instruments to their places, they give

signals to others to begin playing. The path through space traced by soprano and tenor is

therefore a simplified version of their corresponding basic cycle.

4.5.2. Vocal movements through the twelve waves and six bridge sections. The

movements of the soprano and tenor through the twelve wave sections and six bridges are

shown in Example 4.5.2a. Calculations show that an average move from one section to

another for the soprano is 10.3 meters, which is slightly shorter than the average moves in

the E-orchestra over all 12 waves and bridges 5 and 6. The average length of a move by

the tenor is somewhat greater (9.7 meters) than the average move in the M-orchestra but

by less than a meter. In this respect, the individual moves of the tenor and soprano

resemble the spatial activity in the two orchestras.267

267 The distances measured here are the closest possible distances between the singers’ starting and endingpoints. Our measurements do not take into account the fact that the singers will in fact have to walksomewhat further, since they can only move through aisles and not directly across parts of the audience.But, since the distance between starting and ending points is important, we are making a relevant spatial

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Aside from their much slower rate of motion, a crucial parameter sets the movements of

the soprano and tenor apart from the instrumental movement. Several of the vocalists’

moves between instruments are never – or extremely infrequently – traversed during the

waves or bridges. The data in Example 4.5.2b show how many times each move that

occurs by the vocalists happens in the instrumental music. We find that five of the

twenty-two moves (23%) by the soprano and eight of the nineteen moves (42%!) of the

tenor never occur or occur only once in the instrumental motion. Even though the

soprano finds herself on paths that are frequently taken by instrumental melodies towards

the end of her traversal through the space, the tenor walks through more common

instrumental paths towards the beginning of his movement sequence.

These data show exactly what sets the motion of the soprano and tenor apart from the

instrumental ensemble, and in which ways their motion is related. Although the overall

rate of their spatial motion – which is assumed to be so slow that it is not calculated here

– differs considerably from the instrumental motion, we have expressed several crucial

relationships between vocal and instrumental music through our analysis.

4.5.3. Vocal movements in the Ausgang and Schluss-Duett. During the Ausgang, the

vocalists again approach most all of the instrumentalists as they drink from their glasses

of water. Like the Eingang, the vocalists’ motion is a simplified version of their

respective basic cycles. A walk through the entire space would probably take too long

for the vocalists to complete at this point. We may also consider the effect of

performance fatigue; during the wave and bridge sections alone, the singers have each

traversed distances of at least 200 meters while performing difficult music.

During the final Schluss-Duett, the tenor and soprano again sing facing each other near

the center of the hall. Their positions – at locations va3 and va2 – are along the same

measurement.

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horizontal axis that they sang from in the Anfangs-Duett. Once their duet is over, they

too move slowly towards the door and exit the hall. At this point the work LICHTER-

WASSER is over.

4.6. Trends of Spatial Movement in LICHTER-WASSER

4.6.1. Number of Moves Between Instruments. Having measured, analyzed and

described the spatial movement in LICHTER-WASSER, several questions regarding the

use of the space can be explored in further detail. First, how many times is each musician

used through the course of the work? Knowing the answer to this question can tell us

more about the kind of spatial activity that occurs in different parts of the space. In order

to accomplish this, the use of instruments is taken out of time, and summed so that we can

view spatial structure in its totality.

Example 4.6.1a shows the number of times each instrumentalist in LICHTER-WASSER

plays. The graphs beside each instrument show the corresponding number of times that

instrument plays in the Michael orchestra and the Eve orchestra. The black bar shows the

total number of times that instrument plays. The squares around the instrument’s

abbreviation are shaded according to how often the instrument is used; the darker the

shading, the more times that instrument is heard.

This graph opens a remarkable window into the nature of spatial activity in LICHTER-

WASSER. We find that one area of the space is used considerably less frequently than

average. This lower-right quadrant includes the instruments v2, v1, t2 and kb. At the

quadrant diagonally opposite, instruments such as b, va5 and p1 get used much more

frequently than average. The difference is significant: v2 is used only 101 times but b is

called upon 156 times. This represents a difference which is probably noticeable during

performance. While listeners may have the aural impression that one quadrant is used

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more often than another, our data shows exactly how much and where spatial activity

peaks.

Lying in the center of the hall, fa2 receives a considerable amount of spatial attention as

well. In fact, this instrument is the most frequently used of all, totaling 171 times.

Clearly, the second bassoon’s function is more than just one of the twelve instruments in

the E-orchestra; it also has a special importance because of its strategic position in the

middle of the spatial structure. It functions as a direct way of getting from one side of the

space to the other.

Another discovery that can be made from examining Example 4.6.1a relates to the

proportion of times each instrument plays in the two orchestras. We saw in our analyses

in §4.4 that the instrument th is used very often in E-wave structures, despite the fact that

it is technically a member of the Michael orchestra. Our present analysis confirms that

all of the M-instruments play more often in M-structures except for th. While this should

not be surprising considering our findings in §4.4, it is striking to learn the proportion of

times this instrument is used in the E-orchestra: th plays only 21 times in the M-orchestra

but 124 times in the E-orchestra! We may conclude that th is essentially an E-instrument,

not an M-instrument. Examining the E-orchestra, we find that all of the E-instruments

play more often in the E-orchestra except for va5. Va5 is a second “crossover”

instrument; it plays 57 times in the E-orchestra but 93 times in the M-orchestra.268

4.6.2. Amount of Time Each Instrumentalist Plays. Knowing the number of times

each musician plays revealed a surprising amount of information about the way different

parts of the space are used in LICHTER-WASSER. Another measure of spatial density

is the amount of time each instrument is used. Example 4.6.2a, which looks similar to

Example 4.6.1a, correlates this aspect of space and time.

268 The way in which these two instruments are “borrowed” from one orchestra to the other furtherexpresses Stockhausen’s idea of “mystical union” between Eve and Michael.

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Although we are measuring a different aspect of spatial density in this analysis, many of

the statistical data here are similar to the previous graph. The instruments b, fa2, fa1 and

p1 are all used for the most amount of time. Instruments in the quadrant including v1, v2

and t2 are similarly used for a relatively short amount of time. However, two “cold”

spots emerge along the top edge of the space: v5 and v3. Although these instruments are

used at about an average rate, the amount of time they play is considerably lower than

average. Using this measure, the center of spatial activity is located more towards the

middle of the space and along the diagonal extending from fa1 to ob. Considering that

this diagonal is used so frequently during E2, E11 and E12, this finding should not be

surprising; however, the data from both this analysis and the previous one clearly show

that the fa1 – ob diagonal is traversed considerably more frequently and used for a greater

proportion of time than the other diagonal, from v5 to tu.

Examining the data in Example 4.6.2a reveals that there are again some “crossover”

instruments. It should not come as a surprise that th is a crossover instrument by this

measure as well. Va5, which was a crossover instrument from the E-orchestra to M,

plays about the same time in both orchestras, but is slanted slightly towards its “home”

orchestra. Va4 emerges as the third crossover instrument. Although it is in the E-

orchestra, it plays for a little longer in the M-orchestra. This is probably because much of

the E-orchestra activity in its quadrant which might have involved va4 instead goes to th.

However, va4 is not enormously skewed towards the M-orchestra, like th is; it is used

only about five seconds longer in the M-orchestra than in the E-orchestra out of 79.7 total

seconds of playing.

4.6.3. Connectivity. In §4.4, we examined the most common spatial movements in each

wave section individually. Example 4.6.3a is a composite of all those motions. This

example is a kind of weighted graph. Thick lines denote movements that happen

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extremely frequently while thin lines indicate that motion is rare. Moves that occur only

one time are again omitted.

Through examining this graph, we are able to make several positive statements about the

overall trends of spatial motion in LICHTER-WASSER. Many instruments appear to

have different functions judging from the way in which the spatial melodies move to

them, and are subsequently passed on. Fa2, located in the center, is a kind of hub in that

it receives melodic motion many times from a great variety of different places. Fa2 also

tends to send motion out to an equally great variety of places. In contrast to the hub

structure is a conduit. A conduit receives a high percentage of melodic motion from only

one or two sources, and subsequently passes most of that motion on only to one or two

other sources. Instruments around the edge of the space tend to be conduits. P1, v5, v3

and f2 have strong conduit characteristics. While this is partially due to their location in

the space, there was probably a compositional determination made in these functions as

well. The absence of modularity in the space gives these instruments fewer options for

spatial movement.

Further analysis of the shapes in Example 4.6.3a indicates that a great deal of

instrumental motion circles around both the outer ring and inner ring. The instrument th

partly functions as a bridge between the outer and inner rings, and has a characteristic

conduit function in relation to the central fa2 position. This spatial function may help to

explain why th has such an unusual use as a “crossover” instrument: it helps to provide a

way for the E-orchestra to get back and forth between inner and outer ring structures

without having to make large spatial leaps over M-instruments. By keeping the distances

between moves smaller, more spatial continuity is achieved.

We can take our analysis to a deeper level of abstraction by considering each move

between instruments that happens more than once as an edge on a graph. We can

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represent all the edges in the graph with a matrix, which I will call the LICHTER-

WASSER Matrix (hereafter abbreviated “LW Matrix”). The A1 LW Matrix is shown

in Example 4.6.3b, and the A2 LW Matrix is given in Example 4.6.3c.

Analysis of the A2 LW Matrix yields several new insights into the potential connections

that could be made among instruments, given the links Stockhausen chose to make.

Examples 4.6.3d and 4.6.3e interpret the data in the A2 LW Matrix visually. Example

4.6.3d shows the potential connections each instrument might have with other

instruments over pathways of degree 2 (cardinality 3) for motion that begins from the

given instrument. Stockhausen created this structure through the moves that he made

over the course of the work. The darkest instruments in Example 4.6.3d are the ones

from which the greatest potential variety of paths of degree 2 originate, while the fewest

potential degree-2 paths originate at the lightly shaded instruments. The example shows

how “cosmopolitan” or “provincial” each instrumental location is.269 We interpret

Example 4.6.4e in a similar way; here, the darkest instruments are those where the

greatest number of 2-edge paths end, and similarly the fewest number of degree-2 paths

end at the lightly shaded instruments.

Examples 4.6.3d and 4.6.3e show how Stockhausen’s choices of spatial movement

around the hall cause spatial paths which originate at the center instruments, especially

fa2, to have the most potential variety of shape. The spatial shapes that originate in the

corners – that is, v5, fa1, tu, and ob – exhibit the least potential variety of different spatial

shape. This is what we might expect from a space that does not have the property of

“modularity” as we defined it in Chapter 2.270 Under these conditions, edge instruments

simply have fewer places to go to than instruments situated closer to the center. Because

269 While the terms “cosmopolitan” and “provincial” are somewhat prosaic, they effectively express thenumber of inward and outward connections each instrument has with the rest of the graph, much like atransportation network links cities to other cities.

270 If the space were modular, nodes (instruments) located around the perimeter of the graph could moveoff the edges and wrap around to other outlying areas of the graph, allowing them to make as many (ifnot more) potential paths as centrally located nodes.

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of this analysis, we have learned many things not only about the structure of the space in

LICHTER-WASSER, but also the variety of spatial motion that is possible at

various locations.

4.7. Relation of shapes in LICHTER-WASSER to each other

4.7.1. Short paths. Using the A2 LW Matrix, we have made some predictions about the

potential density of degree 2 paths, given the connections of degree 1 that Stockhausen

drew between instruments. What about the actual paths and spatial shapes of degree 2?

Analysis of the wave sections shows that there are approximately 400 different paths of

degree 2 that are traversed two or more times in our LICHTER-WASSER data set. Some

of these paths are related by rotation, reflection, or scaling (“multiplication”). Finding

paths that are related through these operations would reveal any symmetrical structure to

the movements in the piece. Although there are many paths to explore, we can focus our

observations by looking for related paths of degree 2 which start at the corners of the

space. There are two reasons for this. First, we already know that the paths which start

at the four corners of the space are limited in where they can move to, because they have

only a few neighboring instruments, and there is little if any sense of modularity in the

space. This means that we will have fewer paths to compare, making our job easier.

Second, even though the membership of instrumentalists in the E- or M- orchestras is not

symmetrical along the vertical (p1–h2) axis, we know that the E-orchestra and M-

orchestra are arranged symmetrically along a horizontal axis. Therefore, we can

hypothesize that if the melodic motion among instrumentalists in LICHTER-WASSER

exhibits a high degree of symmetry, it will be easiest to identify by comparing movement

around four corners of the space.

Analysis of spatial motion in M1-M10 and E1-E10 shows that surprisingly few paths of

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degree 2 which originate at the four corner instruments are related by rotation or

reflection.271 The paths that are related, along with the number of times they occur, are

shown in Example 4.7.1a. For instance, the path a (v5–k–ob) is related to b (fa1–p2–tu)

reflected in the vertical axis; c (fa1–p1–b) is related to d (tu–h2–va4) reflected in the

horizontal axis, as is e (fa1–sax–fa2) related to f (tu–eu–fa2). G (fa1–sax–p2) is related

to h (ob–va4–k) reflected in the diagonal axis v5–tu; these two shapes bear some

resemblance to i (ob–va2–kb) reflected in the other diagonal axis. But j (va5–v5–f2), a

motive that occurs extremely often (29 times) in the sampled wave sections, has no

corresponding motives related to it by reflection or rotation in the horizontal, vertical, or

diagonal axes. We can conclude from this analysis that although there are some paths of

degree 2 that occur often in LICHTER-WASSER which are related by rotation and

reflection, Stockhausen was probably more concerned with differentiating the spatial

vocabulary in the four corners of the LICHTER-WASSER space than he was with

creating symmetrical shapes. Although the four corner instruments share some

characteristics of spatial motion, these regions do not systematically show a strong degree

of symmetry.

From examining another aspect of the spatial movement of the instrumental melodies, we

can conclude that an obvious compositional strategy in LICHTER-WASSER is to keep

the spatial motion moving continuously around the room. In other words, motion rarely

“turns back” on itself and reverses direction. During M1-M10 and E1-E10 there is only

one cycle of degree 2 that occurs twice (sax–fa1–sax, in M8). No other cycles of degree

2 occur more frequently. Longer cycles, however, do play a significant role in the spatial

language of the work, as we shall see in the following section.

4.7.2. Long paths. Two challenges arise in analyzing larger paths in the space of

271 Because E11, E12, M11 and M12 are so highly repetitive, the paths in these wave sections are omittedfrom the present analysis, lest they skew the results. Aspects of these wave structures are examined inmore detail in §4.7.2.

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LICHTER-WASSER. First, the computational complexity of analyzing and comparing

paths grows exponentially as we increase the cardinality of the spatial shape (or, the

length of the path). This is a difficulty which is unfortunately unavoidable. A second

challenge has to do with the number of elements in a set. When analyzing relations

among pitch sets or pitch-class sets, we rarely have to deal with more than nine or ten

elements in a set, and almost never with twelve-element sets. However this is not the

case with sets of spatial locations, which – at least in LICHTER-WASSER – can involve

up to twenty-nine musicians. Moreover, the angle at which motion moves from one

instrument to another is a property of spatial motion that has no corresponding meaning

in the pitch world. Finally, the large number of intervals between elements in these sets

makes them more difficult to compare than the smaller sets we are accustomed to in

pitch analysis.

However, there are certain relationships among paths of up to seventeen elements in

LICHTER-WASSER. Some lengthy spatial shapes in LICHTER-WASSER are

obviously similar to others. We can identify these because they are so repetitive, thus

avoiding more complex computational analysis. These paths are mostly related in some

way to the two basic cycles that we showed in Example 4.1.3b.

Consider the basic M-cycle and the most common cyclic path in M12. These are shown

in Example 4.7.2a. Both of these paths traverse the same number of instruments –

seventeen. First, both of these paths have a certain aspect of self-similarity; they are

essentially a set of two concentric circles.272 But how are the two paths related to each

other? It is clear that the path in M12 is similar to the basic M-cycle but it is rotated 90º

counterclockwise, with respect to the center node fa2. After rotating the basic M-cycle,

272 The idea of self-similarity is in fact a common theme in Stockhausen’s music. This can be observedmost clearly in the LICHT-project itself, in which the Superformula is usually present at multiplestructural levels (background, middleground and foreground) of musical composition. Self-similaritymight have been a way Stockhausen counteracted criticisms of serialism and comprehensibilityarticulated by Leonard Meyer (Meyer 1994, pp. 245-316).

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as shown in Example 4.7.2b, and overlaying the M12 cycle, we find that the two paths

are similar in shape but not necessarily in direction. The main difference is that the basic

M-cycle traverses the inner circle counterclockwise, while the path in M12 moves around

it clockwise. If we split the M12 path and rotate the inner circle of movements, the basic

M-cycle and M12 are nearly the same. In this way we can combine the operations of

rotation and reflection to relate differences between long spatial paths with some degree

of success.

How might we describe the relationship between spatial shapes with different

cardinalities more precisely? A situation of this kind arises in comparing the spatial paths

in the E11 and E12 waves. These two waves are shown in Example 4.7.2c. Here, the

repeating path in E11 is of degree 13 while E12 has degree 12. But the shapes of the two

paths are very nearly the same; the difference of E11 � E12 is only 3 (h1, va5, va1). A

similarity index correlating the two shapes would have to take into account the distance

each move traverses, the intervals of spatial motion, the angles created by changes in

direction, and possibly even the speed at which the cycle traverses the space. While it is

conceivable that such a similarity index would be possible to calculate, it is beyond the

scope of the current study. Still, the ability to relate shapes in this way would be a useful

tool, and developing such a measure could be a goal of further research.

4.7.3. “Hidden” Shapes. Although superficially the horizontal aisle is an axis of

symmetry with regard to the arrangement of instruments in M- or E-orchestras, our

observations in §4.7.1 showed that little symmetrical motion occurs at the corners of the

space that is related by this axis of symmetry. Based on the arrangement of musicians in

the space, and informal observations from M1-M12 and E1-E12, the two path structures

shown in Example 4.7.3a seem conceivable. This weighted graph shows how many

times each edge in the graph is traversed. Therefore, the shapes in the graph are the

result of a large number of movements over the course of the entire piece. Although

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subpaths of these cycles are very often traversed, the paths themselves are never walked

through as a cycle. However, both of these paths are symmetrical with respect to the

horizontal axis.

The M-path roughly traces out a star with seven points; paths in M7 and M9 traverse

these edges very often. The M-path is appealing because of the way each of the seven

points of the star could represent one of the days of the week in LICHT. However, the

instrument th disrupts the symmetry in the M-cycle, and is therefore shown in black. In

addition, some moves which trace out arms of the star are seldom traversed. The pattern

of E-movements in the example trace out a circle with small bulges at both ends. All of

the E-waves but especially E5, E6, E7, E8 and E10 outline the motions shown in this

example. As we know, the diagonal from NE-SW is traversed much more often than the

NW-SE diagonal, disrupting the symmetry here as well.

While both of these patterns show an appealing symmetry, neither of them are traversed

regularly enough to be considered a strong background structure to the movements of

LICHTER-WASSER. However, both of the structures are present (at least faintly) in the

work’s spatial motion. Both structures also have an appealing relationship to the overall

symmetrical plan of instrument arrangement and thus, the background structure of

spatial motion.

