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UNIVERSITY OF CINCINNATI

Date: August 06, 2007

I, Joseph Williams ,

hereby submit this work as part of the requirements for the degree of:

Master of Music

in:

CCM Division of Composition, Musicology and Theory (Theory)

It is entitled:

Discontinuous Continuity?: Structural Analysis of Sofia

Gubaidulinas String Quartets

This work and its defense approved by:

Chair:Dr. Catherine Losada, Ph.D.

Dr. David Carson Berry, Ph.D.

Dr. Robert Zierolf, Ph.D.



Discontinuous Continuity?:

Structural Analysis of Sofia Gubaidulinas String Quartets

A thesis submitted to the

Division of Graduate Studies and Research

of the University of Cincinnati

in partial fulfillment of therequirements for the degree of

MASTER OF MUSIC

in the Division of Composition, Musicology, and Theory

at the College-Conservatory of Music

2007

by

Joseph Williams

B.M. Stephen F. Austin State University, 2004

Committee Chair: Dr. Catherine Losada, Ph.D.



iii

ABSTRACT

Since the early 1980s, the majority of studies on Russian composer Sofia Gubaidulina (b.

1931) have centered on her historical influence. These writings, although necessary and

interesting, have neglected any detailed analytical approaches in favor of aesthetic descriptions

of her experimental new sounds and manipulations of proportions in large scale form. In

particular, these studies have placed considerable stress on the discontinuous nature of her

compositions, characterized as having episodic forms and lack of development. This study

demonstrates how Gubaidulinas music is not as discontinuous as it first appears. Utilizing the

composers four string quartets as specific examples, the author shows in detail how, although

seemingly episodic on the surface, they each contain streams of continuity as a result of similarly

conceived structural processes that organize the various musical elements, including pitch,

rhythm, and large-scale form. Through an analysis of these quartets, the author illustrates the

processes at work and demonstrates the specific manner in which each process aids in

establishing continuity and coherence throughout.

The first quartet illustrates the process of expansion, in which pitches (and sometimes

rhythms) are organized in one of two ways. First, they may radiate outward in pitch space from

some central point or, second, they may expand outward in pitch space using some interval as a

measure of distance. The second quartet utilizes inversional symmetry as a means of organizing

large collections of pitches about a single asserted axis. This quartet also demonstrates the

process of gap-fill, in which an interval is established and gradually filled in chromatically. The

third quartet presents an organizing principle based on the intervals inherent to the pitch

collection. Specific intervals are assigned particular functions in order to structurally arrange the



iv

individual pitches both linearly and harmonically. In this case, ic5 and ic1 act as structural

markers between sections and as a means of differentiating between individual voices. Finally,

the fourth quartet illustrates an organizing process based on sound masses. These collections are

shown to mutate into one another via transformations.



v



vi

CONTENTS

LIST OF FIGURES viii

CHAPTER ONE INTRODUCTION 1

The Religion of the Avant-Garde: Classical vs. Experimental 2

Gubaidulinas Place in the Avant-Garde 11

Elements of Individual Style 15

The Four String Quartets 21

CHAPTER TWO ANALYSIS OF STRING QUARTET NO. 1 (1971): 24EXPANSION PROCESSES

Wedge Expansion 27

Additive Expansion 32

Conclusion 37

CHAPTER THREE ANALYSIS OF STRING QUARTET NO. 2 (1987): 39

GAP-FILL PROCESSES AND INVERSIONAL SYMMETRY

Section A 41

Section B 46

Conclusion 52

CHAPTER FOUR ANALYSIS OF STRING QUARTET NO. 3 (1987): 53

INTERVALLIC DEFINITION

Hierarchical and Formal Separation 55

Motivic Development and Transformations 61

Conclusion 65



vii

CHAPTER FIVE ANALYSIS OF STRING QUARTET NO. 4 (1993): 67

VERTICALIZED SOUND MASSES

The Primary Pitch Collections 69

Relating Individual Sound Masses 72

Conclusion 79

EPILOGUE CONCLUDING REMARKS 81

BIBLIOGRAPHY 84

MUSICAL FIGURES 88



viii

LIST OF FIGURES

Figure 1.1: Diagram of Avant-Garde dichotomy 88

Figure 2.1: Abstract example of additive expansion 89

Figure 2.2: Viola (mm. 1-2) 90Figure 2.3: Violin II (R1, mm. 1-4) 91

Figure 2.4: Wedge expansion from F#6 (R8, mm. 3-10) 92

Figure 2.5: Wedge expansion from C#4 in violin I, violin II, and cello (R6, mm. 4-7) 93

Figure 2.6: Wedge expansion from E5 in violin II (R7, mm. 2-9) 94

Figure 2.7: Wedge expansion from D4 (R21, mm. 1-15) 95

Figure 2.8a: Violin II and cello (R25, mm. 1-11) 97

Figure 2.8b: Reduction of cello (R25, mm. 1-11) 100

Figure 2.9: Viola (R1, mm. 1-2) 101

Figure 2.10: Violin II, viola, and cello (R4, mm. 1-4) 102

Figure 2.11: Cello (R1, mm. 1-4) 103

Figure 2.12: Additive expansion in rhythm and pitch (R48, mm. 4-15) 104Figure 2.13: Wedge expansion and additive expansion in violin I and cello (R42-43) 106

Figure 3.1: Emphasis on G4/G5 dyad in m. 1 107

Figure 3.2a: Violin I/cello motive (R5, m. 1) 108

Figure 3.2b: Gap-fill process in violin I (R5-15) 108

Figure 3.3: New violin I/cello motive (R15) 109

Figure 3.4: Moment of gap-fill completion in violin I motive (R20) 110

Figure 3.5: R22-25 111

Figure 3.6: R21-22 112

Figure 3.7: Reduction of R21-22 with inversional symmetry in voice leading 113

Figure 3.8: Reduction of R25-27 with inversional symmetry in voice leading 114

Figure 4.1: Formal diagram of Third String Quartet 115

Figure 4.2: Primary set verticalized as sonorities (R49) 116

Figure 4.3: Melody and accompaniment in opening measures 117

Figure 4.4: Violin II accompaniment with second melodic segment (R1-2) 118

Figure 4.5: Alternation of primary-set melodic material and ic5 structural divisions (R13-14) 119