4.8. Group Structure

There are two characteristics of the spatial motion in LICHTER-WASSER that make

developing any kind of group structure challenging. First, there is a lack of any apparent

use of modularity in the space. Modularity is one condition under which mathematical

closure can operate in the pitch domain. As we have seen, Stockhausen was extremely

reluctant to move spatial shapes off the “edge” of the space and back onto the space from

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a different side. Stockhausen sketched the dramatic path p1–h2, which could be

interpreted as a modular move (since it goes clear across the space from the north to the

south location,) always moving through the center of the space, not off the edge. If the

space does not exhibit modularity, spatial movement in closed rings around the center of

the space could create a situation where closure exists. While many such circular or near-

circular paths exist in LICHTER-WASSER, they are by no means the only kind of

motion that we have found.

A second hindrance in developing a mathematical group has to do with the

transformations that would need to be used in order to move musical melodies from one

instrument to another. Some transformations that work well at some points in space

would not be possible if applied to other instruments, since the transformation would

move that location to another location where no instrument is situated.

Stockhausen’s early sketches for LICHTER-WASSER, shown in Example 4.2.1a, exhibit

many symmetrical structures of motion which would have group properties, were they

used consistently. It might be tempting to develop groups of transformations that model

these spatial movements, and then combine them in some way. But these symmetrical

geometric shapes are very seldom used in the actual score to LICHTER-WASSER, as the

individual wave analyses from §4.4 show. Stockhausen seems instead to have been

stimulated by the many possibilities of asymmetrical movement in the space he designed.

If there were a group structure to transformations which model spatial movements in

LICHTER-WASSER, it might look like Example 4.8.1. Unfortunately, the operations in

this structure are very limited; they can only model a few kinds of spatial motion. The

addition of the reflection operations �v, �h, �d1 and �d2 do little to expand the ability of

these operations to model the variety of movements we have found in the piece. The

main problem is the lack of any kind of operation for spatial movement that radiates out

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from or collapses back towards the center. As we saw in Example 2.5.3f, this would

result in transformations that were many-to-one and one-to-many. This kind of

transformation is not part of the definition of a mathematical group.

The evidence presented in this chapter suggests that there is no clear group structure to

the transformations that might operate in the physical space of LICHTER-WASSER.

Perhaps the kinds of spatial motion that would create such a structure were not interesting

to Stockhausen. As mentioned above, Stockhausen seems to have focused on exploring

asymmetrical shapes instead of symmetrical ones.

The lack of a group structure means that a compelling transformational language that

could be isomorphic or at least similar to the structure of pitch transformation is unlikely

to be found in this work. On the other hand, Stockhausen may have had questions of

perception in mind when he constructed the patterns of spatial movement in LICHTER-

WASSER. As we have already theorized, it is doubtful whether a spatial shape that

moves off the edge of a space, and lies “straddled” between two distant regions, will be

perceived as similar to a shape that is the same but occurs in the center of the space. The

fundamental difference between space and pitch is decisive here. Whereas pitch is

customarily perceived as a one-dimensional continuum in Western music, space does not

lend itself as easily to being “wrapped around” in a similar fashion.

The question of perception is one we have purposely not explored, for reasons explained

in Chapter 1. However, the data we have collected and the conclusions we have been

able to draw from our analysis of LICHTER-WASSER provides a basis for investigating

the question of perception of spatial shapes in spatial music. Measuring the kinds of

movement in a radically spatialized composition, as well as investigating the more

theoretical possibilities of movement that are possible will allow us to approach

perceptual questions in a much more informed manner.

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4.9. Conclusions

Stockhausen stated his goals clearly in composing the spatial motion of LICHTER-

WASSER. First, each of the twelve wave sections was to speed up, and symbolically

reflect the speeds of heavenly bodies orbiting the sun, relative to the sun. Second, there

was to be a kind of “mystical unity” between Michael and Eve, each represented by a

vocal soloist and by a group of instrumentalists.

The data which I have analyzed in this chapter show that although there is not a constant

increase in the average velocity of spatial motion in each wave, there is a clear general

trend towards faster motion. I have also uncovered ways in which Stockhausen might

have expressed the “mystical unity”.273 I proposed an exit structure that might subtly

reinforce the concept of mystical unity by interpreting the spatial structure of the

musicians’ exit as a tree graph. By exploring the structure of the instrumental layout

Stockhausen chose, and the shapes he created, I have been able to illuminate some of the

choices Stockhausen made in composing spatial music. Finally, in examining some of

the possible symmetries of spatial motion, I have concluded that even though there is

little reason to believe that transformations relating spatial shapes form a mathematical

group, there are still some possible background structures involving symmetry in the

movements of the instrumental melodies.

The understanding we have gained of the spatial motion in LICHTER-WASSER would

not have been possible had we not measured the shape, speed and direction of spatial

movement throughout the piece. In this way, we have finally made this crucial aspect of

Stockhausen’s music more amenable to analysis. But the approach we took to this work

273 “Crossover” instruments such as th and va5 are the most direct metaphor for “mystical union” in thespatial structure of LICHTER-WASSER. MONDEVA, from Act 1 of the THURSDAY opera, and theHIMMELFAHRT duet at the end of MICHAELS REISE UM DIE ERDE also contain elements of“blending” that foreshadow those in SUNDAY.

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and the knowledge we gained tempt us to apply similar methods to other pieces which

also have sophisticated spatial structures. In the next chapter, we will apply the

techniques we have used here to shed new light on works by Stockhausen, Tallis and

Xenakis, while speculating on further directions of spatial research and analysis.

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Chapter 5.Further Applications of Spatial Analysis and Performance.

5.1. Directions for research in spatial music

In the previous two chapters, we applied the analytical methodology outlined in Chapter

2 can be applied to measure and analyze spatial motion in two later works of

Stockhausen, OKTOPHONIE and LICHTER-WASSER. However, much more remains

to be learned about the structure of spatial movement. This chapter extends the

techniques we have developed so far to three additional areas. First, these methods can

be applied to the analysis of many other works by Stockhausen himself. Second, it is

possible to use the methodology to analyze recent spatial music by other composers

which is similar in the treatment of the spatial domain to Stockhausen’s music. Third, it

is possible to develop a spatial structure in additional works which have certain elements

of spatialization, but for which no specific or overt scheme exists. In this chapter, we

will investigate each of these three areas in turn, with the goal not only to show how

diverse spatial analysis can be, but also to open up new areas and ideas in a field which is

still quite young.

To conclude our study, we will survey some technical innovations in spatialization

available to contemporary composers. Just as the limitations of analogue studios affected

Stockhausen’s spatialization techniques, spatialization in today’s music has particular

characteristics formed mostly thanks to the digital techniques used to create it. Much

modern spatialization owes its existence to decades of research in human perception,

algorithm development, and ever-increasing computational processing power. By

understanding better what techniques are available to composers today, we will identify

specific issues that suggest routes for future research.

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5.2. Other Stockhausen Works

As indicated in Chapter 1, nearly all works of Stockhausen have a specific spatialization

scheme, or are intended to be spatialized through some kind of amplification or “sound

projection”. Many, if not most of these works lend themselves to spatial analysis using

the methods we have developed in Chapter 2. In the following section, we will lay out a

course of analysis for several works that seem to offer the most interesting questions

regarding spatial movement and its connection with compositional structure.

5.2.1. KONTAKTE (1958-60). We briefly described the basic spatial idea of

KONTAKTE in §1.2.1. According to Stockhausen, KONTAKTE includes “six different

forms of spatial movement” realized in four speakers set up around the audience. In

addition, two soloists – one pianist and one percussionist – perform on stage, in front of

the audience. The composer’s extensive sketches to KONTAKTE, which are as yet

unpublished, show in detail how the work was composed.274 The sketches are valuable

since little of the information found within them can be gleaned from the published

scores for the piece.275 A cursory examination of the sketches shows that the relationship

between space and other musical domains in KONTAKTE is unlike of the works we

analyzed in Chapters 3 and 4, OKTOPHONIE and LICHTER-WASSER.

Stockhausen’s sketches for KONTAKTE suggest that he thought of space in an

uncharacteristically close relationship with the other musical parameters. Moreover,

Stockhausen appears to have composed the spatial movements at the same time as other

parameters, which is unusual for him. For example, Stockhausen assigned series to

musical dimensions including intensity, form, register, speed, and instrument, all on the

274 The sketches for KONTAKTE are available for study in the Stockhausen-Archive in Kürten. Theyconsist of 611 pages grouped in ten bundles.

275 While the new 2008 edition of the “Realizationpartitur” for KONTAKTE sheds some light on thespatialization, it lacks the detail present in the sketches. It also seems to have been assembled ratherhastily, lacking the detailed translations and commentary which are often useful in Stockhausen’s otherscores.

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same sketch-book page.276 “Speed” (“Geschwindigkeit”) is the most directly analogous

idea of spatial motion.

In another sketch, Stockhausen developed his ideas concerning the speed and the shape of

the sounds which move around the audience.277 In addition to the other musical

parameters, there are rows labeled “Geschw.” (“Geschwindigkeit”) and “Raum”

(“space”). As a result of seeing spatial information just above other material pertaining

to pitch and rhythm, we are tempted to look for relationships between these different

aspects of the piece. But before any such relationships might be considered, we must

penetrate a special spatial “shorthand” that Stockhausen developed to represent the

spatial movement in this sketch. This shorthand indicates a specific trajectory that

sounds follow as they move around the room. Sequences of spatial moves are

represented by tetrahedrons.278 Two key sketches, reproduced in Example 5.2.1a and

5.2.1b, show this notational shorthand, which is only used in sketches to KONTAKTE.279

276 KONTAKTE sketch III/30.277 KONTAKTE sketch III/16.1.278 It is possible that Stockhausen got the idea for using tetrahedrons to represent spatial movement from

Schaffer and Henry, who in 1950 devised a way of routing five channels of sound to four loudspeakers.The loudspeakers are arranged in a tetrahedral formation. See Barrett 2007, p. 242. The tetrahedronsused to represent the spatial motion in KONTAKTE also bear a resemblance to Walter O’Connell’sgraphical methods of representing all-interval class tetrachords (O’Connell 1968, p. 53ff). Although itseems unlikely that O’Connell was influenced by Stockhausen’s novel use of tetrahedrons since hiswork predates Stockhausen’s, the way that musicians as diverse as Schaffer, Stockhausen andO’Connell all used the same abstract geometrical structure in diagramming musical relationships pointsto the complex relationship that existed between abstract pitch structures and shapes in real physicalspace that developed during the 1960s.

279 The sketch in question is reproduced by Heikinheimo 1972, p. 131. Heikenheimo explainsStockhausen’s shorthand: “The hearing of two consecutive sound-phenomena from the sameloudspeaker does not entail any change. This would have the transformation value ‘0’. The smallestchange for two consecutive sound phenomena is for them to be heard first from one loudspeaker andthen from two loudspeakers...If the first loudspeaker is used only in the first stage we have a largerdegree of change, or transformation, which receives the value 2. The next value, 3, signifies a sound-phenomenon presented first in one loudspeaker and then in three; value 4 signifies a sound transmittedthrough one and then through two other loudspeakers; value 5 signifies a sound transmitted from one tofour other loudspeakers (still with the first, of course); and value 6 signifies the sound phenomenontransmitted from one to three other loudspeakers (the fourth loudspeaker in the diagram is marked withan “×” in the center of the triangle). The transformation alternatives marked in the upper left corner ofthe diagram are calculated on the basis of the transformation beginning in one loudspeaker.”(Heikiheimo 1972 p. 129)

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Analysis of this sketch suggests that certain sounds had particular patterns of

spatialization, opening up the question of whether Stockhausen intended that a specific

location in space is associated with a particular sound or category of sounds. Many

similar sketches follow.280

No published analysis of KONTAKTE has yet analyzed the spatial elements in the

sketches or the score using quantitative methods. It also has yet to be determined the

relationship between the planned spatial movement in the sketches and its ultimate

realization on tape. Further study of the copious sketch material would shed considerable

light on the complex and quite innovative spatial language in this influential work.

5.2.2. Osaka Pieces. In §1.2.2, we summarized some of the conditions in the spherical

auditorium Stockhausen used during the 1970 Osaka world’s fair. Many works of his

were performed there including SPIRAL, HYMNEN, AUS DEN SIEBEN TAGEN,

KONTAKTE, CARRÉ, GRUPPEN, MIKROPHONIE I, PROZESSION,

KURZWELLEN, TELEMUSIK, and STIMMUNG.281 Some of these, such as

GRUPPEN, were played back on electronic tape. Analyzing the three-dimensional

spatial movements Stockhausen improvised at the mixing console would yield further

insight into his techniques of spatialization. As previously remarked, Stockhausen claims

to have created “horizontal circles, vertical circles, below-above, above-below. Or spiral

movements of all different loops”.282

Unfortunately any such analysis would be hampered by the fact that it seems there is no

definitive record of exactly what spatial structures Stockhausen improvised at the

console. If we knew the sequence of the shapes in a spatial improvisation, we could

calculate their size and speed based on the architectural plans for the spherical

280 These include, but are not limited to KONTAKTE sketches III/17, III/19.1, III/20.281 The daily performance schedules can be found in Texte 3, pp. 177-181.282 Cott 1973, p. 46.

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auditorium. But lacking such information, we could still estimate the possibilities by

examining the wiring diagrams,283 interviewing performers, and studying any possible

video footage from the actual performances that might survive.

Despite the difficulties involved in obtaining spatial data for the Osaka project, we could

at the very least gain a better understanding of whether different works were spatialized

in different ways. Did the various acoustic and formal structures in each work cause

Stockhausen to spatialize them differently? Or, did each work performed in Osaka have

a similar spatial structure, despite its different formal and pitch structure? Answering

these questions would help us better to understand how Stockhausen’s idea of

spatialization was linked to the formal structure of a work, or whether he used a

repertoire of spatial movements in a more general way, simply to “intensify” the

experience of listening.

5.2.3. HELIKOPTER STREICHQUARTETT (1992-93). At first glance, it would

appear that analysis of the spatial motion in this work poses challenges for reasons

similar to those examined in the previous section. In this extremely unusual piece, the

four instrumentalists of a string quartet perform in separate helicopters. The musicians

are also linked together electronically, so that they can precisely coordinate their parts

while they are in the air.284 The audience, seated in an auditorium, can see the musicians

through a video link, while the sounds of the rotors and string quartet are projected

around them through eight loudspeakers set up in a circle.

While the movements of the helicopters in the sky are not defined in the score, they do

283 The wiring diagram for Stockhausen’s uncompleted HINAUF-HINAB, which shows how the 50speakers in Osaka were connected to the mixing console, can be found in Texte 3, p. 163.

284 An important structural element of the way Stockhausen interpreted the Superformula segment in theHELICOPTER QUARTET is the almost constant use of a kind of “hocket” technique. Thus, musiciansin the helicopters must coordinate their rhythms very precisely, despite being in different helicopters.This expresses the basic idea of the work – communication and complex musical synchronization overlarge distances.

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have to be planned in advance, in order to conform to flight paths and to avoid venturing

into airspace used by other aircraft. But it is unlikely that the movements of the

helicopters in the air above the performance hall would be of great musical interest, since

the audience does not really perceive their flight paths while seated. The audience gets

only a vague sense of ascent at the beginning of the piece, and descent at the end.

Turning to the score, as we have in previous analyses, we find material which allows us

to make more definite statements about the role of space in this work. The spatialization

of the quartet and helicopter sounds around the seated audience in the hall is musically

relevant. The sounds of helicopter rotors and string quartet are blended together and

spatialized though the eight loudspeaker groups which surround the listeners. Ultimately,

this is the acoustic “experience” that the piece is intended to create. The spatialization

inside the hall, like in Osaka, is essentially an improvisation created at the mixing

console.

Multi-track tapes exist of helicopter and quartet sounds. The spatialization on these tapes

can be inferred by analyzing the peaks and troughs in a simple waveform analysis.285 In

fact, Stockhausen took the unusual trouble to analyze and print the waveforms for the

entire quartet in the score, suggesting that he might have imagined such an analysis

would be possible – and, potentially useful. With careful examination of this waveform

analysis, the improvised spatialization of this work could be interpreted. With this

analysis, we would better be able to characterize Stockhausen’s improvisational practice

in the spatial domain.

5.2.4. COSMIC PULSES (2007). In §1.2.4, we described the basic spatial setup in one

of Stockhausen’s final works, COSMIC PULSES. In brief, eight speakers are set up

around the audience in a circle. The loudspeaker layout is given in Example 5.2.4a.

Through the loudspeaker array, loops of synthesized sound circulate around the audience

285 Thomas Manfred Braun did just this in his analysis of HYMNEN. Braun 2004, pp. 57-58.

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at various speeds. Like OKTOPHONIE, which we analyzed in Chapter 3, COSMIC

PULSES is so strongly oriented towards spatial movement that the actual electronic

sounds themselves are not as interesting as some of Stockhausen’s other synthesized

works. Unlike OKTOPHONIE, a superdense musical texture results from the spatial

overlapping of many similar electronic sounds in COSMIC PULSES. The texture

overwhelms the listener to such an extent that the individual trajectories of each sound

may become lost in hearing. However, Stockhausen subsequently composed a series of

works, each of which is based on three textural layers of COSMIC PULSES, presumably

in an effort to clarify the texture.286

For the time being, we will not be concerned with the actual sounds in the loops

themselves. Instead, we will examine the 241 spatial pathways around the eight

loudspeaker groups.287 These 241 paths are grouped into 24 layers. Each layer includes

between 8 and 12 paths.288 The paths were written out numerically in a series of four

sketches. Stockhausen then drew the pathways around the space in his usual “box

diagram” format in a series of three additional sketches. Example 5.2.4b shows how

paths 1 and 2, which were presumably first worked out numerically, were then converted

to box diagrams. The box diagrams also give information which relates to the amount of

time each path takes to complete one cycle. Although almost all of the paths are

different, a few are the same.289

Since COSMIC PULSES was still a new work for Stockhausen at the time of his passing,

286 COSMIC PULSES, the 13th hour of Stockhausen’s last work-cycle KLANG, gave rise to eight works,each of which is based on three layers of the “parent” work. These pieces, some of which have not yetbeen performed, are called HAVONA, ORVONTON, UVERSA, NEBADON, JERUSEM, URANTIA,EDENTIA, and PARADIES.

287 Stockhausen may have used the number 241 to memorialize the 241 United States service memberskilled in the Beriut bombing on 23 October, 1983. Stockhausen’s comments regarding the September11 World Trade Center attacks, which were widely misinterpreted by the international press, might haveprovided the impetus for this subtle musical reference. Thanks to Dave Headlam for suggesting thisidea.

288 Layer 24 is the only one that includes 8 paths, while layer 2 is the only one with 12 paths.289 For example, paths 192 and 205 trace the same path through space, and each lasts one second.