Figure 4.6: Ic5 structural divider (circled) in viola and cello (R55-56) 120

Figure 4.7: Structural opposition between ic5 (R10) and ic1 (R11) 121

Figure 4.8: Structural opposition between ic5 and ic1 (R15-17) 122

Figure 4.9: Simultaneous structural opposition between ic5 and ic1 (R50) 124

Figure 4.10: Melodic/accompanimental emphasis of ic1 and [G#, A, Bb] (R23) 125Figure 4.11: Linear version of primary set replicating pitch-classes from first sonority 125

at R49 (R52)

Figure 4.12: Violin I melody replicating cello pitches from R52 (R53-56) 126

Figure 4.13: Motivic development using (R58-60) 127

Figure 4.14: Motivic development of violin I melody using (R66-68) 128



ix

Figure 4.15: Replication of primary-set cello pitch-classes from R52 (R71-73) 129

Figure 4.16: Interaction between ic5 and ic1 (R37-38) 130

Figure 5.1: B-viola four-measure ostinato pattern 131

Figure 5.2: Reduction of recorded ensemble pitches into verticalized sound masses 131

Figure 5.3: First occurrence of Live Collection in each voice of live ensemble 132Figure 5.4: Pitch-space graph of violin I (R26) 133

Figure 5.5: Pitch-space graph of violin I (R14) 134

Figure 5.6: Pitch-space graph of live ensemble (R34-41) 135

Figure 5.7: Reduction of lower duet illustrating ascending thirds (R36-41) 137



1Jonathan Kramer, Beyond Unity: Toward an Understanding of Musical Postmodernism, in Concert

Music, Rock, and Jazz since 1945: Essays and Analytical Studies, ed. Elizabeth West Marvin and Richard Hermann

(Rochester: University of Rochester Press, 1995), 23.

1

Chapter 1

Introduction

Since the early 1980s, the majority of studies on Russian composer Sofia Gubaidulina (b.

1931) have centered on her historical influence. These writings, although necessary and

interesting, have neglected any detailed analytical approaches in favor of aesthetic descriptions

of her experimental new sounds and manipulations of proportions in large-scale form. The

formal studies have been informed specifically by the composers intense beliefs in the religious

significance and mystical nature of numbers along with their proportional relations to one

another when applied to rhythmic and sectional durations.

In particular, these studies have placed significant stress on the discontinuous nature of

her compositions. Jonathan Kramer characterized discontinuous music as having episodic

forms and lack of development.1Gubaidulinas musical style has often been described in this

exact manner. The disjunct, micro-chromatic pitch structures and sudden shifts in harmony,

rhythm, and texture create a musical environment that can be disturbing to an unwary listener.

The goal of this study is to demonstrate that this music is not as discontinuous as it appears.

Gubaidulinas four string quartets, although episodic on the surface, contain streams of

continuity as a result of similarly conceived structural processes that organize the various

musical elements, including pitch, rhythm, and large-scale form. Through an analysis of these



2

quartets, I will illustrate the processes at work and demonstrate the specific manner in which

each process aids in establishing continuity and coherence throughout.

The following discussion will place the composer in the proper historical and stylistic

context. It will first present the general attributes of the post-1945 avant-garde movement

followed by an overview of Gubaidulinas life as a Soviet citizen composing in this vein,

including considerations of her particular aesthetic views and a survey of the basic

characteristics of her individual style. Finally, it will address the problems of using conventional

analytic tools in a technical explanation of Gubaidulinas works along with my proposed

solution.

The Religion of the Avant-Garde: Classical vs. Experimental

One of the most salient features of twentieth-century musical aesthetics is the multiplicity

of approaches that arose throughout the era. In earlier historical periods a common aesthetic

approach was often embraced, but aesthetic approaches of the twentieth century seemed to

fragment into countless possible methods of creating artworks. I will focus on the particular

manner in which a dichotomy arose within the avant-garde school of composers in the years

following World War II.

Some composers were determined to find ways to control the musical elements in a strict

fashion in order to create a coherent whole; other composers, however, set their sights on

creating a phenomenological moment-by-moment experience of the work, thereby disregarding

the importance traditionally placed on the artwork as a cohesive whole. Because the former were

more traditional in their overall approach, I refer to them as classical avant-gardists while the



2Leonard B. Meyer,Music, the Arts, and Idea(Chicago: University of Chicago Press, 1967), 210. It should

be pointed out that Meyers duality is not universally accepted within scholarly circles. Ivo Supiispecifically

characterizes the duality as between matter and form rather than between Meyers material and content.

(Matter and Form in Music,International Review of Music Aesthetics and Sociology1, no. 2 (December 1970):

157.)

3

latter group are the experimental avant-gardists. Although this dichotomy over-simplifies a

period of music history that is characterized as one of the most multi-variant, these labels are

meant to distinguish the basic artistic motivations of composers writing in the years following

WWII until 1975. They were either focused on producing a holistically coherent work or they

focused on details while unconcerned with the overall map of the piece. The following

discussion will present a more detailed treatment of this dichotomy along with a general

historical background.

Leonard Meyer, in his discussion of the aesthetic changes in the twentieth century, posits

that a dichotomy arose in the artistic motivations of composers. He states:

Changes in aesthetic attitudes are matters of emphasis, having to do with what is

considered to be the main focus of appreciative attentionthe locus of artistic

significance and value. . . . Aesthetics and criticism were dualistic. On the one

hand were the materials and form of a work of art, the means; on the other hand

were subject matter, expression and meaning, the endor goal. The former were

significant because they made the latter possible. If one asked what a work of art

was about, the answer was almost always in terms of subject matter or whatever

it was thought to express, symbolize, or signify.2

This duality of focus, either on the means or the goal of a work of art, characterized the radical

changes that occurred in compositional practice. Specifically, in the first half of the century, the

two stylistic trends could be distinguished in terms of their relation to past styles. Compositions

organized primarily on tonal procedures were considered traditional while works utilizing novel

strategies were considered avant-garde.



3Ibid., 218.

4Susan Sontag,Against Interpretation(New York: Noonday Press, 1966), 142. Further claims of novelty

as the primary characteristic of avant-garde art can be found in Renato Poggioli, The Theory of the Avant-Garde

(New York: Harper & Row, 1968), 214, and Frank X. Mauceri, From Experimental Music to Musical Experiment,

Perspectives of New Music35, no. 1 (1997): 191.

5Marina Lobanova,Musical Style and Genre: History and Modernity, trans. Kate Cook (Amsterdam:

Harwood Academic Publishers Association, 2000), 22.

4

The primary characteristic of avant-garde art is the focus on novelty. According to

Meyer:

In the absence of viable critical criteria, the one thing that could be assessed was

whether the artists method, style, or idea was new. Novelty, often mistakenlyconfused with originality from which it sprang, gradually became a criterion for

judging works of art. And this gave further impetus to the search for the newfor

novelty for its own sake. . . . The new is valuable because it enlarges the

sensibilities and awareness of the art audience.3

This is further elaborated by Susan Sontag: New art is valued for the novel state it induces in

the spectator and for what it reveals to him about himself, the physical world, or simply his way

of reacting to paintings.4

The intense aesthetic focus on novelty linked composers across national borders creating

what could be called an international style. In other words, the ability to be novel and intriguing

initiated a commonality between artistsunity formed from disunity. In her bookMusical Style

and Genre, Marina Lobanova states that in the twentieth century with the rejection of the

canonical system of thinking in culture, artists acquired an exclusive right to experiment. . . . Art

continued to develop under the banner of discovery.5In fact, this aesthetic carried a mystical

and religious character:

What had unified [the avant-garde composers] was a radicalism of aesthetic

purpose. For the modernist tradition of the nineteenth and twentieth centuries, the

great sin against the human spirit was that of boredom. It was a sin brilliantly and

definitively epitomized by Baudelaire in his great poem, Le Voyage.



6David Osmond-Smith, New Beginnings: The International Avant-Garde, 194562, in The Cambridge

History of Twentieth-Century Music, ed. Nicholas Cook and Anthony Pople (Cambridge: Cambridge UniversityPress, 2004), 361.

7Joaquim M. Benitez, Avant-Garde or Experimental?: Classifying Contemporary Music,International

Review of the Aesthetics of Sociology of Music9, no. 1 (June 1978): 63. The distinction between these two groups

has sometimes been specifically referred to terms of choice vs. chance. (See Glenn Watkins, Soundings: Music in

the Twentieth Century(New York: Schirmer Books, 1988) and Paul Griffiths,Modern Music: The Avant Garde

since 1945(London: J. M. Dent & Sons Ltd., 1981).)

5

Repetitions and returnsforms that achieved closure by recapitulation, by

submitting gratefully to the gravitational force of home keys, styles that

knowingly acknowledged that desire through the manipulation of neo-baroque

and neoclassical topoiwere not merely pardonable indulgences; they were

denials of the quintessential potential of human life. They ministeredeither

resignedly or in calculationto the regressive streak present in every human andmassively encouraged by the popular culture of the years after the Second World

War.6

A new dichotomy emerged in the latter half of the century. While neo-classical stylistic

trends partially continued within certain circles of composers, the general school of avant-

garde composers fragmented into two distinct groupsclassical and experimental avant-

gardists.7This duality was analogous to the split that occurred around the turn of the twentieth

century between the traditionalists and the general avant-gardists. In other words, the classical

avant-garde primarily focused on the end, while the experimental avant-garde leaned toward the

individual means. Figure 1.1 presents a diagram summarizing the previous discussion.

The classical avant-garde, in its earliest form, was associated with total serialism. In

order to present an idea through musical means, complete composer-control over all musical

elements was required:

At the end of World War II the musical world was eager to pursue somewhat

selectively the explorations of the first half of the century. . . . If tonal

Neoclassicism, following a final flourish in the late 1940s, was finally laid to rest

with Stravinskys The Rakes Progress(19481951), the Serialism of Schoenberg

and his followers, which had been pronounced a dead end by many a critic in the

1930s and 1940s, was now taken up with an individualizing touch by composers



8Watkins, 506.

9John Cage, Silence(Cambridge, Mass.: M.I.T. Press, 1967), 69.

6

of various nationalities and stylistic inclinations. The message their new works

carried was that any aesthetic is capable of being reinterpreted and vitalized in

highly personal terms. The early recognition that serial organization need not

imply a style was a revelation to many, and the consequent fervor of the

subscription cannot help but suggest that in part it was a reflection of an almost

compulsive search for order following a period of world-wide chaos.8

In fact, it is somewhat ironic that what originally began as a reaction against the traditional

compositional strategies via focus on altering the musical materials (i.e., the means) eventually

became aesthetically associated with the more content-oriented (i.e., less experimental) artistic

perspective.

The experimental avant-garde, on the other hand, continued to focus on new means of

musical expression without any particular holistic goal. John Cage stated: What is the nature of

an experimental action? It is simply an action the outcome of which is not foreseen. It is

therefore very useful if one had decided that sounds are to come into their own, rather than being

exploited to express sentiments or ideas of order.9To composers of experimental music, the

classical avant-garde tendencies were too stifling and narrow. They believed an artist should not

be concerned with attempting to express some point in the final product; rather, the work of art

should simply be purged of all composer-control.

Following the 1950s, the classical avant-garde gradually began to incorporate more than

just serial techniques. The classical avant-garde, in the specific way I use the term in this

study, came to be associated with any non-traditional method of organizing and structuring

sound. Watkins elaborated: Following a mid-century crisis that seemingly demanded that a

choice be made between Serialism and Neoclassicism, we have come to the point where we are



10Watkins, 688.

11The especially self-conscious nature of the classical avant-garde movement is made evident by the fact

composers began writing more and more about their own music. Essentially, there seemed to be a particular need to

both describe and justify individual organizational techniques. See Lucian Krukowski, Hegel, Progress, and the

Avant-Garde, The Journal of Aesthetics and Art Criticism44, no. 3 (1986), 279 and Watkins, 52023.

12Mauceri notes that the avant-garde is characterized as radically new but it is also posited as an historical

category, a tradition in its own right. But as pointed out by art critic Harold Rosenberg, The new cannot become a

tradition without giving rise to unique contradictions, myths, absurdities. The irony of this historical category is the

attempt to construct a genre out of work that by its own definition is radically different and highly individualistic.

The whole of this study follows from this sentiment in an effort to formulate a fairly complete picture of

Gubaidulinas individual style within the context of her avant-garde organizational procedures.