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he was unable to provide the kind of in-depth analysis of his compositional methods that

exist for many of his other works.290 However, after examining the spatial characteristics

of his earlier works, we can make hypothesizes about how he created the spatial motion

in this late piece. Although it might seem that Stockhausen used a mathematical formula

for systematically deriving the 241 paths through space, this appears to be less likely after

examining them, since some of the paths are same. It is more likely that he improvised

the pathways like so many of his previous spatial structures. Whatever characteristics

Stockhausen was looking for in his paths are, for the time being, elusive.

Preliminary analysis of the distribution around the hall of all 1928 moves291 (241 × 8)

shows that some loudspeaker groups are used more often than others. For example,

loudspeakers 3 and 8 are used only 226 times each, but loudspeaker 4 is used 268 times.

The number of times each loudspeaker group is used can be seen graphically in Example

5.2.4c. From our data, we can conclude that sounds emanate most often from the front

pair of loudspeakers (groups 4 and 5). This would make sense from a psychoacoustic

perspective: human spatial localization ability is most acute in the area directly in front

of us.292

Our brief analysis unfortunately does not take into account the speed at which the loops

move around the listeners. Applying more of the methodology developed in Chapter 2

would allow us to do this. If we add this aspect into the analysis, we may find that the

amount of time sounds are projected from each loudspeaker group may emphasize

different groups in a way that is different from the results obtained by simply tallying up

the number of times each group is used.293 However, limitations of time and space

prevent us from doing this more detailed analysis. Another question that should be

290 However, analyses by Richard Toop, Joachim Haas and Gregori Garcia Karman helped to elucidatemany of the details pertaining to the shapes of the paths during the 2008 Summer Stockhausen Courses.

291 The year 1928 was, of course, the year Stockhausen was born.292 Blauert 2001, p. 41.293 We analyzed spatial motion from both perspectives in LICHTER-WASSER. See Examples 4.6.1a and

4.6.1b.

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considered is the character of each of the 24 spatial layers. Did Stockhausen combine

paths on purpose so that when nine, ten, eleven or twelve of them were heard together in

a layer, the layer itself would have certain distinctive characteristics?

Concerning this piece, Stockhausen remarked that “If it is possible to hear everything, I

do not yet know – it depends on how often one can experience an 8-channel

performance”.294 Further analysis of COSMIC PULSES could help us to find ways of

listening to this difficult piece, while uncovering more layers of structure which would

enrich our understanding of Stockhausen’s use of space.

5.3. Xenakis and Terretektorh

5.3.1. Background. Terretektorh (1965/66) is a work for 88 musicians written by the

Greek composer Inneas Xenakis (1922-2001). The seating arrangement of the work is

extremely unusual. As shown in Example 5.3.1a, 85 instrumentalists are seated among

audience members, while three percussionists play towards the edge of the circular space.

The work was first performed in 1966 at the Royan Festival in France, under the

conductor Hermann Scherchen.295 A circular Casino hall was available for the

premiere.296 Although Xenakis favored a circular hall, he later allowed the piece to be

performed in a more traditional rectangular space.297 The “ideal dimensions” for a

circular hall, according to Hofmann, are a space with a radius of 7 meters and area of 154

m2.298 Approximately 63 m2 is needed for the musicians; 91 m2 is left for the audience,294 Stockhausen 2007a (CD 91), p. 7.295 Harley 2004, p. 46.296 Hofmann 2008, p. 81.297 The English premiere, in Oxford’s Town Hall, was rectangular, with dimensions of approximately 23.8

× 12.7 meters. Thus is slightly smaller than the 14 meter diameter Xenakis intended. Dennis reviewedthe Oxford performance favorably (Dennis 1967).

298 “A large ball-room having (if it were circular) a minimum diameter of 45 yards [sic] would serve indefault of a new kind of architecture which will have to be devised for all types of present-day music...”(Xenakis 1992, p. 236). The 45 yards must certainly mean 45 feet, in which would be equal to about 14meters. Thus, Hofmann’s claim of a radius of 7 meters is correct.

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which could include 290 seats or 546 standing places.299 The dimensions of the space for

Terretektorh are considerably smaller than for Stockhausen’s LICHTER-WASSER, even

though there are almost three times the number of musicians that have to play in

Xenakis’s piece. The loser is the audience; in Stockhausen’s work, there was room for

728 seats (see §4.3.4); there is considerably less space for the audience which presumably

stands during the performance of Terretektorh.

What were Xenakis’s reasons for writing a piece for such an unusual and – in many ways

– impractical spatial arrangement? The composer wrote that

The scattering of the musicians brings in a radically new kinetic conception ofmusic which no modern electro-acoustical means could match... The speeds andaccelerations of the movement of the sounds will be realized, and new andpowerful functions will be able to be made use of, such as logarithmic orArchimedean spirals, in-time and geometrically. Ordered or disordered sonorousmasses, rolling one against the other like waves... etc., will be possible.300

Unlike Stockhausen’s LICHTER-WASSER, sounds in Terretektorh don’t “jump” from

one place to another. Instead, they move – like the sounds in COSMIC PULSES –

around the space like a “searchlight”. This is because each instrumental group

responsible for articulating a part of the spatial shape has a crescendo and diminuendo,

with a loud segment at the middle articulating the central point of the sound. This is

similar to techniques used in spatializing electronic music around listeners and is a clear

example of the cross-fertilization between electronic and acoustic music in the 1960s.

In addition to their normal instruments, each musician has a maraca, a whip, wood-block,

and a small siren-whistle. These are used to create “clouds” of percussive sound which

interrupt the musical texture at certain points in the piece. In an often cited quotation,

299 Hofmann 2008, p. 81.300 Xenakis 1992, p. 236. It can hardly be said that the concept of the Archimedian spiral is “new”, but

Xenakis’s point is that this is not the only way of ordering elements of spatial music – a point which istaken in the current study to mean that many other more complex structures are also possible.

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Xenakis relates how he conceived of the orchestra in this work as a kind of “particle

accelerator” of sounds:

Terretektorh is thus a “Sonotron”: an accelerator of sonorous particles, adisintegrator of sonorous masses, a synthesizer... The orchestral musicianrediscovers his responsibility as an artist, as an individual.301

5.3.2. Analysis of mm. 1-74 of Terretektorh. In the first 74 measure of the work, the

pitch E moves around the space in a series circles. The pitch is played only by string

instruments who, thanks to their ability to create long smooth crescendos and

diminuendos, create the illusion of gradually passing the sound from one group of players

to another. Twice, the motion is overlayed with tutti maraca rattles. An early reviewer of

Terretektorh was able to follow the circles that moved around the room, even though the

piece was performed in a rectangular hall.302 Maria Harley noted the general shapes of

the motion and the rate of motion around the outer edges of the circular space in the first

74 measures, and parsed her data into eight categories of events. We will call these eight

categories “structures” 1-8.

1. mm. 1-9; circle2. mm. 8-24; Archimedean spiral, acceleration3. mm. 23-34; Archimedean spiral, deceleration4. mm. 32-45; hyperbolic spiral, acceleration5. mm. 45-47; angular motion in group H6. mm. 51-60; logarithmic spiral, acceleration, new direction7. mm. 60-65; similar logarithmic spiral with steeper curvature8. mm. 65-74; six logarithmic spirals with increasing curvature, increasing

acceleration of movement.303

The spirals Xenakis uses – Archimedean, hyperbolic, and logarithmic – each are

301 Xenakis 1992, p. 237.302 However, it was easier to hear the revolutions when standing within the orchestra: “This kind of linear

circular movement with the instruments of a given kind not grouped together but situated at regularspacings around the instrumental arena could not nearly so easily be assimilated from the gallery, sothat the second performance was for me, having moved from my former position on the floor, definitelyless effective.” Dennis 1967, p. 27.

303 Harley 1994, p. 305.

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characterized by different mathematical formulas, and are shown in Example 5.3.2a.

Harley’s analysis of these shapes led her to draw the graph reproduced in Example

5.3.2b. However, it is crucial to note that the spatial motions Harley identified are not

literally present in the space. Rather, the “spirals” describe the rate of acceleration or

deceleration of the sounds. All the sounds, with only one exception, move in circular

motion around the center point of the space.

Much like we did in LICHTER-WASSER, we can estimate the size of the space in

Terretektorh using Hofmann’s and Xenakis’s suggestions, taking a diameter of 7 meters

to represent the ideal dimension of a circular space. Then, we can identify the

coordinates of each instrument. This analysis is shown in Example 5.3.2c.304 While these

coordinates would be useful for precisely analyzing spatial motions later in the work, it is

more practical to find the central point of each slice of the space, since Xenakis typically

used two or three instruments in each wedge for each element of his spirals in the first 74

measures. Example 5.3.2d shows the central coordinate of each slice of each wedge. It

also shows the midpoint between some areas, since Xenakis often coupled two slices of

the space together.

We know that each page of Xenakis’s score equals 30 seconds, so a half-note ideally lasts

one second. With this information, we can calculate the time-point where each

crescendo/diminuendo structure peaks by examining Xenakis’s score notation. In

Examples 5.3.2e and 5.3.2f, we estimate the time-points at which sounds peak in volume

at each place in space. Also, the density of orchestration in each wedge is indicated by

shading different slices of the space, though this does not significantly affect our

present analysis.

Finally, we can combine the coordinates of each area in space with the time-points we

304 Among other things, this shows that the arrangement of musicians is asymmetrical. We will return tothis in §5.3.3.

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just found. These calculations are shown in Examples 5.3.2g-h. They show some

extraordinarily rapid movements. In segment 8e, the calculated speed is an astonishing

253 m/sec. Partly as a result of the very short rhythmic durations involved, Xenakis

appears to have reached the limits of his notational precision around structure 8c.305 Past

this point, the speed of motion can be estimated, but not precisely calculated. When the

motion is so rapid around the space, the musicians themselves must be relied upon to

coordinate the spatialization. Thus, Harley’s diagram (previously shown in Example

5.3.2b) is an idealization of what is supposed to happen in structures 8c-8f, and does not

reflect any rhythmic notation in the score.

5.3.3. Comparison of Terretektorh with Stockhausen’s LICHTER-WASSER. By

measuring the speeds of spatial music in LICHTER-WASSER and Terretektorh, we can

make several comparisons which allow us to begin to identify stylistic traits in the

treatment of the spatial domain. Although the top speeds in Terretektorh are faster than

in LICHTER-WASSER, the average speeds are similar. There is less variety of spatial

movement in Terretektorh than in LICHTER-WASSER, since the movement in

Xenakis’s introduction is almost always in a circle. Stockhausen’s arrangement of

musicians is symmetrical, but Xenakis’s is not – though it is close. His “quasi-stochastic

sprinkling”306 of musicians would create many irregular shapes were we to follow

through and analyze further sections of spatial movement, whereas Stockhausen’s

symmetrical arrangement allows for more regular shapes to be brought into play.

Whereas Stockhausen “jumps” his melodies from one instrument to another, Xenakis

blends one into another, utilizing the technique Stockhausen employed in GRUPPEN and

COSMIC PULSES. This probably makes the effect of continuous space much more

305 Obviously, Xenakis could have notated his music more precisely. However this would probably havenecessitated very small rhythmic divisions, which would not fit on a page that was calculated to holdexactly 30 seconds of music. Xenakis’s priority was clearly not to notate his final few accelerationswith utmost precision, rather to create a score that was more readable.

306 Xenakis 1992, p. 236.

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directly perceptible in Terretektorh. The audience might not at first hear the complex

spatial movements in LICHTER-WASSER as a result of the absence of any

blending technique.

Ultimately, the two composers’ aesthetic goals are different, but they use similar means

to achieve them. Xenakis essentially wanted a listener to think he or she is “either

perched on top of a mountain in the middle of a storm which attacks him from all sides,

or in a frail barque tossing on the open sea, or again in a universe dotted about with little

stars of sound, moving in compact nebulae or isolated”.307 On the other hand,

Stockhausen wanted to express the awe of contemplating the beauty and complexity of

our solar system, with all its simultaneously rotating moons, planets and other objects.

Although their aesthetic goals are different, the means by which Xenakis and

Stockhausen composed in the spatial domain have many similarities.

5.3.4. Conclusion. The techniques we have used to measure the rate of spatial motion in

OKTOPHONIE and LICHTER-WASSER can be used to calculate spatial motion in

other contemporary music. Data gleaned from score analysis can then be used to

compare composers’ techniques in the spatial domain. In works where space plays such

a prominent role, the data we have collected and the conclusions we have drawn are

critical to the makeup of the work; in works where space is not the principal focus of

interest but still plays a part, spatial data can still add richness and subtlety to a

musical analysis.

5.4. Spatialization in other musical repertoires: Tallis’s Spem in alium

5.4.1. Premise. Although our analytical techniques have so far been applied to

spatialization in contemporary music, the methods we have used have quite deliberately

remained general in scope. Might we be able to engage music of other historical periods

307 Ibid., p. 237.

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or cultures using these techniques?308 To better explore the limits of our methodology,

we will investigate a work of Renaissance polyphony in the following section. But

instead of analyzing the spatialization scheme provided by the score as our starting point,

we will work in the opposite direction: starting with the score, we will develop a spatial

scheme from it. In this way, we will propose a way of manifesting a complex musical

structure that seems to be ideally expressed through spatial means.

5.4.2. Development of Spatialization in Western Music. Evidence of spatialization can

be found in Greek ampitheaters. Typically, bronze or ceramic urns were placed around

the Greek stage in various configurations to resonate with the singers’ voices. The urns

helped to amplify the sound on stage. Written evidence of this practice can be found in

the writings of the Roman architect Vitruvius.309 Physical evidence of resonating urns or

cavities can be found in Medieval churches from the British isles as far east as Russia.

Unfortunately, most urns were melted down for cannon in the Renaissance; however, the

holes or niches that held them in place still exist and their relative size can be estimated

based on measurements. Research into the placement of the urns in Greek ampitheaters

and Medieval churches suggests that urns that resonated at specific frequencies were

arranged according to coherent spatial plans.310

Evidence of spatialization can be found in many different traditions of early music. In

the early Christian era, the practice of antiphonal psalm singing developed. This

technique of performing liturgical chant allowed for the alternation between two choirs

which could have been spatially distant from one another. As polyphony developed in

the Middle Ages, methods of spatialization did not lag far behind. Research has

suggested that the widespread medieval practice of hocket developed out of the

improvised singing of Gregorian chant melodies. However, scholars have neglected to

308 Blesser and Salter 2007 suggest a cross-disciplinary approach towards spatial research.309 Lewcock 2001, pp. 81ff.310 Arns and Crawford 1995.

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point out that hocketing adds a distinct element of spatial interest to Medieval

polyphony.311 The Renaissance technique of cori spezzati (“broken choirs”) is a later

development of experiments with composing for two or more independent choirs in order

to create special effects.312 By splitting choirs, spatial relationships could emerge in the

polyphonic texture even if the choirs were not separated from each other by any great

distance. This practice was so common that by the time of Gabrieli, almost all Italian

composers were actively using cori spezzati. While the technique of cori spezzati

originated in Italy, English composers developed the technique in ways that suggest even

more sophisticated control over the spatial design of their music.313

In the Baroque, Classical, and Romantic periods, composers’ attention generally focused

on elements of music such as harmony, orchestration, and form. While seating plans in

orchestras occasionally allowed for antiphonal effects in orchestral music, overtly spatial

music such as Berlioz’s Requiem were exceptional. In other works of the nineteenth

century, off-stage soloists or ensembles create an added feeling of distance or separation

between the music and the audience, or add an extra dimension to the dramatic effect of

opera.314 While some writers suggest that spatialization plays a role in large orchestral

works of the late nineteenth- and early twentieth centuries,315 the technical breakthrough

of magnetic tape and the loudspeaker spawned an interest in creating more abstract and

sophisticated structures in space during the 1950s.

311 Dalglish 1969, Sanders 1974, and Dalglish 1978. Although Dalglish writes that “vocal performancepractice of the late Middle Ages was one characterized by fantasy, variety and color” (Dalglish 1978, p.20), neither he nor the Medieval writers he quotes realize the implicit spatial aspect in the practice ofhocketing.

312 Willaert pioneered the technique of cori spezzati in the mid-sixteenth century and Zarlino defined theterm in Le istitutioni harmoniche. Bryant 1981; see also Arnold and Carver 2008.

313 Rastall argued that spatial design is important in works of English Renaissance consort music. Hesuggested that in works printed in a “Tafelmusik” layout, a spatial structure is implied. Composers suchas Dowland, Shepherd, Gombert, Tye and Byrd all printed music in this format. See Rastell 1997.

314 The sonic effect of distance in Mahler, noted by Adorno in the beginning of the first symphony, ismore overtly exploited in the fifth movement of the second symphony (Adorno 1991, p. 4). InGötterdämmerung, Wagner intensified the dramatic effect by employing several off-stage stierhorns.The technical issues involved with spatializing the Decca recording are vividly described in HumphreyBurton’s 1965 documentary “The Golden Ring”.

315 Brelet 1967.

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From this brief historical sketch, we can see that the idea of spatialization is not new. In

fact, it goes back to the very origins of the Western musical tradition. Indeed, the

historical foundation on which contemporary spatialization is based is so widespread and

rich that it seems likely that earlier repertoires of music may be amenable to the

techniques of measurement and analysis we have developed and applied to

Stockhausen’s works.

5.4.3. Tallis’s spem in alium. Thomas Tallis’s (c. 1505 – 1585) famous 40-part motet

Spem in alium is a well-known example of a remarkable and effective use of multiple

choirs in the English Renaissance.316 Using some of the techniques we have applied to

Stockhausen’s spatial music, we can analyze a spatial arrangement of the choirs in

Tallis’s work. This will allow us to evaluate the effectiveness of one particular spatial

design, while also learning something of the spatial structure of the work itself.

5.4.4. Methodology and Spatialization. Tallis’s choral pieces raises many questions

related to the use of space. Do we focus attention on each choir as a self-contained

spatial source, or do we analyze particular registral lines? The tradition of cori spezzati

suggests that each choir should be treated as one independent sound-source complete in

and of itself.317 This is the track we will take in the following analysis. However, when

listening to Spem, we are often most aware of the highest note in the composite texture.

Future analyses may interpret the spatial structure in connection with register. In both

methodologies, sounds typically “jump” from one place to another and are not really

316 Asked to compose a work rivaling Striggio’s 40-part Ecce beatam lucem, Tallis responded – at theencouragement of the Duke of Norfolk – with a work that utilizes eight choirs of five voices each. Thefirst performance of Spem was probably in the late 1560s in the Arundel house, which stood betweenthe Strand and the Victoria Embankment in present-day London (Stevens 1982, p. 175ff.) The room,called the “Long Gallery”, was large enough to provide space for all the singers plus a number ofinstrumentalists. A drawing of the long gallery at Arundel can be found in Parry 1980. Coope 1986describes the more general physical characteristics of long galleries in Elizabethian England.

317 In Tallis’s spem in alium, each choir generally sings in its full five-part texture. The only notableexception occurs in the soprano’s solo in choir 7 in mm. 50-56.

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meant to move continuously.

Although it is not known exactly how the eight choirs were arranged in Tallis’s day, we

can infer by the shape of sixteenth-century long galleries what the original spatialization

of Spem might have been. The choirs, along with whatever instrumentalists might have

doubled the parts, were probably arranged along a line down one side of the room.