7

comfortable with the wealth of our various alternatives, and an anything goes attitude

prevails.10

The primary aesthetic focus on novelty, however, also posed new compositional

problems that required resolution. Experiments in new sounds and performance strategies forced

composers to create novel means of organizing sound. On a local level, melodic and harmonic

parameters required new systems of arrangement in order to sustain any sense of coherence. At

the same time, large-scale properties such as rhythmic and metric proportions as well as formal

dimensions also demanded some type of organized structure. After World War II, locating a

means of unifying the various elements in a work instantly became the primary concern of

classical avant-garde artists everywhere.11Although it is impossible to formulate a specific

stylistic trend,12I will attempt to describe certain overarching concerns that arose in the

compositional strategies in post-1945 classical avant-garde musical works.

In structuring local-level elements (i.e., melody and harmony), an interesting pattern of

stylistic influence arose among composers. Because of the nearly infinite possibilities of creating

art, a similarly infinite array of potential organizational apparatuses could be conceived. The

modernist movement saw a number of these enacted at some time; however, memories of past



13Lobanova, 158 (italics original).

14Harold Bloom, The Anxiety of Influence: A Theory of Poetry(London: OUP, 1973), 5.

8

structural means were often invoked. In other words, although composers were constantly

seeking novel musical properties, they often reached, either consciously or unconsciously, to the

past for inspiration. Lobanova calls this mixed style:

The change of the musical paradigm . . . affected the new style conception:

from polystylistics proper where one could detect fairly precise popular

principles of quotation and allusion, composers moved to a different

interpretation of style. The new stage was connected with the search for organics:

tradition, which was offered point-blank in stuck-together collages and lost its

meaning in them, was included more and more mediately and subtly in the style

model of the composition. It was precisely on this difficult path of reflected

transformation, accumulation of the traditional, that innovative treatments were

found. The overcoming of distancing in the view and interpretation of the other

style provided a vital link of the ages within an artistic work. Today the mixed

styleis not a means of polemic with attitudes of the preceding period, but a newsystem of artistic communicationwhich admits of a multitude of individual

treatments while preserving a connection with tradition.13

Thus, although the avant-garde aesthetic insisted on novelty and creative originality at the center

of the role of art, it was ironically dependent on the art of the past. The property of intertextuality

between the present and the past was originally posited by Harold Bloom in The Anxiety of

Influence: We need to stop thinking of any poet as an autonomous ego, however solipsistic the

strongest of poets may be. Every poet is a being caught up in a dialectical relationship

(transference, repetition, error, communication) with another poet or poets.14In other words, for

any poet (or composer), the interpreter (or analyst) must always realize how the works from the

past both influence and even define the works of the present.

The mixed style and anxiety of influence can be illustrated with serialism. Although

the pitches are arranged in such a way as to avoid any sense of pitch centricity or tonality,



15Quoted in David Cope,New Directions in Music, sixth ed. (Dubuque, IA: Brown & Benchmark, 1993),

66.

16See Joseph Straus,Remaking the Past: Musical Modernism and the Influence of the Tonal Tradition

(Cambridge, Mass.: Harvard University Press, 1990); Kevin Korsyn, Towards a New Poetics of Musical

Influence,Music Analysis10, no. 12 (MarchJuly 1991), 372; and Mark Evan Bonds,After Beethoven:

Imperatives of Originality in the Symphony(Cambridge, Mass.: Harvard University Press, 1996).

17For a discussion of the various types of utilizing past works, see Meyer,Music, the Arts, and Ideas,

193208.

9

composers continued to limit themselves with their reliance upon the same pitches and rhythms

used for centuries prior. In fact, the necessity of arranging the pitches as a linear row

automatically retains a sense of hundreds of years of melodic organization. Ben Johnston goes so

far as to present the twelve-tone pitch limit itself as unnatural:

Over the whole of the historical period of instrumental music, Western music has

based itself upon an acoustical lie. In our time this liethat the normal musical

ear hears twelve equal intervals within the span of an octavehas led to the

impoverishment of pitch usage in our music. In our frustration at the complex

means it takes to wrest yet a few more permutations from a closed system, we

have attempted the abandonment of all systems, forgetting that we need never

have closed our system. . . . In our laziness, when we changed over to the twelve-

tone system, we just took the pitches of the previous music as though we were

moving into a furnished apartment and had no time to even take the pictures offthe wall.15

Most avant-garde composers attempted to negate this misreading of the past, but

present-day music theorists have demonstrated how modern art music is reliant upon the past

traditions of tonality.16Other composers, however, deliberately attempted to exploit the past in

an attempt to create novelty. This could be done by simulating or modeling a past style, or even

borrowing or quoting directly from a specific composition.17

Gubaidulina, in her creative efforts, has written works that avoid past influence as well as

works that consciously utilize themes or styles from the past. For instance, The Seven Last

Words of Jesus Christ(1982), Offertorium(1980), andMeditation on the Bach Chorale Vor



18Sofia Gubaidulina and Vera Lukomsky, Sofia Gubaidulina: My Desire is Always to Rebel, to Swim

Against the Stream!Perspectives of New Music36, no. 1 (1998): 26.

19Lobanova, 12223 (italics mine). This lack of cause-and-effect connections adds to the notion that

avant-garde music is fragmented and discontinuous, a position to which this study stands antithetically.

10

deinen Thron tret ich hiermit(1993) all feature a quoted statement from J. S. Bach (16851750)

or Heinrich Schtz (15851672). For Gubaidulina, these quotations are meant to symbolize some

aspect of the work rather than to imitate a past style: I used this theme [of King Frederick the

Great of Prussia in Offertorium] not to refer to the style of Frederick the Great (or Bach, who

developed the same theme in hisMusical Offering), but to symbolize the idea of sacrifice.18

As for the large-scale formal properties of avant-garde music, ingenuity abounded.

Although mixed procedures were quite evident, rhythmic and formal attributes were subjected

to numerous experimental attempts at novel organization. Lobanova states:

The search for new musical syntax was connected with the idea of thefundamental unlimitedness of musical and objective time and the full or partial

openness of musical form. This is an important symptom of the rejection of

classical conceptions of time organisation [sic] and a single interpretation of

artistic time as opposed to physical time. As a result there arose a plurality of

individual treatments which broke completely or partially with the familiar cause-

and-effect connections of the classical type. . . . The consecutive application of

simultaneous forms meant the interconvertibility of many musical parameters, the

formation of new dimensions, and a new attitude to musical space and time

structures.19

In other words, the musical elements on the local level were more limited and dependent on

previous compositional thought. Formal proportions, being temporal in nature, were more easily

transformed in the classical avant-garde because of recent philosophical trends that had

reconsidered the nature of time itself.

To summarize the discussion thus far, the main concept of the classical avant-garde

aesthetic was based on creation and novelty; the means of achieving it, on the other hand, were



20Griffiths, 15.

11

not so straightforward. The musical elements basically necessitated recourse to the past because

of the nature of music itself. Composers were forced to conceptualize new methods of

presentation and organizational procedures. Formal manipulation was also necessary in order for

classical avant-garde composers to produce novel works with little or no connection to previous

thinking. Into this musical environment, Sofia Gubaidulina was born.

Gubaidulinas Place in the Avant-Garde

For a composer interested in the avant-garde aesthetic, Soviet Russia was a less than

ideal environment. Paul Griffiths describes compositional practice in the years immediately

following World War II as such:

In the U. S. S. R., too, the symphony was being cultivated assiduously, though as

a genre suited to the optimistic and aesthetically conservative doctrine of socialist

realism . . . . The newly severe interpretation of socialist realism, announced to

musicians by Andrey Zhdanov in January 1948, obliged composers not only in

the U. S. S. R. but throughout the communist bloc to address their music to the

people in the simplest and most direct terms. Thus musicians from eastern Europe

were effectively prevented from participating in avant-garde movement for some

years. . . . But equally and more widely significant was the tension set up betweenmusical radicalism and commitment to political revolution.20

Further description is given by Lobanova:

The concept of modern music, like that of world musical culture, hung around in

the Soviet Union for a painfully long time: the rich contacts of the twenties were

replaced by a long bout of delirium. After the Russian Association of

Proletarian Musicians turned into the official censor in 1930, having established

together with the NKVD (later the KGB) and the Communist Party control over

musical culture, achieved the banning of ideologically hostile organisations

(first and foremost, the Association of Contemporary Music), publishers,magazines and compositions, and repressed some of its political opponents as

enemies of the people, while forcing others to renounce their convictions,



21Lobanova, 29.

22The latter three are collectively referred to as the great three. However, both Volkonsky and Karamanov

were essential in cultivating the experimental school of composers and must also be included in any list of

influential avant-garde Soviet composers. See Levon Hakobian,Music of the Soviet Age: 19171987(Stockholm,

Sweden: Melos Music Literature, 1998), 25661.

23Svetlana Savenko, Valentin Silvestrovs Lyrical Universe, in Underground Music from the Former

U.S.S.R., ed. Valeria Tsenova, trans. Romela Kohanovskaya (Amsterdam: Harwood Academic Publishers, 1997), 67.

12

connections with the West were broken, as were the threads linking the past,

present and future in Russian culture. All modernistic, formalistic,

decadent, religious, bourgeois, and reactionary works were banned from

cultural usage. . . .East-Westwas interpreted by official ideology in accordance

with the canons of mythological description as them and us, left and right,

bad and good.21

Thus, experimental art of any type was strictly forbidden and censored from both publication and

performance.

Despite the stifling socio-political environment, the avant-garde aesthetic continued to

thrive in some academic circles. Specifically, five graduate students came to be known as the

primary leaders of avant-garde music in Moscow: Andrey Volkonsky, Alemdar Karamanov,

Edison Denisov, Alfred Schnittke, and Sofia Gubaidulina.22It was they who began the intensive

assimilation of the avant-gardist musical idioms. The Sturm und Drang of the young composers

allowed within a short span of time to overcome the local confinement of Soviet music and

disrupt its actually existing isolation despite the world-wide recognition of Dmitry Shostakovich,

Sergei Prokofiev, and Aram Khachaturyan.23

The influence of Western compositional practice is apparent in the works of these

composers. Nevertheless, the Soviet classical avant-garde composers, in general, acquired their

own unique stylistic traits. Joel Sachs attributes this to the fact that they received the new



24Joel Sachs, Notes on the Soviet Avant-Garde, inRussian and Soviet Music: Essays for Boris Schwarz,

ed. Malcolm H. Brown, Russian Music Studies, vol. 11 (Ann Arbor: UMI Research Press, 1984), 307. He clarifies

his claim further, stating: They did not have strong preconceptions, because they did not participate in the agony

associated with the evolution in the West of those techniques, and thus escaped the strongly partisan atmosphere so

often transmitted in composition training. [They] seem, therefore, to have bypassed the aggressive modernity that

can be found in many Western composers of their generation.

25

Gubaidulina, Lukomsky, My Desire is Always to Rebel, 16.

26Gubaidulina gives specific examples of blatant censorship of both Schnittkes and her own music in Sofia

Gubaidulina and Dorothea Redepenning, Sofia Gubaidulina: An Interview with Dorothea Redepenning, in

Contemporary Composers on Contemporary Music, ed. Elliott Schwartz, Barry Childs, and James Fox (New York:

Da Capo Press, 1998), 44950 and idem, Sofia Gubaidulina: Into the Labyrinth of the Soul, in The Voice of Music:

Conversations with Composers of Our Time, ed. Anders Beyer and Jean Christensen (Aldershot: Ashgate, 2000),

4547.

13

Western techniques more or less in one burst, at a time when they, already experienced young

composers, could pick and choose from them at will.24

Of the five composers listed above, Gubaidulina is the only one who never held a

university position. She was encouraged to pursue a career in composition, despite her affinity

toward anti-communist artistic trends:

And although we were accepted to the graduate school [in 1954], the

Conservatory officials declared that, despite our giftedness and capacity for hard

work, we had chosen the wrong way, or what they called a false way (the right

way, of course, meant Socialist Realism). . . . [Shostakovich] told me personally:

Everybody thinks that you are moving in the wrong direction. But I wish you to

continue on your mistaken path.25

Upon completing her compositional studies at the Moscow Conservatoire in 1963, she worked as

a freelance composer developing film scores. Her mistaken career path allowed her numerous

opportunities to experiment with unusual collections of instruments (both conventional and

exotic) and unique methods of producing sounds.