However, we do not know exactly whether the choirs were arranged in a row (I next to II,

etc.) Whittaker, a noted English musician in the first half of the 20th century, mounted a

performance of Spem in 1929. He described his reactions to the work and his

spatialization in a brief published report.318 He also drew a diagram showing exactly

when each choir enters and how long it sings for. This useful diagram is shown in

Example 5.4.4a.

Spatialization emerges as a virtual necessity in Spem because of the great density of the

contrapuntal texture. Spatialization also adds an extra dimension of musical interest

which would substitute for the overall lack of harmonic fluctuation in the motet.319 In his

scheme, Whittaker clumped his choirs together at one end of the performance space.

This arrangement would tend to muddle the texture in a way that fails to do justice to

Tallis’s choral writing. Whittaker’s spatialization is shown in Example 5.4.4b.

Other spatial arrangements which take advantage of Tallis’s specific compositional

structure are possible. During the first 35 measures of Spem the choirs enter one by one,

and generally stop singing when a third choir enters. This sequential opening strongly

suggests that arranging the choirs in a circle around the audience would create an

effective spatial effect. Lending support to this notion is the complex antiphonal passage

in mm. 87-105, where the effect of alternating choir groups across the space would be

318 Whittaker 1940, pp. 86-89.319 The motet essentially circles around the triads of G major/minor, D major/minor, A major/minor, and

includes a certain amount of C major, F major, B-flat major and e minor.

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heightened in a circular arrangement. Measures 87-105 are diagrammed in great detail in

Example 5.4.4c. After examining the spatial structure of this passage, it is clear that

choirs 1 and 2 are paired, as are 3 and 4, 5 and 6, and 7 and 8. This analysis emphasizes

the fact that Tallis tends to pair choirs that would be next to each other in a circular

arrangement.320 One surmises that Whittaker’s arrangement of choirs would do violence

to Tallis’s choral pairings. As our analysis suggests, an effective spatial arrangement for

the choirs in Spem is a circle, as shown in Example 5.4.4d.321

Constructing a spatial layout for the eight choirs can be aided by further examining the

way Tallis employs the groups of singers. There are essentially three different

techniques. The first is a staggered or sequential entry, such as in the opening. This

texture is always contrapuntal. The second technique is a block texture where all eight

choirs sing together. It is treated either contrapuntally or homophonically. The third

technique makes the most direct reference to the cori spezzati tradition in its antiphonal

treatment of choir groups. In this technique, the texture is almost always homophonic.

Tallis thus mediates between textural and spatial extremes, and also associates certain

contrapuntal techniques with spatial techniques. The tutti choir is usually reserved for

climactic moments, or when a particular word of text (such as “respice”) is emphasized.

5.4.5. Analysis. Following our observations from the previous section, we can divide

the spatial structure of Spem into four large sections. Each section, except for the first,

begins with all 40 parts singing together. The four sections are diagrammed in Examples

5.4.5a-b. Although only the main spatial structures are shown, they correlate closely with

Whittaker’s analysis of movement.

Assuming the spatial setup is a circle, section 1 is dominated by two slow rotation

320 However, the way in which notes are passed around within the paired choir groups adds complexity tothe treatment of space.

321 Evidence from Rastall’s study of Elizabethan consort music indicates that arranging musicians in acircle (around a table) was not at all uncommon at the time.

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sequences. In the first spatial sequence, music moves around the audience in a clockwise

direction. After moving clockwise almost one complete time around the circle, the

spatial motion is interrupted briefly by a “flip”. Here, two choirs unrelated to the

clockwise movement briefly sing. This is followed by a brief 40-part tutti section. A

counterclockwise rotation then follows and slowly swings the music back to the same

point where the piece began – choirs 1 and 2. A 40-part tutti announces the second

section, which begins at m. 69 and lasts only to m. 85. This section features paired choirs

whose spatial relationships can be described as flips across the circle or rotations around

it. Section 3 begins with another choral tutti, and, in exploring a more elaborate cori

spezzati technique serves as evidence of more “flip” movements. The fourth and final

section, which again starts with a choral tutti, draws a close parallel with section 2.

Measures. 110-121 and 81-85 pair the same choirs and same spatial motion, drawing

long-range connections in the spatial domain. The motet closes with a full tutti section

on the text “humilitatum nostrum”.

If choirs are set up in a circle around the audience, we find that the movement of musical

material around the space can be modeled by a set of rotation and reflection (flip)

transformations. These sixteen transformations form a Dihedral group of order 8. The

operations, as well as the group table can be found in Example 5.4.5c, which is the final

graphical example of the dissertation.

5.4.6. Conclusion. While there is no strong evidence to support the idea that the choir

groups were arranged in a circle around the audience in Tallis’s time, we find that the

composer’s strategies for dealing with eight choirs in Spem can be interpreted in a way

that brings us surprisingly close to the spatialization employed by composers in the late

20th century. Spatializing Spem in the way described not only clarifies the texture, but

also articulates a layer of structure – spatial structure – in a work which seems to call for

it. Spatialization makes the experience of hearing the piece richer and more rewarding

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perceptually. We also can find a wonderful connection between modern performance

practice and Renaissance practice in the fruitful combination of old and new. Other

works of Renaissance polyphony, while perhaps not as sophisticated as Tallis’s, could

also conceivably benefit from careful analysis of the advantages or disadvantages of

spatialization in performance.322

5.5. Electronics and Spatialization

5.5.1. Role of Electronic Hardware in Spatial Music. In music since 1950, electronic

apparatus has played the most important role in developing spatial music in the history of

Western culture. These technical developments in turn have inspired composers to

experiment with new configurations of instrumentalists in performance spaces, initiating

a kind of feedback loop from which both electronic and instrumental music has

developed. But in order to get a more complete picture of how spatialization worked in

its early years, we must understand better the electronic devices that were in use during

that period. Their limitations define the boundaries of what was possible, while spurring

technicians and composers to push beyond the limits and invent new procedures.

Learning about these electronic techniques is difficult because few of the devices survive,

having generally been relegated to the trash heap whenever newer technologies replaced

them.

One aspect of Stockhausen’s scores to which we have given too little attention is the

many wiring diagrams he published. Stockhausen was both a “composer” in the

traditional sense of the word, and a “sound-engineer”; the links between these two roles

are vividly represented by his wiring diagrams. Though we briefly alluded to

Stockhausen’s diagrams in connection with the massive speaker array that was installed

322 Other works of this period that profit from spatialization include the 40-part Missa Salisburgensis(attributed to Biber), Josquin’s 24-part Qui habitat, Striggio’s 40-part Ecce beatam lucem, Gabrieli’s16-part Exaudi me Domine, and various works from the Eton Choirbook by Browne and Wylkynson.

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in Osaka, they contain a wealth of information that has received little, if any attention in

the published literature. Yet the fact that they are included in so many Stockhausen

scores is a testament to how important the composer believed they were. Research into

Stockhausen’s technical means, and the way in which he employed them, would no doubt

add depth to our understanding of his spatialization practices.

5.5.2. Software for Spatialization. We summarized and described one popular

approach to electronic spatialization, called ambisonics, in §2.2.1. Engineers at the

Institute for Computer Music and Sound Technology in Zurich, Switzerland have

developed software that allows ambisonic spatialization for one of the most popular

composition packages available, Max/MSP. Their software includes a graphical user

interface called an “ambimonitor” which allows both real-time control of spatialization

and out-of-time, algorithmic control. It also allows sounds to be spatialized in three

dimensions.323 These tools are appealing from the perspective of the composer since they

are readily available and make possible complex spatial patterns that would be difficult to

achieve otherwise.

One of the most important electronic studios in the world is IRCAM (“Institut de

Recherche et Coordination Acoustique/Musique”) in Paris, France. Engineers at IRCAM

have developed software called Spat (“Spatialisateur”) which allows composers to

spatialize sounds regardless of the type of method used to synthesize them. Spat allows

users also to specify room acoustics, setting various characteristics such as reverberation

and reflection. Spat is versatile enough to be used in real-time environments, as post-

production, or in “virtual reality” situations. Like the software designed in Zurich, Spat

will also work with Max/MSP.324

323 http://www.icst.net/files/ICST_Ambisonics_ICMC2006.pdf. Accessed on 2 November 2008.324 http://www.ircam.fr/745.html?&L=1&tx_ircamboutique_pi1[showUid]=199&cHash=8094c8f4ed.

Accessed on 2 November 2008.

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Other engineers have developed less ambitious, but useful tools for spatialization as well.

Karen Bryden, at the Communications Research Centre of Canada, developed simple

software to spatialize sounds in a 2-dimensional sound field, heard through headphones.

This was accomplished by taking measurements made of dummy heads at the MIT Media

Lab in Cambridge, Massachusetts and utilizing an HRTF (“Head-Related Transfer

Function”) to to calculate the spectral distortions needed for the illusion of spatial

movement.325 Even though Bryden’s software is less sophisticated than that developed in

Zurich and Paris, it shows that there is great interest in spatialization among audio

engineers, and that a variety of software is available for use by the general public.

In COSMIC PULSES, Stockhausen employed a piece of software called the OCTEG

(“OCTophonic Effects Generator”), which was an outgrowth of the earlier hardware-

based QUEG (“Quadrophonic Effects Generator”). Stockhausen used the QUEG in

SIRIUS and in OKTOPHONIE. The OCTEG was created at the Experimental Studio for

Acoustical Art (formerly known as the Heinrich Strobel Institute) in Freiburg, Germany,

when it was under the direction of André Richard. Although the OCTEG is not available

for the public to use at present, it is yet another example of a sophisticated piece of

software developed at a professional sound studio for use in the spatialization of

contemporary music.

5.6. Concluding Remarks

5.6.1. Directions for Further Study. As we find ourselves towards the end of the first

decade of the twenty-first century, it could be argued that “the novelty of the early

decades – of listening to strange sounds emanating from loudspeakers – has passed”.326

Although audiences of contemporary music may have indeed become inured to “new

sounds emanating from loudspeakers”, it could also be argued that the experience of

325 http://web.ncf.ca/aa508/Software/spatial. Accessed on 2 November 2008.326 Barrett 2007, p. 232.

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space in music still has the potential to shock and delight. The “drastic change” in

spatialization that occurred during the 1950s327 still remains relatively unexplored terrain

in composers’ vocabulary in comparison to pitch, rhythm and timbre. Yet, as we have

seen, composers have created complex compositional structures not only by spreading

musicians around (or within) the audience in various ways but also through sophisticated

electronic and computer techniques. Today, software allows for the creation of intricate

spatial designs even on a moderately-priced computer apparatus.

A program of further research should focus on four areas. First, more emphasis should

be placed on cognition. Unfortunately, navigating this enormous field is made more

difficult because of the extremely complex interrelated variables that room acoustics,

pitch, rhythm, and timbre introduce in spatial localization and perception. Still, the

question remains: how much spatial movement can listeners perceive in a composition?

In developing this area of knowledge, we should not become mired in minute technical

details, but rather seek to devise some general rules or guidelines by which composers

could know better what audiences would expect in certain acoustic environments with

certain sounds. These guidelines would allow a composer to make a rough estimate the

general percepts of an audience when listening to his or her composition. As we research

this question, it is necessary to consider architecture and other musical parameters as they

relate to spatial perception.

Second, more spatial compositions dating from circa 1950 to the present day could be

analyzed using the methods we have developed. As we have seen, a number of concrete

differences emerged when contrasting data gleaned from spatial works by Xenakis and

Stockhausen. These contrasts allowed us better to appreciate each composer’s individual

approach to space. How might these composers’ spatial compositions compare to music

by Edgar Varèse or Henry Brant? Or, how might they compare with other Stockhausen

works? Because of the simplicity and versatility of our analytical methods, they can be

327 Reynolds 1978, p. 181.

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applied to a great deal of spatialized music.

Third, a great deal of knowledge can be gleaned by examining technical data from

composers’ experiences in creating or synthesizing spatialized music. Stockhausen’s

works are an ideal place to begin, since he notated and published so many of his wiring

diagrams in scores. These questions can be answered through historical study of audio

engineering, which – though not typically a field studied by music theorists – can and

must play a greater role in understanding how the technical limitations influenced the

development of spatial music in the 1950s, 60s and 70s. It is not inconceivable that

composers’ experience with spatialization influenced their composition in pitch, rhythm,

or other musical domains.

Finally, works from the past could be brought into dialog with modern spatial

technologies. How might polychoral works from the centuries-old cori spezzati tradition

be performed today using modern techniques? While the integrity of the original work

must of course be respected, we can find new inspiration in these centuries-old works by

spatializing them in ways their composers never could have imagined. This dialog

between old and new would both enhance our appreciation for masterpieces of the past

while making them more relevant for today’s listeners.

5.6.2. Conclusion. Although the field of spatialization has been around for centuries,

composers, researchers and theorists have begun to develop this aspect of musical

experience in radically new ways only during the past few decades. Their efforts promise

to open up new realms of experience and enjoyment for the listener. They also enable

more sophisticated compositional structures which have yet to be fully appreciated or

even described. In this dissertation, I have proposed several simple and general methods

for measuring spatial motion as a first step towards enlarging our knowledge of past

compositions, and stimulating ideas for new ones. They have been applied to works of

168

Stockhausen and, all too briefly, to two other composers. The results have shown that a

great deal can be learned by measuring spatial motion in these compositions –

information that cannot be obtained through any other means. While telling us much

about these compositions that we did not yet know, my analyses open up a number of

questions which are relevant not only to music of the present and recent past, but for

works composed centuries ago. It seems clear that this field has an exciting future ahead

of it, for it already has an impressive and fascinating body of music behind it.

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Afterword: Spatial “Serialism”

The title of this dissertation, “Stockhausen and the Serial Shaping of Space”, contains the

word “serial”. “Serial music”, according to Paul Griffiths, is “a method of composition

in which a fixed permutation, or series, of elements is referential (i.e. the handling of

those elements in the composition is governed, to some extent and in some manner, by

the series).” Griffiths continues, “Most commonly the elements arranged in the series are

the 12 notes of the equal-tempered scale.” He adds that “[Babbitt, Boulez, Nono and

Stockhausen] sometimes extended serialism to elements other than pitch, notably

duration, dynamics and timbre.”328

Stockhausen claimed unequivocally throughout his life that he was a “serial” composer.

Yet in this dissertation we have not made much study of referential series of elements, or

the “serial number squares” that feature so prominently in many excellent analyses of his

works.329 Nor have we dealt with issues of pitch, duration, dynamics or timbre to any

significant degree. Instead, we have investigated spatial motion around listeners,

analyzing and quantifying the structures it creates. In the course of analysis, we found

that the shapes, densities and speeds of spatial motion have their own structures, which

are governed more by intuitive or improvisatory processes, not strict permutations of

series of elements. Does this mean that spatial motion is “not serial”?

I suggest that the answer to this question is no. Spatialization is Stockhausen’s music is

indeed “serial”, but the composer’s spatial structures often behave in a more stochastic,

or probabilistic way. As I have shown, the motion of sounds in space is such a crucial

part of Stockhausen’s compositional project that it cannot be neglected in favor of other,

328 Griffiths 2001, p. 116ff. Other definitions, for instance in Musik in Geschichte und Gegenwart, tend toemphasize parametricization in serialism more than Griffiths does, but agree in general with basicprinciples (Frisius 1994).

329 It might be argued that the “basic cycles” in LICHTER-WASSER are referential in some sense (see§4.1.3). Still, these cycles are not permuted in the same way that Stockhausen typically made use ofnumber squares and other serial ordering techniques.

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more familiar parameters such as dynamics, rhythm, pitch or timbre.330 The fact that the

structure of spatial movement in Stockhausen’s serial music behaves differently from the

structure of other musical aspects suggests that Griffiths’ definition could be more

nuanced.

By considering intuitive spatial structures along with more intellectual ordering

principles, we set up a dualism. The tension between more ordered aspects and more

improvisatory ones helps to mitigate the cold logic of mathematics in serialism with the

human and organic process of artistic creation. Stockhausen often masterfully negotiated

the boundaries between these two extremes, creating music that is on the one hand

logically structured, but on the other hand filled with unexpected twists and turns of

originality that break established patterns. While this dualism is present in Stockhausen’s

pitch and formal structures,331 it is experienced most directly in his spatial composition.

Integrating this dualism into a more nuanced definition of serialism will help us to

appreciate the variety of techniques and the complexity of thought that went into

designing these intriguing musical compositions. I hope that by examining the practices

of “serial spatialization”, I have added a wrinkle to future discussions of this music,

which continues to fascinate many perceptive listeners today.

330 As quoted in §1.1, Stockhausen once said that “instrumental and electronic spatial composition havebeen my artistic mandate since 1951” (Nauck 1997, p. 173).

331 This is particularly true of GRUPPEN. See Misch 1998a and 1999b.

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Scores, CDs, and Other Sources

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———. 2001d. HELIKOPTER-STREICHQUARTETT (score). Kürten: StockhausenVerlag.

———. 2007a. Compact Disc 91: KLANG: Thirteenth Hour, COSMIC PULSES.Kürten: Stockhausen Verlag.

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———. 2007c. Text-CD 13: Vier Kriterien der Elektronischen Musik (Four Criteria ofElectronic Music), Lecture 1963. Kürten: Stockhausen Verlag.

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Stockhausen Verlag. 2008. Stockhausen Work List. Kürten: Stockhausen Verlag.

Stockhausen and the Serial Shaping of Space

by

Paul Miller

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Dave Headlam

Department of Music TheoryEastman School of Music

University of RochesterRochester, New York

2009

Part 2Examples

184

Example 1.2.1a. Seating Plan for GRUPPEN.

187

Example 1.2.2a.Spatialization in Osaka.

Example 2.2.1a. Dolby 5.1 Surround Sound

Example 2.2.1b. Dolby 7.1 Surround Sound

both diagrams taken from Dolby’s official web site,http://www.dolby.com/consumer/home_entertainment/roomlayout2.html

195

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v18

v19

v20

v21

v22

v23

v24

v25

v1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1

1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0

0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0

0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1

1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1

1 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0

1 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0

0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0

0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 0 1

0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0

0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1

0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 1

1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1

1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0

0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0

0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1

1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0

Example 2.4.6.3a. Adjacency Matrix A for Graph 1

v1 v2 v3 v4 v5

v6 v7 v8 v9 v10

v11 v12 v13 v14 v15

v16 v17 v18 v19 v20

v21 v22 v23 v24 v25

202

Example 2.4.6.3b

8 4 3 3 4 4 2 2 2 2 3 2 1 1 2 3 2 1 1 2 4 2 2 2 2

4 8 4 3 3 2 4 2 2 2 2 3 2 1 1 2 3 2 1 1 2 4 2 2 2

3 4 8 4 3 2 2 4 2 2 1 2 3 2 1 1 2 3 2 1 2 2 4 2 2

3 3 4 8 4 2 2 2 4 2 1 1 2 3 2 1 1 2 3 2 2 2 2 4 2

4 3 3 4 8 2 2 2 2 4 2 1 1 2 3 2 1 1 2 3 2 2 2 2 4

4 2 2 2 2 8 4 3 3 4 4 2 2 2 2 3 2 1 1 2 3 2 1 1 2

2 4 2 2 2 4 8 4 3 3 2 4 2 2 2 2 3 2 1 1 2 3 2 1 1

2 2 4 2 2 3 4 8 4 3 2 2 4 2 2 1 2 3 2 1 1 2 3 2 1

2 2 2 4 2 3 3 4 8 4 2 2 2 4 2 1 1 2 3 2 1 1 2 3 2

2 2 2 2 4 4 3 3 4 8 2 2 2 2 4 2 1 1 2 3 2 1 1 2 3

3 2 1 1 2 4 2 2 2 2 8 4 3 3 4 4 2 2 2 2 3 2 1 1 2

2 3 2 1 1 2 4 2 2 2 4 8 4 3 3 2 4 2 2 2 2 3 2 1 1

1 2 3 2 1 2 2 4 2 2 3 4 8 4 3 2 2 4 2 2 1 2 3 2 1

1 1 2 3 2 2 2 2 4 2 3 3 4 8 4 2 2 2 4 2 1 1 2 3 2

2 1 1 2 3 2 2 2 2 4 4 3 3 4 8 2 2 2 2 4 2 1 1 2 3

3 2 1 1 2 3 2 1 1 2 4 2 2 2 2 8 4 3 3 4 4 2 2 2 2

2 3 2 1 1 2 3 2 1 1 2 4 2 2 2 4 8 4 3 3 2 4 2 2 2

1 2 3 2 1 1 2 3 2 1 2 2 4 2 2 3 4 8 4 3 2 2 4 2 2

1 1 2 3 2 1 1 2 3 2 2 2 2 4 2 3 3 4 8 4 2 2 2 4 2

2 1 1 2 3 2 1 1 2 3 2 2 2 2 4 4 3 3 4 8 2 2 2 2 4

4 2 2 2 2 3 2 1 1 2 3 2 1 1 2 4 2 2 2 2 8 4 3 3 4

2 4 2 2 2 2 3 2 1 1 2 3 2 1 1 2 4 2 2 2 4 8 4 3 3

2 2 4 2 2 1 2 3 2 1 1 2 3 2 1 2 2 4 2 2 3 4 8 4 3

2 2 2 4 2 1 1 2 3 2 1 1 2 3 2 2 2 2 4 2 3 3 4 8 4

2 2 2 2 4 2 1 1 2 3 2 1 1 2 3 2 2 2 2 4 4 3 3 4 8

Adjacency Matrix A2 for Graph 1

1 2 3 2 1

2 2 4 2 2

3 4 8 4 3

2 2 4 2 2

1 2 3 2 1

Example 2.4.6.3c.

(Total number of paths of cardinality 3 = 64.)

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v18

v19

v20

v21

v22

v23

v24

v25

v1

Analysis of path outflow for v13 in Graph 1

203

Example 2.4.6.4a

0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1

1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0

0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0

0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1

1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1

1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0

0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0

1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 0 0 0 0

0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0

0 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1

0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0

0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1

1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1

1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0

0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0

0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1

1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0

Adjacency Matrix A for Graph 2

v7

v11

v15

v10 v12v9 v13

v3

v19

v4v2

v6

v14

v18 v20

v8

v16

v1 v5

v17 v21

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v18

v19

v20

v21

v1

205

Example 2.4.6.4b.

7 3 2 3 4 3 1 2 2 1 0 1 2 3 1 2 4 2 2 2 2

3 7 3 3 3 1 1 2 1 2 1 1 1 2 2 1 2 4 2 2 2

2 3 6 3 2 1 2 1 0 1 1 1 0 1 3 1 2 2 4 2 2

3 3 3 7 3 2 1 1 1 1 1 2 1 1 2 2 2 2 2 4 2

4 3 2 3 7 2 1 3 2 1 0 1 2 2 1 3 2 2 2 2 4

3 1 1 2 2 7 2 4 3 1 1 2 2 3 1 2 3 2 1 1 2

1 1 2 1 1 2 6 2 1 1 2 1 1 1 3 1 1 2 3 2 1

2 2 1 1 3 4 2 7 2 2 1 1 3 2 1 3 2 1 1 2 3

2 1 0 1 2 3 1 2 6 2 1 3 4 3 1 2 2 1 0 1 2

1 2 1 1 1 1 1 2 2 6 2 3 3 1 1 2 1 2 1 1 1

0 1 1 1 0 1 2 1 1 2 4 2 1 1 2 1 0 1 1 1 0

1 1 1 2 1 2 1 1 3 3 2 6 2 2 1 1 1 1 1 2 1

2 1 0 1 2 2 1 3 4 3 1 2 6 2 1 3 2 1 0 1 2

3 2 1 1 2 3 1 2 3 1 1 2 2 7 2 4 3 1 1 2 2

1 2 3 2 1 1 3 1 1 1 2 1 1 2 6 2 1 1 2 1 1

2 1 1 2 3 2 1 3 2 2 1 1 3 4 2 7 2 2 1 1 3

4 2 2 2 2 3 1 2 2 1 0 1 2 3 1 2 7 3 2 3 4

2 4 2 2 2 2 2 1 1 2 1 1 1 1 1 2 3 7 3 3 3

2 2 4 2 2 1 3 1 0 1 1 1 0 1 2 1 2 3 6 3 2

2 2 2 4 2 1 2 2 1 1 1 2 1 2 1 1 3 3 3 7 3

2 2 2 2 4 2 1 3 2 1 0 1 2 2 1 3 4 3 2 3 7

2

4

2

2 21 1

1

1

11

1

1

1 1

1

1

0 0

0 0

Adjacency Matrix A2 for Graph 2 Example 2.4.6.4c.

Example 2.4.6.4d.

1

1

2

2 12 3

1

1

21

2

4

2 1

3

7

2 3

2 3

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v18

v19

v20

v21

v1

Analysis of path outflow for v11 in Graph 2

Analysis of path outflow for v16 in Graph 2

206

Example 2.4.6.5a

0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1

1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0

1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0

1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0

0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0

0 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 0

0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0

0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0

0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0

0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1

0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1

0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1

1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0

Adjacency Matrix A for Graph 3

v9

v4

v14

v8 v10

v6v5

v12 v13

v2

v1

v3

v11

v16

v17

v15

v7

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v1

207

Example 2.4.6.5b.

4 0 0 0 2 2 1 0 1 0 1 0 0 1 2 2 0

0 4 1 2 0 0 0 2 1 0 2 0 1 0 1 0 2

0 1 4 2 0 0 2 0 1 2 0 1 0 0 0 1 2

0 2 2 4 1 1 0 2 2 2 0 1 1 1 0 0 1

2 0 0 1 4 2 2 1 2 1 0 2 1 1 0 1 0

2 0 0 1 2 4 0 1 2 1 2 1 2 1 1 0 0

1 0 2 0 2 0 4 0 1 1 0 2 0 0 0 2 1

0 2 0 2 1 1 0 4 2 1 1 1 1 2 2 0 0

1 1 1 2 2 2 1 2 8 2 1 2 2 2 1 1 1

0 0 2 2 1 1 1 1 2 4 0 1 1 2 0 2 0

1 2 0 0 0 2 0 1 1 0 4 0 2 0 2 0 1

0 0 1 1 2 1 2 1 2 1 0 4 2 1 0 0 2

0 1 0 1 1 2 0 1 2 1 2 2 4 1 0 0 2

1 0 0 1 1 1 0 2 2 2 0 1 1 4 2 2 0

2 1 0 0 0 1 0 2 1 0 2 0 0 2 4 1 0

2 0 1 0 1 0 2 0 1 2 0 0 0 2 1 4 0

0 2 2 1 0 0 1 0 1 0 1 2 2 0 0 0 4

8

2

2

2 2

22

2 2

1

1

1

1

1

1

1

1

Adjacency Matrix A2 for Graph 3

Example 2.4.6.5c.Analysis of path outflow for v9 in Graph 3

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v1

208

cl1

vn1

vn3

vn4 vn2

vla1vla4

vla3 vla2

trb3

tr1

tb1

cl2

tr2

tr2

cl3

tr3

vn=violinvla = viola

tr =trumpettb = trombone

cl = clarinet

vn1

vn3

vn4 vn2

vla1vla4

vla3 vla2

Example 2.4.7a.

cl1 cl2

cl3

trb3

tr1

tb1

tr2

tr2

tr3

Example 2.4.7b. An alternative graph structure based on a placement of different instruments in Graph 3.

Spatial orchestration.

210

Example 2.5a.

Graph 1 Graph 2 Graph 3

Number of vertices

Number of edges

Type of graph

Sets of cardinality 1

Sets of cardinality 2(A Matrix)

Sets of cardinality 3(A2 Matrix)

25 21 17

200

regular irregular

25 21 17

200

1600

Sets of cardinality 4(A3 Matrix)

12,800

136

irregular

892

5868

72

320

1408

Comparison of set-generating potential in Graphs 1, 2, and 3.

136 72

211

Example 2.5.1. Matrix representations of traditional transformations as applied to pitch-class

Transpose (T)

Type ofTransformation

Algebraic Notation Matrix Notation Example

Invert (I)

Multiply (M)

x9 = x + a

x9 = s ? x s

0

Invert a pitch-class xaround PC b

x91

10

a1

x1

Transpose PC 5by 9.

10

91

51

= 5+91

21

=

Multiply the PC5 by 9.

01

51

=90

mod 12(45)1

=

0

1

Description

result: 2

result: 9

=x9

1

x

1

Multiply PC x by ascalar s.

Transpose a pitch-class xby an interval a

x9 = mod 12(x - 2b) 10

-2b1

=x91

x1

Invert PC 5 aroundPC 9.

10

-181

51

= mod12(5 -18)1

11

=

result: 1 (sum 6 relationship)

91

=

Invert a pitch-class xaround PC 0 (or 6)

x9 = mod 12(-x) -10

01

=x91

x1

Invert PC 5 aroundPC 0.

-10

01

51

= mod 12(-5)1

71

=

result: 7

213

Example 2.5.2a. Matrix representations of four basic transformations in a plane

Translate (T)

Type ofTransformation

Coorelationwith pitch

Algebraic Notation Matrix Notation Example

Transpose (T)

Rotate(R)

Reflect (F) Invert (I)

Scale (S) Multiply (M)

x9 = x + ay9 = y + b

x9 = xcosu - ysinuy9 = xsinu + ycosu

In general, invert ineither of the two axes,using the unit vector(ux, uy)

Or, where u = the anglefrom the x axis, and theline goes through theorigin

=cosusinu

0

-sinucosu

0

x9 = sx ? xy9 = sy ? y

sx

0

0

0

sy

0

2u2x - 1

2uxuy

0

2uxuy

2u2y - 1

0

cos2usin2u

0

sin2u-cos2u

0

Reflect across the line x = y

010

100

Reflectacross the line x = -y

0-10

-100

x9y91

100

010

ab1

xy1

Translate (2,1)by (3, -1):

100

010

3-11

211

=2+31 - 1

1

501

=

001

Rotate (2,1)90 degreescounterclockwise:

010

-100

001

-121

clockwise Rotate (2,1)90 degreesclockwise:

cosu-sinu

0

sinucosu

0

001

0-10

100

001

1-21

x9 = xcosu + ysinuy9 = -xsinu + ycosu

Reflect (2,1) around the x-axis (using the

unit vector )

001

100

0-10

001

2-11

Reflect (2,1) across adiagonal line 26.566degrees fromthe x-axis;should producean involution:(sin 53.132 = 0.800; cos 53.132 = 0.600)

Reflect (2,1)across the linex = y:

001

001

010

100

001

121

Reflect (2,1)across the linex = -y:

0-10

-100

001

-1-21

Scale (multiply) thepoint (2,1) by 2:

020

001

211

=200

421

0.6000.800

0

0.800-0.600

0

1.2 + .81.6 - 0.6

1

=

x9y91

xy1

=x9y91

xy1

=x9

y9

1

0

0

1

x

y

1

=

0

0

1

=x9y91

=x9y91

=x9y91

xy1

xy1

xy1

none

none

Description

x9 = xcos2u + ysin2uy9 = -xsin2u + ycos2u

x9 = yy9 = x

x9 = -yy9 = -x

result: (5, 0)

211

=

result: (-1, 2)

211

=

result: (1, -2)

211

=

result: (2, -1)

001

=211

result: (2, 1)

211

=

result: (1, 2)

211

=

result: (-1, -2)

result: (4, 2)

=x9

y9

1

x

y

1

Scales the point withrespect to the origin

Move a point to the righta units, and up b units.

Rotate the pointcounterclockwise withrespect to the origin

Rotate the pointclockwise withrespect to the origin

counter-clockwise

10

x9 = x(2u2x - 1) + y(2uxuy)

y9 = x(2uxuy) + y(2u2y - 1)

214

Example 2.5.3a. A space with four sound sources

Example 2.5.3b. Transformations in the space of Example 2.5.3a.

1

2

3

4

1 2 3 4

1 2 3 4

r0 r1 r2 r3

f1 f2 f3 f4

( ) 1 2 3 4

2 3 4 1( ) 1 2 3 4

3 4 1 2( ) 1 2 3 4

4 1 2 3( )1 2 3 4

4 3 2 1( ) 1 2 3 4

2 1 4 3( ) 1 2 3 4

3 2 1 4( ) 1 2 3 4

1 4 3 2( )

Rotation:

Flip:

216

Example 2.6.2a.

r0 r1 r2 r3 f1 f2 f3 f4

r0

r1

r2

r3

f1

f2

f3

f4

r0 r1 r2 r3 f1 f2 f3 f4

r1

r2

r3

f1

f2

f3

f4

r2

r0

r2r0 r1

r3 r1

r3 r0 f1f4 f3 f2

f2 f1 f4 f3

f3 f4 f2 f1

f1

f3 f2 f4

f4 f1 f3

f2 f4 f1

f3 f2

r0 r2 r1 r3

r2 r0 r3 r1

r3 r1 r0 r2

r1 r3 r2 r0

The graph and its transformations are defined in Example 2.5.3b.

e a aa b bb c cc d dd

e

a

aa

b

bb

c

cc

d

dd

e a aa b bb c cc d dd

a

aa

b

bb

c

cc

d

dd

aa e d cc b dd c bb

e a c dd d bb b cc

d c ebb

e

e

e

e

e

b

cc

c

dd

dd a cc aa

cc dd aa d a c

b d dd aa bb a

dd bb a d aa b

c b cc a bb aa

bb cc aa c a b d

Example 2.6.3a.

Group table of the dihedral group D4.

Group Table of Translation Group

219

Example 3.1.1a.Nuclear Formulas in Stockhausen’s LICHT

220

Example 3.1.1b.

Monday

Wednesday

Saturday

Tuesday

(Tuesday) Thursday

ThursdayFriday

Sunday

Superformula for LICHT

221

Example 3.1.2. “Study for OKTOPHONIE.”

222

1st Invasion

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Example 3.1.3a. Formal Structure of OKTOPHONIE, Part 1

= crash = explosion

Part 1

shots:

bombs:

Time(minutes):

1st Luftangriff/ 1st Air Strike 2nd Luftangriff/ 2nd Air Strike

23

0

1 10 20 24 25 26 27 28 29 30 31 32 33 34 35 37 40 45 46 50 55 60 6544

A thick line indicates the bomb takes longer to descend

1 2 3 4 10 13 14 15 16 17 18 19 20 21 22 23 24

shots:

2nd Invasion

(continued)

18

Kampf/Battle

22 23 24 25 26 27 28 29 30 3619 20 21 31 32 33 34 3536:18

25 26 27 28 30 34 37 38 39 41 42 43 44 45 48

223

1 2 9 10 11 12 13 14

shots:

3rd Invasion

3 4 5 6 7 8

Pietà Jenseits/Beyond...

Time (minutes):

13:0010:00

3rd Invasion with “Explosions”

Part 2

Example 3.1.3b. Formal Structure of OKTOPHONIE, Part 2

= crash = explosion

3215 16 23 24 25 26 27 2817 18 19 20 21 22 29 30 31

Synthi-Fou(Klavierstück XV)

Abschied, Spiegelwelt/Farewell, Mirror World17:14 22:36 (31:54)

(continued)

...Jenseits/Beyond

0

224

1 5 6 9 11 15 16174

225

Example 3.2.1a. Sketch for the spatialization of OKTOPHONIE

Example 3.2.2a. Bomb Analysis 1

228

Example 3.2.2b. Bomb Analysis 2

229

Example 3.2.2d.Combined Scatter Plot of Bombs in OKTOPHONIE.