Eventually, the oppression of censorship became too much for the composer. Not only

was it impossible to arrange a performance of her works, she was finding it increasingly difficult

to be creative. As a result, her compositional output began to suffer.26To relieve herself of



27Quoted in Josiah Fisk, ed., Composer to Composer: Conversations About Music(St. Leonards, NSW,

Australia: Allen & Unwin, 1993), 460.

28Quotes concerning Gubaidulinas thoughts on ethical composing are countless in number. Some examples

can be found in: Fisk, 461; Gubaidulina, Labyrinth of the Soul, 50; Sofia Gubaidulina, Vera Lukomsky, The

Eucharist is My Fantasy: Interview with Sofia Gubaidulina, Tempo206 (Sept. 1998): 31; Claire Polin, The

Composer as Seer, but Not Prophet, Tempo190 (Sept. 1994): 1617.

14

compositional sterility, she traveled to Germany, visiting for long periods of time starting in

1985. In 1992, she permanently relocated to Hamburg.

Aesthetically, for Gubaidulina, the liberty discovered in musical experimentation was

religious in nature. In describing what caused her to become a composer, she resorts to mystical

language, stating:

A human being, even under the most difficult conditions, even in a murderous

atmosphere, must have something to hold sacred. I remember extremely well my

impressions at five years of age, when I began to have some understanding: all

my life was gray and I felt good only when I entered the door of the music school.

From that moment I felt myself in a sacred space. I heard the sounds coming from

the halls, I felt the bond that united the students, and everything came together in

that polytonal harmony of sounds, and I wanted to live in that world.27

Based on her experiences as a child, Gubaidulina turned to religion as a means of moral strength,

allowing her beliefs to form her aesthetic attitude toward music, experimental music in

particular. Because of this religious focus, she regards the act of composition as an ethical

process; thus, much of her compositional output centers on a religious theme.28

Because of the intense socio-political situation, Gubaidulinas artistic opportunities were

limited. Her compositional ventures were tightly controlled by those who believed her music

would not suit the common class, whether because of difficulty in comprehension or simply

because of radical individualism. Either would contradict the socialist ideology. For these

reasons, she considered her own compositions to be a sacred and holy object. To her, they were a



29Hakobian, 287.

30Gubaidulina, Lukomsky, My Desire is Always to Rebel, 9.

15

means by which to escape the mundane into a divine presence, requiring the sacrifice of the

composer and giving a moral quality to the process of composing.29

In the end, Gubaidulinas experimentation with musical style can be understood and

appreciated only if her motivations are perceived. In the context of the above discussion, she can

be placed firmly in the classical avant-garde: We must worry about the incarnation of our idea,

about meaning and formation. The contemporary artist is faced with an extremely important

task: finding a correlation between intuition and intellectual work.30For Gubaidulina, the

composer must find new musical materials and organizational means in order to create a work of

art containing specific spiritual content. Based on her own admission, her compositions are

formed with a particular end in mind, placing her squarely in the arena of the classical avant-

garde aesthetic.

Elements of Individual Style

Throughout her career, Gubaidulina has experimented with and incorporated various

post-1945 avant-garde trends into a characteristically individual style. I will discuss these in four

categories: melody, vertical dimension (harmony), timbre/texture, and rhythm/form.



31Joseph N. Straus, The Music of Ruth Crawford Seeger(Cambridge, Mass.: Cambridge University Press,

1995), 18.

32Gubaidulina, Lukomsky, My Desire is Always to Rebel, 11.

16

Melody

Gubaidulinas melodies are especially characterized by their intense chromaticism. Any

tonal implication is quickly subsumed by a shift toward extra-diatonic/atonal pitch collections. In

Ruth Crawford Seegers music, Joseph Straus refers to this process as dissonation: Melodies,

. . . should be systematically dissonated, that is, organized to suppress any traditional tonal or

triadic implications. Since any consonant interval might be heard to suggest a major or minor

triad, it will normally be followed by a dissonant interval, to undermine that implication.31

Although Gubaidulina utilizes triads to some extent, the succeeding sonorities do not reference a

tonal syntax.

Another melodic characteristic involves the use of quarter-tones. The composer describes

her view:

I understand it as a unification of two spaces: the first is the twelve-semitonal

space, and the second is another twelve-semitonal space a quarter tone higher. For

me this is a metaphor of the image and its shadow, or a day and a night. From my

point of view, in the twelve-tone compositions of the twentieth century,

everything is as in the daytime; everything is enlightened and reationalized; there

is no place for night. Night existed as a supplement of the diatonic system:the diatonic sphere was day, whereas the chromatic sphere was night: one

could go there and return. That blessed situation gave us classical and romantic

composers. In twelve-tone compositions we lost night: everything became

day. But within the twenty-four tone scale, we may have not only a day, but

also a night.32



33See Dorothea Redepenning, Staccato Existence: Russian Women Composers in Germany, in Women

Composers in Germany, ed. Roswitha Sperber (Bonn: Inter Nationes, 1996), 101.

34Gubaidulina, Polin, 1516.

17

From her own words, it can be deduced that the traditional twelve tones make up the tonal

system while those pitches micro-tonally related to the former are chromatic. This binary

opposition is elsewhere referred to in terms of consonant and dissonant.33

Finally, glissandi are used fairly often, further adding to the ambiguous nature of her

melodic material. Although upper and lower limit pitches may be notated, the glissandi serve to

downplay the notated pitchs importance by creating a continuous melodic line lacking precision

in its placement in pitch space.

In general, Gubaidulina avoids long melodic lines, preferring to link several short

melodic segments together. In this way, cadential implications are circumvented, allowing for an

open-ended quality. The segments are often sequenced through some systematic variation,

yielding a dynamic melodic process as opposed to a static melodic object.

Vertical Dimension (Harmony)

Because of the detailed stress placed on melodic structure, Gubaidulinas harmonies

almost always arise from some linear stratification. In fact, at times she has even described the

individual lines and voices of her music as characters intersecting and crossing at specific

points.34For this reason, the term vertical dimensions seems best suited in describing the

chordal character.