231

id NW (II) NE (III) SW (I) SE (IV)

1 5 5 5 5 (0, 0)2 -3 -1 5 -9 (-3.1, -1.7)3 0 6.5 -12 -4 (2.6, 4.6)4 6 -2.5 -12 -10 (-3.3, 5.8)5 -3 -4 5 3 (-0.7, -3.3)6 2.5 4 -7 -0.5 (1.3, 2.7)7 -5 -9 3.5 2 (-0.7, -4.1)8 3 -10 -15 5 (3.5, -2.9)9 -16 -7 -8 2.5 (3.8, -3.6)

10 -5 2.5 4 -6 (-0.8, -0.6)11 0 -10 3 -6 (-3.9, -1.1)12 -16 -1 -4 2.5 (3.6, -1.9)13 3 2 3.5 2.5 (-0.4, -0.2)14 -7.5 3 7 -12 (-6.1, -5.4)15 5 -11 -8 -1 (-3.8, 3.3)16 2 -7 -12 5 (2.6, -1.9)17 -10 -10 3 3 (0, -5.1)18 -12.5 3 -5 0 (4.2, 0.4)19 6 -8 2.5 5 (-2, -1)20 3 -8 -10 3.5 (0.5, -0.2)21 -7.5 1.5 5 -6 (-1.7, -2)22 1 -10 0.5 -8 (-4.1, 0)23 -14 -14 0.5 -2 (-1.1, -4.5)24 -8 -9 2 1.5 (-0.2, -4.2)25 -12 -10 -6 2.5 (3, -4.1)26 1.5 1 3 1.5 (-0.4, -0.4)27 -8 -8 3 -5 (-2.7, -3.5)28 -12.5 -10 0 5 (1.8, -5.5)29 -7 -11 2 2 (-0.2, -4.4)30 6.5 -4 -3 6 (-0.5, 0.2)31 0 0 2 -2.5 (-1, 0.2)32 -11 3 -12 3 (5.5, 0)33 -10 0.5 2 -9 (-0.7, -0.9)34 -10 -8 2.5 2.5 (0.1, -4.7)35 10 10 -4 5.5 (0.9, 4)36 7 -7 7.5 5 (-3.1, -1.6)37 -6.5 -5.5 3 5 (0.7, -4.4)38 -12.5 -8 5 5 (0.1, -6)39 2.5 2.5 3 4 (0.2, -0.4)40 6 6 6 7.5 (0.4, -0.4)41 1 0 4 1.5 (-0.7, -0.9)42 -10 -10 -1 7.5 (3.5, -6.2)43 -25 -25 -1 0 (0.6, -5.9)44 -9 2.5 4 -5 (-0.4, -1.2)45 -5 -9 -10 4 (4, -3.8)46 -10 2.5 1 -7 (1, 0.5)47 2 -7 -10 4 (1.5, -1)48 -11 5 -12 5 (6.3, 0)49 3.5 -11 -12 -2 (-3.3, 3.4)50 -2 -12 5 -2.5 (-3.3, -3.1)51 -13 -14 -2 1 (1.2, -4.6)52 4 3.5 1 2 (0.1, 0.9)53 -11 -1 -14 2.5 (4.9, -1.2)54 -2.5 -1 -4 1 (1.6, -0.3)55 1 1 0 0 (0, 0.4)56 -10 -11 0 1 (0.3, -4.3)57 0 -12.5 2.5 -7 (-4.3, -1)58 0 -7.5 -8 2 (0.8, -0.7)59 -2.5 -4 0 0 (-0.3, -1.7)60 -1 1 -5 3 (2.2, 0.1)61 -11 -6 -9 7 (6, -5.3)62 -12.5 0 -10 0.5 (4.4, -0.3)63 -3 -2.5 2.5 2.5 (0.1, -2.4)64 -10 -10.5 4.5 4 (-0.2, -5.8)65 -12 -6 -6 1 (2.7, -2.7)

intersectionpoint

Example 3.2.2e.Decibel settings

Coordinates of Bombs

232

Example 3.2.3b. Shot Analysis 1

234

Example 3.2.3c. Shot Analysis 2

235

Example 3.2.3d. Shot Analysis 3

236

Example 3.2.3e. Combined Scatter Plot of Shots 1-48

237

id NW (VI) SE (VIII)

1 0 0 0 0 0 0 0 (0, 0)2 -8 2 4.2 -4 -9 3 -15 (-4.7, -2.4)3 3 -5 -3.6 -8 -2 -16 3 (4.8, -1.6)4 -6 2 3.5 -4 -6 4 2 (-0.8, -3.5)5 -7 7 6 -13 5 -4.5 -12.5 (6, 6)6 5 -12.5 -6.6 -8 5 -14 5 (6.1, 0.1)7 5 -4 -4 -17 -3 5 5 (0.5, -5.3)8 -12 0 4.5 5 1.5 -16 -2 (-0.4, 4.6)9 -4 -0.5 1.7 4 -14 -6 5 (0.5, -1.7)

10 2 -11 -5 -16 -4 1 -1 (0.1, -3.5)11 2 -4 -2.8 -10 1.5 -2 3 (3.2, -1.2)12 3 4 0.4 7 1 5 1 (-2.1, 0.6)13 5 1 -1.6 -17 1 5 1 (0.2, -3.5)14 -6 5 4.8 -1 -7 8 2.5 (-2.4, -4)15 0 0 0 -2 -8 -6 3 (1.8, -2.2)16 5 0 -1.9 -10 -2.5 5 3 (-0.1, -4.3)17 -2 2.5 2 -14 -4.5 -2.5 3 (2.6, -3.4)18 4 -4 -3.5 -4.5 -11 -2 4 (1.6, -4)19 -2.5 -1 0.8 2.5 -7 3 -2.5 (-3.2, -0.6)20 -2.5 0 1.4 0 -25 -5 2.5 (0.3, -2.5)21 0 -5 -2.5 -10 -2.5 5 3 (-0.1, -4.3)22 -12.5 0.5 4.8 1 2 5 0 (-0.8, -0.6)23 -4 0 2 -12.5 -1 5 -11 (-4.6, -4.9)24 -2 -1 0.5 -12.5 -7 4 -9 (-4.2, -4.8)25 5 -5 -4.4 0 0 6 6 (0, -2.4)26 4 -2 -2.6 0 -7 -12 5 (3.4, -2.6)27 -10 2.5 5 -10 0 2 -10 (-1.1, -1.1)28 5 5 0 -4 -2 -5 0 (1.8, -0.5)29 -9 -5 1.3 -4 -11 -1 -6 (-2.2, -1.5)30 3 -5 -3.6 -3 -7 -12 -1 (1.3, -0.4)31 3 -7 -4.3 0 0 0 0 (0, 0)32 0 3 1.1 -25 -10 -1 -1 (0.4, -4.5)33 -5 4 4 0 0 0 0 (0, 0)34 0.5 -5 -2.6 -11 -25 -2 -2 (-0.4, -4.1)35 2 -7 -3.9 -7 -7 0 -25 (-3.9, -1.8)36 0 5 1.9 -3 -5 -19 0 (2.6, -0.4)37 1.5 2.5 0.4 -6 -6 0 0 (0, -2.8)38 1 -8 -3.8 -3 -8 -13 -1 (1.3, -0.5)39 7 4 -1.3 -9 -3 -1 -9 (-0.8, -0.8)40 -7 -2 2.1 -5 -3.5 -5 0.5 (1.8, -1.3)41 -2 -12 -3.4 -11 -4 -12.5 -1 (3.2, -1)42 3 -0.5 -1.4 -10 -2 -5 1.5 (3, -1.6)43 -11 1 4.6 -6 -12 -1 -4 (-1.6, -2.3)44 -2 1 1.5 -12 -6 -2 -4 (0, -2.1)45 0 -12 -4.5 -14 -3.5 -7 0.5 (3.2, -1.9)46 0 -5 -2.5 -12 0 -6 1 (3.8, -0.7)47 -5 2.5 3.4 -1.5 -21 -5 -8 (-3.1, 0.5)48 -0.5 0.5 0.5 -6 0 -7 -7.5 (1.8, 2.2)

Frontleft (II)

Frontright (III)

stereolocation

NE (VII)

SW (V)

intersectionpoint

Example 3.2.3f. Decibel settingsDecibel settings

Coordinates ofshots 1-48

238

Example 3.2.3g. Shot analysis 1 for shots in the second half of OKTOPHONIE.

239

Example 3.2.3h. Shot analysis 2 for shots in the second half of OKTOPHONIE.

240

Example 3.2.3i. Combined Scatter Plot of Shots in the second half of OKTOPHONIE

Starting points of the first group (shots 1-10, floor) Starting points of the second group (shots 11-18, front)

Endingpointsof shots 1-18.(celing)

241

Example 3.2.3j. Coordinates of shots 1-10 in the second half of OKTOPHONIE

ID

1 -16 0 0 -15 0 0 -1 0 (0, 5.1) (-0.3, -0.3)2 -19 -2.5 0 -11 0 -6 -9 -13 (1.4, 4.3) (-3.3, -1.8)3 -15 0 -7 -28 -12.5 -7.5 -5 0 (-2.9, 4.5) (3.1, -1.5)4 -10 0 -2 -16 -4 -11 -7.5 0 (-1.1, 4) (1.7, -2.9)5 -13 -6 0 -12 0 -0.5 0 0 (2.2, 3.6) (0.1, -0.1)6 0 -7.5 -7 -5 -5 -10 -7.5 0 (-1.6, -2.3) (1.9, -2.7)7 -7 -18 -16 0 0 -4 -12 -12 (2.6, -4.3) (-3.7, -1.5)8 -7 -18 -16 0 0 -4 -12 -12 (2.6, -4.3) (-3.7, -1.5)9 -9 0 -5 -11 -19 -8 -9 -4 (-1.9, 3.1) (1.7, -0.2)

10 -14 -7.5 0 0 0 -4 -13 -9 (4.1, 0.3) (-3.4, -1.7)

bottom swI

bottom nwII

bottom neIII

bottom seIV

top swV

top nwVI

top neVII

top seVIII

bottomintersection

point

topintersection

point

Example 3.2.3k. Coordinates of shots 11-18 in the second half of OKTOPHONIE

id

11 -2.5 0 -0.5 -2 0 -7 -7 0 (-0.1, 1.1) (0, -3.2)12 -12 -2.5 0 0 -1 -6 -4 -8 (2.7, 1) (-1.2, -0.7)13 -5 -8 0 -3 -5 -10 -12 -2 (2.3, 0.7) (0.7, -2.6)14 -8 0 -14 0 -11 -10 -2 0 (-0.4, -0.4) (3.7, -0.8)15 0 -10 -2 -13 0 -10 -9 0 (-1.4, -1) (0.1, -3.9)16 0 0 0 0 0 -10 -9 0 (0, 0) (0.1, -3.9)17 -5 0 0 -5 -3 -15 -15 -3.5 (0, 2.5) (-0.1, -3.4)18 -15 -7 0 0 -3 -15 -15 -3.5 (4.1, 0.4) (-0.1, -3.4)

left bottomII

left topVI

right topIII

right bottomIV

top swV

top nwVI

top neVII

top seVIII

frontintersection

point

topintersection

point

Decibel settings

Decibel settings

242

Example 3.2.4a. Crash Analysis

243

Example 3.2.4b. Combined scatter plotof crashes

Starting points(on the ceiling)

Ending points(on the floor)

244

id

1 6 -2 -2.5 6 0 0 0 0 (-0, -3.6) (0, 0)4 3 10 -1 0 4 -4 -6 2 (-3.7, 2.4) (-0.8, -3.5)

15 -7.5 0 -6 -15 -6 -2 -8 3 (-2.5, 3) (1.8, -2.2)17 -5 5 -3 -8 -2.5 -14 -4.5 3 (-2.9, 3.7) (2.6, -3.4)18 0 0 -1 0 -2 -4.5 -11 4 (-0.3, -0.3) (1.6, -4)19 -16 -9 0 0 3 2.5 -7 -2.5 (4.5, 0.3) (-3.2, -0.6)20 -5 -9 0 -14 -5 0 -25 2.5 (1.8, 2.6) (0.3, -2.5)21 -12 -6 0 -7 5 -10 -2.5 3 (2.3, 2.7) (-0.1, -4.3)22 -5 -11 -8 0 5 1 2 0 (2, -2.9) (-0.8, -0.6)23 -27 -15 4 -7 5 -12.5 -1 -11 (6, 4.4) (-4.6, -4.9)24 -10 0 0 -8 4 -12.5 -7 -8 (0.1, 3.7) (-4, -4.7)

bottom swI

bottom nwII

bottom neIII

bottom seIV

top swV

top nwVI

top neVII

top seVIII

bottomintersection

point

topintersection

point

Example 3.2.4c. Coordinates of the crashes

Example 3.2.4d. Example 3.3.1a. Alternative loudspeaker arrangement for OKTOPHONIE.

Decibel settings

Shape of a descending crash

245

0

2

4

6

8

10

12

0

50

100

150

180

Example 3.4.2e. Plot of distances and angles of the 65 bombs

8 16 24 32 40 48 56 64

8 16 24 32 40 48 56 64

dist

ance

sepa

ratin

g co

nsec

utiv

e bo

mbs

(in m

eter

s)

Angl

es b

etw

een

sets

of t

hree

bom

bs(in

deg

rees

)

Bomb number

Bomb number 249

0

2

4

6

8

10

12

0

50

100

150

180

8 16 24 32 40 48

8 crashes

Example 3.4.3e. Change in distance and angle for the endpoints of the first 48 shots

Dist

ance

bet

wee

n co

nsec

uriv

e sh

ots

(in m

eter

s)

Angl

es b

etw

een

sets

of t

hree

shot

s(in

deg

rees

)

8 16 24 32 40 48

Shot number

Shot number 253

Crash Time ptof crash

Preceding Bomb

Time ptof bomb

Coordinates ofstart of crash

Coordinates ofbomb

Time frombomb tocrash (m:s.s)

Distancefrom bombto crash (m)X-cord Y-cord X-cord Y-cord

1 1:00 1 1:00 0 0 0 0 00:00.004 4:46.8 23 4:46.7 -0.8 -3.5 -1.1 -4.5 00:00.10

01.04

15 7:17.6 24 5:36 1.8 -2.2 -0.2 -4.2 00:41.60 2.8317 10:42.1 34 10:42.1 2.6 -3.4 0.1 -4.7 00:00.00 2.8218 11:22.4 36 11:04 1.6 -4 -3.1 -1.6 00:18.40 5.2819 12:29.5 43 12:25 -3.2 -0.6 0.6 -5.9 00:04.50 6.5220 13:09.8 45 12:56.3 0.3 -2.5 4 -3.8 00:13.50 3.9221 14:16.9 51 14:11.8 -0.1 -4.3 1.2 -4.6 00:05.10 1.3322 14:57.2 54 14:41.2 -0.8 -0.6 1.6 -0.3 00:16.00 2.4223 16:04.3 61 15:59.2 -4.6 -4.9 6 -5.3 00:05.10 10.6124 16:44.6 64 16:38.4 -4 -4.7 -0.2 -5.8 00:06.20 3.96

Example 3.4.4a. Distance from preceeding bomb to the next crash

256

Example 4.1.3a.

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

M13

The Kernformeln or “Nuclear Formulas” in LICHTER-WASSER (central pitches)

260

E1

E2

E3

E4

E5

E6

E7

E8

E9

E10

E11

E12

Example 4.1.3a (continued).The Kernformeln or “Nuclear Formulas” in LICHTER-WASSER (central pitches)

261

Example 4.1.3b. Division of the 29 musicians into two “orchestras”.

Eve instruments are in light grey (normally, green); Michael instruments are in dark grey (normally, blue).

These cycles are refered to as the “basic cycles” throughout.

262

Example 4.1.3c. Formal Structure and Pitch Structure of LICHTER-WASSER

Anfangs-Duett

EingangM1*

Wave sections are abbreviated M1, E1, etc. Nuclear formula material is abbreviated M1NF, M2NF, etc. v = “vertical”; h = “horizontal”

M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

E1* E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12

1st Bridge 2nd Bridge 3rd Bridge

4th Bridge

1stAnnouncement

4th Bridge(repeated)

2ndAnnouncement

3rdAnnouncement

5th Bridge 6th Bridge

Ausgang Schluss-Duett

M1NFE1NF

M1NFE1NF

M2NFE1NF

E3NFM4-5NF

M6-8NFE5-9NF

M9NF?E10NF

M9-10NFE10NF

M9-10NFE10NF

E10NF(M9NF)

M9NF

M10NFE10NF

M12NFE12NF

M1NF (v)M13NF (h)

E1NF (v)E12NF (h)

M1NFE2NF

* Each wave section (”M1”, “E1”, etc.) is based on its corresponding nuclear formula. The nuclear formulas are given in Example 4.1.3a.

263

Example 4.2.1a. Various spatial shapes Stockhausen considered using in LICHTER-WASSER

264

Example 4.3.3a. LICHTER-WASSER, mm. 139-147.

265

266

Example 4.3.3b. LICHTER-WASSER, mm. 398-401.

Example 4.3.3c. LICHTER-WASSER, mm. 249-253.

267

268

Example 4.3.3d. LICHTER-WASSER, mm. 336-340.

Example 4.3.3e. LICHTER-WASSER, mm. 324-328.

269

mwave1(106) f2 v5 k va5 k va3 b v5 f2 v5 va5 k va3 b k f2 v5 b va5 t1 va3 k ob v4 h1 h2 va4 v2 th fa2 ob v4 t2 v4 ob k v5 f2

v3 t2 f1 v3 va1 va2 v1 v2 th va3 t1 eh f2 v5 k ob v4 t2 f1 v3 eh t1 va3 th v2 v1 va2 va1 eh v3 fa1 f1 t2 h2 v4 ob v4 h2

t2 f1 fa1 v3 eh t1 va3 t1 th v2 t1 v1 va3 v2 v1 va2 va1 eh t1 va3 th v2 v1 va2 v1

ewave1(101) p2 p2 h2 p2 fa1 p2 th fa1 sax p1 b t2 va2 f1 va1 v3 f2 va5 b p1 sax fa1 p2 fa2 b va5 b p1

sax fa1 p2 fa2 b p1 sax fa1 p2 fa2 b va5 h1 fa1 p1 h1 k va5 b t1 va3 th va4 h1 va5 b va3 th va4 h2 p1 h2 fa2 sax

p1 b va5 h1 va4 h2 fa2 p2 fa1 sax p1 b va5 h1 va4 h2 fa2 p1

mwave2(106) v5 f2 k ob v4 th v4 ob va4 h1 th ob h1 k t1 fa2 va3 t1 eh va1 t2 v1 v2 fa1 v3 f1 v5 f2 k f2 t1 eh va2 v1 v2 v1 va2 va1

eh t1 va3 th v2 t2 f1 v3 p1 f2 v5 va5 b t1 eh va1 v3 eh t1 va3 th v2 h2 v4 va4 ob h1 va5 v5 f2 p1 eh f1 fa1 f1 v3 f1 p2 f1

t2 f1 v1 f1 va2 f1 va1 f1 eh f1 va1 t1 va2 va3 v1 t2 v2 th v4 ob k f2 v5 va5 va5

ewave2(105) p1 fa1 f1 sax p1 b h1 va5 f2 k va3 th va4 fa2 h2 va2 va1 sax fa1 va2 p2 t2 f1 fa1 sax fa2 va1 f1 t2 h2 v2 v1 fa2 eh

v3 p1 t1 v3 eh va3 b va5 v5 k h1 th va4 v4 v1 h2 p2 sax v3 fa1 va1 fa2 va2 h2 p1 b h1 th fa2 h2 p2 fa1 p1 eh va1 sax

va2 t2 p2 fa1 eh sax p1 eh va5 v5 f2 t1 ob h1 k va3 v4 va4 h2 t2 p2 fa1

mwave3(85) k va5 v5 f2 b t1 va3 th v4 k ob va4 h2 v2 fa2 va1 eh t1 v3 fa1 sax f1 va2 v1 t2 h2 t2 v1 v1 va2 f1 p2

f1 sax fa1 v3 sax fa1 p1 v3 eh p1 fa2 eh v2 fa2 h2 v2 v4 h2 ob va4 v4 ob h1 va4 th h1 va3 th t1 va3 b t1 f2 b v5 va5 f2 v5 k va5

h1 k h1 k f2

ewave3(44) h2 fa2 p1 va2 p2 h2 p1 fa1 va1 sax fa1 p1 fa2 p2 va2 eh va3 k b p1 eh fa2 va5 b p1 fa2 th h1 fa2 va3 k va4 h2 th

va3 k h1 th

mwave4(116) f2 v5 va5 k h1 ob v4 h2 t2 va2 f1 fa1 v3 p1 v5 b v4 sax eh t1 va3 th v2 v1 va2 va1 fa2 va3 t1 va1 va2 v1 v1 v2 th b va4

p1 v3 sax fa1 f1 p2 t2 h2 v4 h1 k va5 v5 f2 p1 v3 fa1 f1 sax va1 eh t1 b f2 v5 va5 k va3 fa2 va2 p2 t2 v1 v2 v4 va4 h1

ob v4 h2 t2 p2 f1 fa1 sax va1 fa2 th va4 ob h1 k va5 v5 f2 p1 v3 eh b eh t1 va3 fa2 va2 v1 v2 v4 h2 eh

ewave4(117) th h1 fa1 h2 va4 h1 va3 th h2 va4 h1 fa2 h2 va4 h1 th fa2 h2 k h1 th fa2 b va4 h1 th va3 fa1 h1 k b t1 f2 p1 va5 va3 p1 va3 b

va3 va5 v5 f2 p1 va1 fa1 va3 k va5 v5 b p1 eh f2 p1 fa1 sax p2 va2 fa2 va1 eh p1 fa1 p2 va2 fa2 eh p1 fa1 sax p2 sax va1 fa2 p1 fa1 sax

p2 va1 p1 sax fa1 p2 h2 fa2 p2 fa2 va2 p2 h2 fa2 va2 p2 h2 fa2 va2 p2 h2 fa2 va2 p2 p1

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Example 4.3.4a. Sequence of instrumental moves in wave sections (page 1 of 5)

270

For a key to the abbreviations, see Example 4.3.4g.

mwave7(402) ob v4 va4 k va3 va5 v5 f2 t1 eh v3 f1 va1 va2 t2 f1 v1 v2 va4 v4 ob k va3 va5 v5 f2 t1 eh v3 f1 va1 t2 v1 va2 v2 v4 va4 ob k

va3 va5 eh t1 va5 v5 f2 eh v3 f1 va1 va2 t2 v1 v2 va4 v4 ob k va3 va5 v5 f2 t1 eh v3 va1 va2 f1 v1 t2 v2 v4 va4 ob k va3 va5 v5 f2

eh t1 v3 f1 va2 v1 t2 v2 v4 va4 ob k va3 k va5 v5 f2 eh t1 va1 v3 eh va1 f1 va2 t2 v1 v2 v4 ob va4 va3 k t1 va5 v5 f2 eh v3

va1 f1 va2 v1 t2 v2 v4 ob va4 k va3 t1 va5 v5 f2 eh va1 v3 fa1 f1 va2 t2 v1 v2 v4 va3 va4 ob k va5 v5 t1 eh f2 v3 va1 va2 f1 t2

v1 v2 v4 ob va4 k va3 va5 t1 v5 f2 eh v3 va1 va2 f1 t2 va2 v1 v2 v4 va4 ob va3 k va5 v5 f2 t1 eh v3 va1 f1 t2 va2 v1 f1 v2 v4

ob va4 va3 k va5 v5 f2 eh t1 va1 v3 f1 va2 v1 t2 v2 v4 ob va4 k va3 va5 v5 f2 t1 eh va1 va3 va2 f1 v1 t2 v2 v4 ob va4 k va5 v5

f2 t1 va1 v3 eh va2 f1 t2 v1 v2 v4 ob va4 k va3 va5 v5 f2 t1 eh v3 va1 f1 va2 v1 t2 f1 v2 v4 va4 va3 ob k va5 v5 f2 v3 eh t1 va1

f1 va2 t2 v1 v2 va4 v4 ob k va3 va5 v5 f2 eh t1 va1 v3 f1 va2 v1 v2 va3 va1 t2 v4 va4 ob k t1 va5 v5 f2 eh v3 f1 va1 va2 v1 p2 t2

v2 h2 v4 th fa2 va3 va4 ob h1 k t1 va5 b v5 f2 eh p1 v3 va1 va2 f1 t2 v1 v2 va2 v4 ob va4 k va3 va5 t1 b v5 f2 eh v3 f1 va1

va2 t2 v1 v2 v4 ob va4 k va3 t1 b va5 v5 f2 eh v3 f1 va1 va2 v1 t2 v2 fa2 va3 k th va4 v4 ob h1 va5 t1 b v5 f2 v3 fa1 p1

eh f2

ewave7(99) fa2 b p1 fa1 sax p2 h2 h1 th fa2 va3 va5 b p1 fa1 sax p2 h2 h1 th fa2 va3 va5 v5 b p1 fa1 sax p2 h2 v4 va4 h1

th fa2 va3 va5 t1 b p1 fa1 sax p2 h2 fa2 th h1 va4 b p1 fa1 sax p2 h2 fa2 b p1 sax fa1 p2 h2 fa2 th h1 va3 va5 b

p1 sax fa1 p2 fa2 va1 va2 h2 va4 h1 th va3 t1 va5 b p1 sax

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Example 4.3.4b. Sequence of instrumental moves in wave sections (page 2 of 5)

mwave6(91) k va5 v5 f2 v3 f1 va1 v1 va2 v2 ob v4 va4 va3 ob k va5 v5 f2 v3 va1 f1 va2 v1 v2 v4 va4 va3 k va5 v5 f2 v3 f1 va1 v1 va2

v2 ob v4 va4 k va3 va5 v5 f2 v3 f1 va1 va2 v1 t2 v2 v4 ob va4 va3 va5 v5 k f2 v3 eh f1 va1 va2 v1 v2 v4 va4 ob k va3 va5 v5

f2 v3 eh va1 v3 f1 va2 v1 t2 ob

ewave6(83) fa1 sax p2 h2 va4 h1 b fa2 p1 fa1 sax fa2 fa1 p2 h2 va4 th fa2 b h1 th p1 fa1 sax p2 h2 fa2 th va4 h1 va3 va5 b p1

sax fa1 p2 h2 th h1 fa2 b p1 fa1 sax p2 fa2 h2 th va4 b va5 b sax p1 fa1 p2 h2 fa2 th h1 b p1 sax fa1 p2 fa2

h2 fa2

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mwave5(68) eh t2 f1 v3 va1 va2 v1 v2 eh f2 b t1 eh va4 th h1 k va5 v5 f2 p1 v3 fa1 f1 p2 t2 p2 t1 p1 va2 va1 sax fa2 eh v2 h2 v4 va4

th va3 t1 b va5 k h1 ob v4 h2 h2 v2 va4 h1 th v1 t1 va2 p2 v2 va3 k t1 va1 eh v3

ewave5(134) b p1 fa2 th h1 h2 fa1 sax fa1 fa1 p1 sax fa2 b h1 th h2 p2 sax fa1 fa2 sax p1 b fa2 p1 b fa2 th h2

p2 fa1 sax fa2 p1 fa1 b h1 fa2 th h2 fa2 p2 sax fa1 p1 b fa2 h1 th h2 p2 fa1 p1 sax h2 va4 h1 th fa2 va3 va5 b p1 fa1

p2 fa2 th h2 h1 b sax p1 fa1 p2 p2 fa2 p2 fa2 h2 th b h1 va5 b p1 sax fa1 p2 h2 fa2 th h1 b va5

p1 fa1 sax fa2 va2 p2 h2 th h1 b p1 fa1

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271

Example 4.3.4c. Sequence of instrumental moves in wave sections (page 3 of 5)

mwave9(140) va1 f1 va1 va2 t2 va1 v1 v2 va1 v4 ob va4 va3 va1 t1 k va5 v5 eh va1 f2 v3 sax f1 t2 va2 va1 v1 v2 v4 ob va4 va3 va1 t1 k va5 v5

f2 eh v3 sax va1 va2 f1 t2 v1 v2 v4 ob va4 t1 va1 va3 va5 k v5 f2 eh v3 f1 sax va2 va1 t2 v1 v4 ob va4 k va3 t1 va1 va3 sax

f1 va2 t2 v1 v2 v4 ob va4 va3 va1 eh t1 k va5 v5 v3 f2 sax f1 va2 t2 va1 v1 v2 v4 va4 ob va3 k va5 v5 f2 t1 eh v3 va1 sax f1

va2 t2 v1 v2 v4 ob va4 va3 k va5 v5 f2 t1 va1 sax v3 f1 va2 fa2

ewave9(124) fa1 sax th b p1 fa2 p2 h2 va4 h1 fa2 sax th b p1 fa1 p2 h2 va4 h1 fa2 sax th b p1 fa1 p2 h2 va4 h1 fa2 sax th

b p1 fa1 p2 h2 va4 h1 fa2 sax th t1 sax p2 h2 va4 h1 fa2 sax th b p1 fa1 p2 h2 va4 h1 fa2 sax th b p1 fa1 p2 h2

va4 h1 fa2 sax th p1 fa1 p2 h2 va4 h1 fa2 sax th b p1 fa1 p2 h2 va4 h1 fa2 sax th b p1 fa1 p2 h2 va4 h1 fa2 sax

th p1 sax p1

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ewave8(214) sax b fa2 p1 th h1 h2 p2 fa1 p1 b fa2 th h1 h2 p2 fa1 sax p1 b fa2 th h1 h2 va2 p2 va1 sax fa1 p1 eh b va5 va3

th h1 h2 p2 sax fa1 p1 b fa2 th h1 va4 ob v4 h2 p2 sax fa1 f1 sax fa1 f1 sax fa1 p1 b va5 va3 th h1 va4 h2 t2 p2

va2 v1 v2 fa2 va1 va2 v1 sax p1 fa1 sax p1 b va5 va3 fa2 th h1 va4 v4 h2 t2 p2 v1 v2 fa2 va2 va1 sax p1 b va5 va3 fa2 th h1 va4 h2

p2 sax fa1 p1 b va5 fa2 va3 th h1 h2 va2 p2 va1 sax fa1 p1 eh fa2 b p1 fa2 h1 th h2 p2 sax fa1 sax p1 b fa1 sax p1

b th h1 h2 p2 t2 p2 fa1 p1 b fa2 k va3 th h1 va4 fa2 h2 p2 va2 va1 sax fa1 sax fa1 p1 b va5 k t1 eh fa2 va3 h1 va4 th

h2 v2 v1 p2 va2 va1 sax fa1 p1 va5 b fa1

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mwave8(255) f2 v5 va5 k ob v4 va4 ob k f2 v5 f2 va5 t1 va3 k ob v4 va4 ob k v5 f2 v3 p1 f2 eh t1 v5 va5 va3 v2 v4 va4 va3 va5 v5 f2 v5 va5

k t1 va3 v4 ob va4 k va3 t1 va5 v5 f2 v3 eh f2 v5 va5 k t1 va3 v4 va4 ob k va5 v5 v3 va1 va2 v1 v4 va4 ob k va3 t1 va5 b eh v3

p1 eh b v5 va5 k va3 va4 ob v4 v1 v4 ob va4 f1 va3 k va5 t1 va5 eh b f2 v5 va5 k ob v4 va4 va3 t2 v1 va2 va1 f1 v3 f2

v5 va5 k ob v4 t2 f1 v3 eh t1 va3 v1 va2 va1 f2 v5 va5 k ob v4 t2 f1 v3 eh t1 va3 v2 v1 va2 va1 f2 v5 va5 k ob va4

t2 f1 v3 eh t1 va3 v1 va2 va1 f2 v5 va5 k ob v4 t2 f1 v3 eh t1 va3 v2 v1 va2 va1 f2 v5 va5 k ob v4 t2 f1 v3 eh t1 va3 v2

v1 va2 va1 f2 v5 va5 k ob v4 v2 t2 f1 sax eh t1 va3 v2 v1 va2 va1 f2 v5 va5 k ob v4 t2 f1 v3 eh t1 va3 v2 v1 va2 va1 f2 v5 va5

k ob v4 t2 va1 v3 eh t1 va3 v2 v1 va2 va1 va1

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272

Example 4.3.4d. Sequence of instrumental moves in wave sections (page 4 of 5)

mwave11(211) v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh t1 va3 v2

t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4

v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 ob k v4 v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh t1 va3 v2

t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4

v1 va2 va1 eh t1 va3 v2 t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh va3 v2 t2 f1 v3 f2 v5 k ob v4 v1 va2 va1 eh va3 v2 t2 f1

v3 f2 v5 k ob v4 v1 va2 va1 t1 f1

ewave11(166) va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b va5

h1 va4 th fa2 sax fa1 p2 h2 p1 b va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b va5 h1

va4 th fa2 sax fa1 p2 h2 p1 b va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b va5 h1 va4

th fa2 sax fa1 p2 h2 p1 b va5 h1 va4 th fa2 sax fa1 p2 h2 p1 b v5 va5 h1 fa1 th h2 p1 b v5 va5 h1 fa1 th

h2 p1 b v5 va5 b

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tu eu tu eu tu eu

tu eu tu eu tu eu

tu eu tu eu tu eu

tu eu tu eu tu eu tu eu

mwave10(156) fa2 t1 va3 v2 v1 va2 va1 eh f2 v5 k ob v4 t2 f1 v3 fa2 t1 va3 v2 v1 va2 va1 eh f2 v5 k ob v4 t2 f1 fa2 v3 t1 va3 v2 v1 va2

va1 eh f2 v5 k ob v4 fa2 t2 f1 v3 t1 va3 v2 v1 va2 va1 eh f2 v5 k fa2 ob v4 t2 f1 v3 t1 va3 v2 v1 va2 va1 fa2 eh f2 v5 k

ob v4 t2 f1 v3 t1 fa2 va3 v2 v1 va2 va1 eh f2 v5 k ob fa2 v4 t2 f1 v3 t1 va3 v2 v1 fa2 va2 va1 eh f2 v5 k ob v4 fa2 t2

f1 v3 f2 v5 k fa2 ob v4 t2 f1 v3 fa2 f2 v5 ob k v4 fa2 t2 f1 v3 fa2 f2 k v5 fa2 ob v4 fa2 fa2 t2 fa2 fa2

ewave10(292) p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 fa1

p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 fa1

p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 p1 fa1 b h1 th h2 fa2 p2 p1 fa1 b h1 th h2 fa2 p2 fa1

p1 b h1 th h2 fa2 p2 fa1 p1 b h1 th h2 fa2 p2 fa1 p1 fa1 sax p2 h2 va4 th h1 va5 b p1 fa1 sax p2 h2 va4

th h1 va5 b p1 fa1 sax p2 h2 va4 th h1 b va5 p1 fa2 sax p2 h2 va4 th h1 va5 b p1 fa1 sax p2 h2 va4 th h1 va5 b

p1 fa1 sax p2 h2 va4 th h1 va5 b p1 fa1 sax p2 h2 va4 h1 th va5 b p1 fa1 sax p2 h2 va4 th h1 va5 b p1 fa1 sax p2

h2 va4 th h1 va5 b p1 fa1 sax p2 h2 va4 th h1 va5 b p1 fa1 sax p2 h2 th h1 va5 b p1 fa1 sax p2 h2 va4

th h1 va5 b p1 fa1 sax p2 h1 eh

kb kb

kb kb

kb kb kb

kb kb kb

tu eu tu eu tu eu tu eu

tu eu tu eu tu eu tu eu

tu eu tu eu tu eu tu eu

tu eu tu eu eu tu eu tu

eu tu eu tu eu tu

eu tu eu tu tu eu

eu tu eu tu eu tu tu eu tu

eu tu

273

fa2 fa2 v1 fa2 fa2 fa2 eh v1

eh th eh h1 eh p1 eh b eh va5 sax eh va5

f1 t2 ob k f2 t1 eh f1 t2 ob k f2 t1 eh f1 t2 k ob k f2 t1 ob f1 ob k f2 k f2

b h1 fa2 h2 th p2 sax fa1 p1 b h1 fa2 h2 p2 sax fa1 p1 th b h1 fa2 h2 th b

5th_bridge_mwave(12) eu eu eu eu

5th_bridge_ewave(13)

6th_bridge_mwave(31) kb kb kb

6th_bridge_ewave(29) eu tu eu tu eu

Example 4.3.4f. Sequence of instrumental moves in bridges 5 and 6

mwave12(360) f2 h1 ob v4 t2 v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 t2 v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 v4 f1

v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 t2 v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 t2 v1 v1 t2 t1 eh va2

f1 v3 f2 v5 va5 k h1 ob v4 t2 v1 v2 v3 v3 f2 v5 va5 k h1 ob v4 t2 v1 t1 eh ob v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5

k h1 ob v4 t2 k t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 f2 t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 t1 f1 v3 f2 v5 va5 k

h1 ob v4 t2 t1 eh f1 v3 f2 v5 va5 k h1 ob v4 t2 ob f2 v5 va5 k h1 ob v4 t2 k f2 t1 k h1 ob v4 t2 v1 v2

eh f1 t2 f1 v3 f2 v5 va5 k h1 ob v4 t2 v1 v2 ob k f2 v5 va5 k h1 ob v4 t2 v1 v2 va3 t1 eh va2 f1 v3 f2 k h1 ob

v4 t2 v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5 t1 v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 t2 ob t1 eh va2 f1 v3 f2 v5

va5 k h1 ob f1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 t2 v1 v2 va3 t1 eh va2 f1 v3 f2 v5 va5 k h1 ob v4 t2 v1 v2 va3 ob

va2 eh t1 va3 v2 v1 t2 v4 ob h1 k va5 v5 f2 v3 va2 f1 eh t1 va3 v2 v1 t2 v4 ob h1 k va5 v5 f2 v3 f1 va2 eh t1 va3 f2 k

ewave12(281) b p1 h2 p2 fa1 sax va1 fa2 th va4 b p1 p2 fa1 sax va1 fa2 th va4 b p1 b p2 fa1 sax va1 fa2 th va4 b p1 h1 p2 fa1 sax va1 fa2

th va4 b p1 h2 p2 fa1 sax va1 fa2 th va4 b p1 h2 fa2 sax va1 fa2 th va4 b p1 h2 h2 p2 fa1 sax va1 fa2 th va4 b p1 h2

th sax va1 fa2 th va4 b p1 h2 p2 fa1 sax va1 fa2 th va4 b p1 h2 p2 fa1 sax va1 fa2 th va4 b p1 h2 sax fa1 sax va1 fa2 th

va4 b p1 h2 fa1 p1 b h1 fa2 h2 fa2 th va4 b p1 h2 p2 fa1 sax va1 fa2 th va4 b p1 h2 sax fa1 sax va1 fa2 th va4 b p1 h2

p2 p1 fa1 sax va1 fa2 th va4 th b p1 fa2 h2 p2 fa1 sax va1 fa2 th va4 b p1 fa2 h2 p2 fa1 b p2 fa1 sax va1 fa2 th va4 b p1

fa2 h2 p2 fa1 h1 va1 p1 fa1 sax va1 fa2 th va4 b p1 fa2 h2 p2 fa1 sax va1 fa2 th va4 b p1 h2 p2 fa1 sax va1 fa2 th va4

b p1 h2 h2 fa2 th va4 b p1 h2 p2 fa1 th va1 fa2 th va4 b p1 h2 p2 p2 fa1 sax va1 fa2 b p1 h2 p2 fa1 sax va1

p2

kb kb

kb kb

kb kb kb

kb kb

kb kb kb kb kb

kb kb kb

kb kb

kb kb kb

kb kb

eu tu eu

eu tu eu eu tu

eu eu tu eu tu

eu tu eu tu

eu tu eu tu

eu tu eu tu eu tu

eu eu eu tu tu eu eu

Example 4.3.4e. Sequence of instrumental moves in wave sections (page 5 of 5)

274

E-Wave 1Average speed: 8.9 m/sec

10:52

E-Wave 2Average speed: 9.4 m/sec

E-Wave 3Average speed: 9.0 m/sec

6:46

M-Wave 1Average speed: 9.4 m/sec

M-Wave 2Average speed: 16.1 m/sec

M-Wave 3Average speed: 13.8 m/sec

M-Wave 4Average speed: 9.4 m/sec

M-Wave 5

1st Bridge 2nd Bridge

E-Wave 4

3rd Bridge

0

50

100

150

200

0

50

100

150

200Anfangs-Duett

0 1:53

Eingang

12:53 14:07

15:13 16:29 17:41 19:03

0

50

100

150

200

0

50

100

150

200

10:30 15:13

20:34

Example 4.4.1a. Speed of motion in LICHTER-WASSER (page 1 of 2)

0 20 40 60 80 10010 30 50 70 80

Scale1 minute

Seconds

Velo

city i

n m

/sec

Velo

city i

n m

/sec

Time (in minutes)

Data have been coorelated with the timings on the CD recording.