One important aspect in Gubaidulinas music must be stressed. As in the music of Edgard

Varse (18831965) and Gyrgy Ligeti (19232006), octave equivalence is rarely granted a



18

prominent place in Gubaidulinas works. That is, notes an octave apart are denied equivalent

status, and thus pitch-class space is generally irrelevant. Rather, the musical action occurs in

pitch space. In this way, for example, the occurrence of C3 is a qualitatively different event than

the occurrence of C4.

Based on the organizational principle of pitch space, two other types of vertical

dimensions are possible: sound masses and pitch clusters. Although similar, for this study I will

distinguish between these two. I define sound mass as a group of pitches (melodically distributed

or simultaneously occurring) irrespective of proximity to one another in pitch space. A pitch

cluster, on the other hand, consists of pitches in extremely close pitch-space proximity

performed simultaneously (i.e., as an actual sonority).

Texture/Timbre

As implied above, Gubaidulina often organizes a composition into an intricate web of

independently moving lines intersecting at various points. Because of the intensity of the

counterpoint, thick pitch clusters and sound masses are inevitable; however, the density of these

sections is almost always counterbalanced by sections containing a lighter texture and, often,

solo instruments. Once more, it is apparent how much stress is placed on individual lines.

Register plays an important structural role in the composers works. Registral extremes

are often favored in order to produce some dramatic effect, such as melodic climax and

heightened contrast between her self-described characters. Also, register is intimately linked

with concerns of timbre. Gubaidulina often employs string harmonics in melodic structures. In

her own words: Sound can have an earthly, only too human expressiveness. And yet if you



35Quoted in Hakobian, 288.

36Susanne Langer,Feeling and Form(New York: Scribners, 1953), 110.

37Sofia Gubaidulina and Vera Lukomsky, Hearing the Subconscious: Interview with Sofia Gubaidulina,

Tempo206 (July 1999): 30.

19

touch the same spot of the string in another way, if you change a bit your attitude, you are

carried away from earth to heaven. The ordinary, the trivial becomes heavenly, if not sacred.35

Thus, timbre is directly linked to the composers aesthetic values.

Gubaidulina also experiments with unconventional performance methods. Performers are

often instructed to manipulate the natural sound of their instruments in order to create a specific

auditory effect. Along with experimental acoustic sounds, the composer has also worked with

electronic sounds and devices.

Rhythm/Form

According to Gubaidulina, temporal factors are the most important aspects of a musical

work. In her view, numbers contain mystical properties that can be manifested within the music.

Similar to Susanne Langer, Gubaidulina believes that music makes time audible36and that it is

the composers ethical duty to embody the spiritual within a musical form. This is achieved by

means of accurate proportions and rhythmic durations. In her own words: I form a certain

profile of numbers. But in general, there is a beautiful picture of rhythmic calculation,

proportionality, mathematic exactness in the large-scale formal organizationover the absolute

freedom of all other musical elements: melody, harmony, rhythm.37

More specifically, she is most concerned with the Fibonacci sequence, a series of

numbers in which each member is formed by the addition of the previous two members. The



38See Valeria Zenowa,Zahlenmystik in der Musik von Sofia Gubaidulina, trans. Hans-Joachim Grimm and

Ernst Kuhn (Berlin: Verlag Ernst Kuhn, 2001). Although I admit the possibility of the formal and rhythmic inclusion

of this property in the string quartets, I have failed to locate any specific example. Also lacking is any reference bythe composer in her interviews to this formal aspect in the quartets. The discussion is included in this chapter

primarily in an attempt to describe the composers stylistic tendencies as completely as possible.

39See Swetlana Sarkisjan, Die Streichquartette Sofie Gubaidulinas als Versuch der Erschleissung des

Sonoristischen Raumes,International Review of the Aesthetics and Sociology of Music36, no. 2 (2005): 27186.

40Gerard McBurney, Encountering Gubaidulina, The Musical Times129, no. 1741 (March 1988): 123.

20

sequence as a whole approximates a series of increasingly refined golden means since each

member is related to its neighbor in a golden ratio. For Gubaidulina, this produces the most

perfect and natural relation between two parts; thus, any musical form manifesting a golden-

mean proportion is spiritually and physically balanced. Based on her aesthetic views, beauty is

characterized by this perfect proportion.38

In a more practical sense, texture and timbre, more than any other musical element, are

the most convenient means of articulating formal sections lacking a form of syntax. Sudden

changes in texture or timbre generate an immediate sense of restarting (i.e., a new section). In

fact, as will be illustrated shortly, she employed this method of formal division in three of her

four string quartets.39

As for rhythmic properties in general, Gubaidulina places significant emphasis on the

possibility of utilizing rhythm as a structural entity as opposed to mere local figuration:

In the light of the very frequent speculation about the different ways in which

modern composers try to control rhythm and proportion in a work, it is worth

noticing one important and distinctive part of Gubaydulinas [sic] attitude to the

problem. In conversation she is most keen to stress that she cannot accept the idea(a frequent post-serial one) of rhythm or duration as the material of a piece. . . .

To her, rhythm is nowadays a generating principle as, for instance, the cadence

was to tonal composers of the Classical period; it therefore cannot be the surface

material of a work. . . . [S]he expresses her impatience with Messiaen, whose use

of rhythmic modes to generate local imagery, she feels, restricts the effectiveness

of rhythm as an underlying formal level of the music.40



21

In other words, durations segment and organize time; thus, rhythm cannot be musical material on

its own. It is a structural coordinator that arranges other musical parameters.

In sum, Gubaidulinas music is structured in terms of melodic properties that are

organized by rhythmic proportions. Vertical dimensions arise out of melodic interaction while

formal units similarly arise out of textural interactions. For her, the music is not a static object;

rather, it is a living dynamic organism that grows and changes as it is performed. Melodies do

not exist in themselves; they are formed through a process by which a small segment is expanded

and developed. Formal divisions are not objectively present; rather, they arise out of the

interaction of different textures and shifts in timbre. A formal scheme does not structure music;

instead, the music produces a form as it grows in time.

The Four String Quartets

The quartets were written over a period of thirty-two years. Following post-graduate

studies in Moscow, Gubaidulina began working in the Moscow experimental studio for

electronic music (196970). The first quartet, composed in 1971 (significantly earlier than the

remaining three), was one of her earliest instrumental pieces not designed in this electronic

studio. The remaining three were composed within a shorter time-span and betray influence from

her involvement with the Astrea improvisational team (197580 and 1991 to present). The

second and third were composed the same year (1987) and the fourth was composed six years

later (1993).