276

E-Wave 5 E-Wave 6 E-Wave 7 E-Wave 8 E-Wave 9

M-Wave 6 M-Wave 7 M-Wave 8

E-Wave 10 E-Wave 11 E-Wave 12

M-Wave 9 M-Wave 10 5th Bridge M-Wave 11 6th Bridge M-Wave 12

20:34 21:29 22:56 24:34 25:56 27:30 28:40 29:45

36:26 37:42 39:27 40:09 41:21 41:5445:47

Ausgang

48:09

Schluss-Duett

49:53

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

36:26

4th Bridge

Example 4.4.1b. Speed of motion in LICHTER-WASSER (page 2 of 2)

277

Example 4.5.2a (continued). Movement of singers through all wave and bridge sections (chart)

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

E1 E2

B1 B2 B3

E3 E4 E5 E6 E7 E8 E9 E10 E11 E12

B4/1A1 A2 B4/2 A3 B5 B6

tu

f2

va1soprano position

tenor position h1

v1 t2 eu th f1 k b fa2 sax fa1 v3 f2 p1 eh v5 va5 k-h1-ob-v4

k va5 b t1 eh fa1 p2 kb v2 va2 va3 va1 fa2 eu v1 ob v3

296

1 19.76

12 6.25

26 6.25

28 4.48

6 6.25

29 4.78

0 13.98

33 12.5

0 13.98

14 8.84

1 8.84

1 12.5

4 11.55

26 6.25

5 12.5

1 6.25

0 18.75

H1

K

Va5

B

T1

Eh

Fa1

P2

Kb

V2

Va2

Va3

Va1

Fa2

Eu

V1

Ob

F2

H1

K

Va5

B

T1

Eh

Fa1

P2

Kb

V2

Va2

Va3

Va1

Fa2

Eu

V1

Ob V3 0 31.25

Moves byTeonor

Number of CorrespondingInstrumental

Moves

DirectDistance(meters)

Averages:

0 18.75

5 8.84

13 8.29

3 4.48

13 13.98

0 20

0 25.77

1 9.57

9 12.5

25 12.5

41 5.18

4 6.25

35 12.5

7 6.25

9 6.25

0 13.98

42 6.25

38 6.25

26 6.25

24 6.25

63 6.25

Number of CorrespondingInstrumental

Moves

DirectDistance(meters)

Va1

V1

T2

Eu

Th

F1

K

B

Fa2

Sax

Fa1

V3

F2

P1

Eh

V5

Va5

K

H1

Ob

Va1

V1

T2

Eu

Th

F1

K

B

Fa2

Sax

Fa1

V3

F2

P1

Eh

V5

Va5

K

H1

Ob

V4

Tu

Moves bySoprano

Averages:

Example 4.5.2b.Comparison of instrumental and vocal motion

17.1 10.3m

10.4 9.65m

297

Example 4.6.1a.

Key 2

= Instrument in Eve Orchestra

= Instrument in Michael Orch.

= Location of a loudspeaker

0

50

100

150

200 = times used in M-Orch.

= times used in E-Orch.

= total times used

100 110 120 130 140 150 160 170 180

Key 1 Shading For Each Instrument Based on the Number of Times it is Used

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

01

23

45

67

89

1011

1213

1415

1617

1819

2021

2223

2425

2627

2829

30

V5

F2

P1

V3

Fa1

B

Sax

Va5

Eh

F1

T1

Va1

K

Va3

Fa2

Va2

P2

H1Th

V1V2

T2

Va4

Eu

Ob

V4

H2

K b

Tu

conductor

mixing console

synth.

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

2000

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

2000

50

100

150

200

0

50

100

150

200

0

50

100

150

2000

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

Number of times each instrumentalist is usedin LICHTER-WASSER (according to data from Examples 4.3.4a-e)

298

Example 4.6.1a (continued).

V5F2P1V3Fa4BSaxVa5EhF1T1Va1KVa3Fa2Va2P2ThV1H1V2T2Va4EuObV4H2KbTu

Data For Each Instrument

violin 5flute 2

trombone 1violin 3

bassoon 4bass clarinet

saxophoneviola 5*

english hornflute 1

trumpet 1viola 1

clarinetviola 3

bassoon 2viola 2

trombone 2tenor horn**

violin 1horn 1

violin 2trumpet 2

viola 4euphonium

oboeviolin 4horn 2

e-flat clarinettuba

instrument abbreviation M Orch. E Orch.

86

1514

1401361215714

68

401228

13922

132124

6104

48

80138

25

1326

127

117122

14108

14201693

109116111

93128107

32103

921

1054097

10056

8121117

1697

6

125128165112154156137

150123122119133140135171125141

145111144101108136146123122148103133

totals

*Although this the viola 5 is technicallyin the Eve-Orchestra, it actually plays morenotes in the Michael-Orchestra.

**Although Tenor Horn is in theMichael-Orchestra, it actuallyplays more notes in the Eve-Orchestra.

299

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

01

23

45

67

89

1011

1213

1415

1617

1819

2021

2223

2425

2627

2829

30

V5

F2

P1

V3

Fa1

B

Sax

Va5

Eh

F1

T1

Va1

K

Va3

Fa2

Va2

P2

H1Th

V1V2

T2

Va4

Eu

Ob

V4

H2

K b

Tu

conductor

mixing console

synth.

Example 4.6.2a.

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

1500

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

0

50

100

150

Key 2

= Instrument in Eve Orchestra

= Instrument in Michael Orch.

= Location of a loudspeaker

0

50

100

150

= times used in M-Orch.

= times used in E-Orch.

= total times used

60 70 80 90 100 110 120 130 140

Key 1 Shading For Each Instrument Based on the Amount of Time it is Used

Amount of Time (in seconds) each instrumentalist isused in LICHTER-WASSER (according to data from Exs. 4.3.4a-e)

300

V5F2P1V3Fa4BSaxVa5EhF1T1Va1KVa3Fa2Va2P2ThV1H1V2T2Va4EuObV4H2KbTu

Data For Each Instrument

violin 5flute 2

trombone 1violin 3

bassoon 4bass clarinet

saxophoneviola 5

english hornflute 1

trumpet 1viola 1

clarinetviola 3

bassoon 2viola 2

trombone 2tenor horn*

violin 1horn 1

violin 2trumpet 2viola 4**

euphoniumoboe

violin 4horn 2

e-flat clarinettuba

instrument abbreviation M Orch.(in sec.)

E Orch.(in sec.)

6.416.9

116.210.3

115.8100.2

88.662.514.811.5

4.531.914.526.8

101.323.5

109.076.3

5.784.2

1.75.5

37.1123.0

0.44.4

94.06.1

95.3

54.478.917.056.112.916.216.053.286.078.172.963.579.458.833.953.6

7.424.161.119.065.376.5

42.65.7

90.869.916.885.7

7.3

60.895.8

133.266.4

128.8116.4104.6115.7100.9

81.677.495.493.985.6

135.277.1

116.4100.4

66.8103.2

67.082.0

79.7128.7

91.274.3

110.891.7

102.6

totals(in sec.)

Example 4.6.2a (continued).

*Th is used for more time in the Eve-Orchestra than the Michael-Orchestra, although it is aMichael instrument.

**Va4 is used for slightly more time inthe Michael-Orchestra than theEve-Orchestra, although it is anEve instrument.

301

Example 4.6.3bv1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v18

v19

v20

v21

v22

v23

v24

v25

v1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0

0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0

1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 0

0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0

1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0

0 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0

0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0

1 1 0 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0

0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 0 0 0

1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1

0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1

0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1

0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0

0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0

0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 1 1 0 1 0

0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1

0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1

0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 0 0 1 0

0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1

0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0

The A Matrix for LICHTER-WASSER.

v26 v27 v28 v29

v26

v27

v28

v29

0 0 0 0

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

1 0 1 0

0 1 1 1

0 0 1 0

0 0 0 1

1 1 0 1

0 0 1 0

0 1 0 0

1 1 1 0

0 0 1 1

1 1 1 0

0 1 0 1

1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1

0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1

0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0

0 1 1 0

1 0 1 1

1 1 0 1

0 1 1 0

V5 F2 P1 V3 Fa1 B Sax Va5 Eh F1 T1 Va1 K Va3 Fa2 Va2 P2 Th V1 H1 V2 T2 Va4 Eu Ob V4 H2 Kb Tu

V5

F2

P1

V3

Fa1

B

Sax

Va5

Eh

F1

T1

Va1

K

Va3

Fa2

Va2

P2

Th

V1

H1

V2

T2

Va4

Eu

Ob

V4

H2

Kb

Tu

303

Example 4.6.3cv1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25

v2

v3

v4

v5

v6

v7

v8

v9

v10

v11

v12

v13

v14

v15

v16

v17

v18

v19

v20

v21

v22

v23

v24

v25

v1 4 4 4 1 2 2 2 3 4 1 5 1 3 4 3 0 1 0 0 4 0 0 2 0 1

4 8 4 4 3 5 3 5 5 3 5 3 3 5 6 3 1 0 1 4 1 0 1 0 2

3 5 9 4 4 6 4 3 7 5 6 6 3 5 7 3 5 4 1 5 2 2 3 5 2

3 5 5 7 3 6 4 4 6 4 4 6 2 4 5 6 4 2 2 4 2 2 0 2 2

2 4 5 3 8 3 5 4 5 3 4 4 1 4 9 5 4 2 1 4 2 3 3 4 2

3 6 7 7 4 1 4 6 6 5 5 7 7 6 8 6 3 5 2 5 2 2 2 5 5

1 5 6 4 4 4 6 2 5 5 4 7 0 3 7 6 5 3 2 4 2 5 0 3 2

4 6 3 5 3 7 2 8 4 2 5 3 5 5 7 3 1 2 2 3 2 0 3 2 4

4 6 4 4 4 5 5 6 9 3 6 6 3 5 8 4 4 3 5 4 2 3 3 3 4

1 3 5 3 3 2 3 1 4 8 5 7 1 3 8 5 4 2 4 2 3 3 1 5 2

5 8 6 4 3 6 4 6 7 4 10 6 4 6 8 4 4 2 4 5 4 4 2 4 5

3 4 3 3 1 7 1 5 4 1 4 3 7 5 4 2 1 4 3 4 1 1 4 3 5

3 4 3 3 1 7 1 5 4 1 4 3 7 5 4 2 1 4 3 4 1 1 4 3 5

5 6 5 1 1 5 2 4 5 2 7 2 4 10 11 4 3 5 5 9 3 4 5 2 7

4 5 8 5 6 7 3 7 6 6 7 7 6 7 15 8 6 5 5 6 6 5 6 8 4

1 3 4 2 2 2 3 1 2 4 5 5 1 3 6 7 4 2 4 2 5 6 0 4 3

1 3 4 4 4 3 4 2 5 4 3 6 1 2 6 7 8 4 3 4 2 5 0 5 2

2 4 3 2 4 6 3 5 5 1 3 3 4 6 10 4 3 8 4 6 2 2 6 5 6

1 2 2 2 1 2 2 1 5 4 3 5 2 3 6 6 5 3 7 1 4 6 1 4 3

3 4 7 1 1 6 3 3 3 1 6 3 4 7 7 2 4 6 1 9 3 2 5 3 4

0 2 4 1 0 5 1 1 1 2 5 3 3 5 8 3 1 5 4 6 6 3 5 4 5

3 3 2 2 3 1 2 3 4 4 3 4 1 4 10 6 4 3 6 2 3 7 1 4 4

2 3 4 0 2 4 2 4 2 0 5 2 3 6 8 1 2 5 3 7 4 2 7 4 6

0 2 4 1 3 3 3 1 3 3 3 4 1 2 6 5 4 3 2 4 3 4 2 6 1

1 1 0 0 1 3 0 3 0 0 2 1 3 3 5 0 0 4 2 3 1 0 3 2 5

The A2 Matrix for LICHTER-WASSERv26 v27 v28 v29

v26

v27

v28

v29

2 0 0 0

1 1 0 0

1 2 3 3

1 2 1 2

2 4 1 2

2 4 3 3

1 3 2 4

2 2 1 0

3 3 4 1

2 1 4 3

3 1 4 1

2 3 3 1

2 3 3 1

7 7 5 4

6 6 5 6

2 3 3 5

1 4 4 4

4 7 7 2

2 4 5 4

6 4 4 5

6 5 6 4

4 4 4 3

6 5 4 4

2 4 5 5

3 3 3 0

0 1 3 0 1 4 1 1 1 1 3 1 3 5 6 2 1 4 2 5 4 3 4 4 4

0 2 3 1 3 5 2 2 3 0 2 1 2 5 9 4 3 7 3 5 2 4 4 3 6

0 1 2 0 1 1 1 0 2 2 3 2 2 3 5 4 3 5 4 4 3 3 4 5 3

0 1 6 1 1 3 3 0 2 4 3 5 1 2 6 4 5 4 3 4 4 5 2 5 2

5 5 4 4

4 10 5 4

3 4 7 3

3 4 4 8

V5 F2 P1 V3 Fa1 B Sax Va5 Eh F1T1 Va1 K Va3 Fa2 Va2 P2 Th V1 H1 V2 T2 Va4 Eu Ob V4 H2 Kb Tu

V5

F2

P1

V3

Fa1

B

Sax

Va5

Eh

F1

T1

Va1

K

Va3

Fa2

Va2

P2

Th

V1

H1

V2

T2

Va4

Eu

Ob

V4

H2

Kb

Tu

Totals

Totals

53

81

118

100

103

141

105

96

124

98

134

92

92

140

181

94

105

127

96

117

104

104

107

89

52

82

104

80

95

63 111 125 75 77 131 79 96 119 83 130 116 87 133 208 116 94 106 88 129 79 87 83 107 106 88 108 104 86

304

Example 4.6.4d. Connectivity out, based on LW A2 Matrix

V5 P1 V3 Fa1

B Sax

Va5 Eh F1

T1 Va1

K Va3 Fa2 Va2 P2

H1

Th V1

V2 T2

Va4 Eu

Ob V4 H2 Kb Tu

F2

50 19070 90 110 130 150 170

V5 P1 V3 Fa1

B Sax

Va5 Eh F1

T1 Va1

K Va3 Fa2 Va2 P2

H1

Th V1

V2 T2

Va4 Eu

Ob V4 H2 Kb Tu

F2

Example 4.6.4e. Connectivity in, based on LW A2 Matrix

The darkest instruments sendmelodic movement to thegreatest variety of places for pathsof length 2.

Here, the darkest instruments receivemelodic movement from thegreatest variety of sources for pathsof length 2.

Key for both examplesnumber of moves in/out

305

314

Example 5.3.1a. Front page of Xenakis’s score to Terretektorh

Example 5.3.2a. Three spirals used by Xenakis in Terretektorh.

Archimedian Spiral

Logarithmic Spiral

Hyperbolic Spiral

r = a + bu

r = a/u

r = abu

Formulas are given in polar coordinates, where radius r is a continuous function of angle u.

Distance between arms is constant.

Often found in nature.

Inverse of Archimedian spiral

315

316

Example 5.3.2b. Maria Harley’s analysis of spatial motion in mm. 1-74 of Terretektorh.

319

1 2.53

2

A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

Example 5.3.2g. Calculations of speed in Terretektorh, mm. 1-66.

DistanceTraversed(meters)

Time(seconds)

Speed(m/sec)

Structure SpatialAreas

A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

3 A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

4 A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

5.06

2.00

2.00

2.00

2.00

2.00

2.00

6.67

5.83

5.00

3.33

2.50

1.83

2.00

1.00

1.50

1.50

1.83

2.17

3.00

3.67

4.17

10.75

7.00

2.50

1.00

0.36

0.14

0.51

5 H5/6 - H1

H1 - H2

H2 - H3

H3 - H4

H4 - H5

H5 - H6

H6 -H5

H5 - H4

H4 - H3

H3 - H2

H2 - H1

0.49

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.76

0.87

1.01

1.21

1.52

2.02

2.77

5.06

3.37

3.37

2.77

2.33

1.69

1.38

0.47

0.72

2.02

5.06

14.06

36.14

9.92

4.5

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

9.18

4

4

4

4

4

4

4

4

4

4

6 A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

7.20

4.47

2.83

1.63

1.20

0.55

0.37

.70

1.13

1.79

3.10

4.22

9.20

13.68

7 A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

4.5

2.75

1.75

1

0.63

0.37

0.25

1.24

1.84

2.89

5.06

8.03

13.68

20.24

2.53

2.53

2.53

2.53

2.53

2.53

DistanceTraversed(meters)

Time(seconds)

Speed(m/sec)

Structure SpatialAreas

322

Example 5.3.2h. Calculations of speed in Terretektorh, mm. 65-74

8a A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

2.75

1.92

0.83

0.63

0.37

0.25

0.11

1.84

2.64

6.10

8.03

13.68

20.24

46.00

8b A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

1.75

1.00

0.65

0.48

0.12

0.25

0.08

2.89

5.06

7.78

10.54

42.17

20.24

63.25

8c A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

1.1

0.5

0.5

0.25

0.13

0.12

0.10

4.06

10.12

10.12

20.24

42.17

38.92

50.60

8d A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

0.62

0.38

0.27

0.10

0.10

0.07

0.08

8.16

13.32

18.74

50.60

50.60

72.29

63.25

8e A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

0.5

0.02

0.1

0.1

0.23

-0.08

0

–

–

10.12

253.00

50.60

50.60

22.00

8f A5/6 - B5/6

B5/6 - C5/6

C5/6 - D5/6

D5/6 - E5/6

E5/6 - F5/6

F5/6 - G5/6

G5/6 - H5/6

5.06

5.06

5.06

5.06

5.06

5.06

5.06

-0.4

0.13

0.12

0

0.08

0

0.07

–

38.92

42.17

–

–

63.25

72.29

DistanceTraversed(meters)

Time(sec.)

Speed(m/sec.)

Struc-ture

SpatialAreas

DistanceTraversed(meters)

Time(sec.)

Speed(m/sec.)

Struc-ture

SpatialAreas

(!!)

323

Example 5.4.4b.

audience

conductor

VI VII VIII

III IV V

I II

Example 5.4.4a. Whittaker’s diagram of Tallis’s Spem in Alium

Whittaker 1940 (1929), p. 89.

324

Whittaker’s 1929 spatialization

9285 86 87 88 89 90 91 93 94 95 96 97

8

7

6

5

4

3

2

1

98 99 100 101 102 103 104 105 106 107 108 109 110

Domine Deus, Creator caeli et terrae respice

Example 5.4.4c. Detailed diagram of mm. 85-110, showing paired antiphonal passages.

I

II

III

IV

V

VI

VII

VIII

audience

Example 5.4.4d.

Key

contrapuntal texture

homophonic texture

325

Circular arrangement of choirs