Also, during this later time in the composers life, her interests turned toward symphonic

textures, giving more equality to the individual instruments than do her initial works. This shift



22

toward symphonic textures can be seen in Stimmen . . . Verstummen . . .(1986) andPro et contra

(1989). As in most of her orchestral works, all four quartets are designed to allow the maximum

freedom for each member of the ensemble, but the later they were written, the more experimental

they became. The composer not only found new possibilities for organizing sound, she also

became more comfortable with her choices.

Attempting to analyze a piece organized entirely on the basis of an individually created

style that lacks any conventional syntactical system poses a problem. Understandably,

endeavoring to employ traditional analytic tools on such idiosyncratic music proves fruitless. In

the case of Gubaidulinas four string quartets, conventional set-class analysis cannot explain the

pitch organization. In general, her music does not depend on similar pitch groupings. Discrete

passages often organize the same pitches in ways that are fundamentally different. This requires

two different interpretations of the same pc-set class, severely limiting the amount and

importance of the information a pc-set class analysis can convey.

On the other hand, separate passages contain radically different pitch collections

organized in a similar fashion. Once again, a pc set-class analysis would fail to yield valuable

information concerning the comparative structure of these passages. The lack of available tools

that might aid in a valid understanding of the work actually supports the notion that the

composition is fragmented and lacks continuity. I argue that these quartets are coherent works

and do retain some form of continuity throughout. I will demonstrate my point by exploring four

different processes of organization unfolding within the quartets.

The first quartet illustrates the process of expansion, in which pitches (and sometimes

rhythms) are organized in one of two ways. First, they may radiate outward in pitch space from



23

some central point or, second, they may expand outward in pitch space using some interval as a

measure of distance. The second quartet utilizes inversional symmetry as a means of organizing

large collections of pitches about a single asserted axis. This quartet also demonstrates the

process of gap-fill, in which an interval is established and gradually filled in chromatically. The

third quartet presents an organizing principle based on the intervals inherent in the pitch

collection. Specific intervals are assigned particular functions in order to structurally arrange the

individual pitches both linearly and harmonically. In this case, ic5 and ic1 act as structural

markers between sections and as a means of differentiating between individual voices. Finally,

the fourth quartet illustrates an organizing process based on sound masses. These collections are

shown to mutate into one another via transformations.

The analyses that follow are not intended to be exhaustive, nor even complete. Instead,

my primary purpose is to demonstrate continuity in these well-designed compositions by

presenting specific processes that organize the musical elements throughout each individual

work.



1These properties are discussed at length in chapter 1.

24

Chapter 2

Analysis of String Quartet No. 1 (1971):

Expansion Processes

Unlike the remaining three quartets composed by Gubaidulina, the first does not

conveniently divide into large formal sections. Rather, the work seems to play out as a chain of

small, often unrelated ideas formally separated only by rests or sudden shifts in texture. This

stream-of-consciousness property only adds to the accepted notion that this composers works

are discontinuous and unfold in a series of moments.1However, I propose to demonstrate how

one particular process of organization assists in uniting the various formal portions of the work

and aids in establishing continuity. This process, which I will call expansion, coordinates the

seemingly random pitches and rhythms into an organized and cohesive structure.

The expansion process that occurs throughout the work can be divided into two basic

categories. The first, which I call wedge expansion, begins with the assertion of a single pitch.

From this starting point, the pitch space is gradually opened up by the statement of pitches on

either side of the initial pitch. The size of the intervals between each of the outer pitches and

the central pitch (and necessarily between the outer pitches themselves) gradually increases. This

wedge can be achieved via contrary motion between two voices: as one voice states the

various pitches below the initial pitch, an alternate voice balances the wedge with pitches above

the initial pitch. The same process is also possible by means of a single voice. In this case, an



2See especially chapter 3 for a discussion of Gubaidulinas use of inversional symmetry.

3Jonathan Bernard, Inaudible Structures, Audible Music: Ligetis Problem, and His Solution,Music

Analysis6, no. 3 (1987): 226.

25

initial pitch is followed by alternate statements of pitches of gradually increasing intervals in

both directionsfrom the original pitch.

One caveat must be included with this discussion of wedge expansion. Although

symmetrical structures do play an important part in the later quartets,2the presence of exact

inversional symmetry is interestingly lacking in this work. Here, the process of expanding or

opening up pitch space rarely corresponds to exact interval sizes traversed in both directions, yet

the wedge expansion does describe the specific manner in which the various pitches are

organized in pitch space. Jonathan Bernard, in his discussion of LigetisKammerkonzert,

presents a description that also applies to this quartet: As the dimensions of occupied space

continue to spread outward, the general idea of a symmetrical centre is effectively conveyed, but

the actual, specific location of that centre is constantly in flux.3In other words, although exact

inversional symmetry is rarely presented, the pitch organization mimics a symmetrical layout. In

order to avoid implying an exact symmetrical center, however, I will refrain from using the term

axis in reference to the initial pitch.

In contrast to the wedge category, the second category of expansion, additive expansion,

is more unidirectional. Instead of organizing pitches around a central tone, an interval distance is

utilized as the starting point from which the rest of the process stems. This is best explained

through demonstration. Figure 2.1 presents a chain of four pitches: E4, G4, C5, G5. The first



4Throughout this study, a number placed within square brackets will always refer to an exact interval size in

semitones (e.g., [13] = 13 semitones). If the interval is more clearly represented as part of a class of intervals, this

will be notated with the prefix ic (e.g., ic1 = 1 semitone, mod-12 and regardless of direction). Therefore, [13] =

ic1, depending on the contextual emphasis. Finally, anytime it is necessary to define or clarify an interval size (exactor interval class) with exact pitches, the following shorthand will be utilized: [13]: (C4C#5) or ic1: (C4C#5). For

both, the interval size before the colon is specifically defined by the pitches in the post-colonic parentheses.

5The idea of transformations as performed on pitches and intervals was originally posited by David

Lewin. See Transformational Techniques in Atonal and Other Music Theories,Perspectives of New Music21, no.

12 (1982): 31271 and Generalized Musical Intervals and Transformations(New Haven: Yale University Press,

1987).

26

interval, E4G4, is [3], that is, the pitches are separated by three semitones.4This stated interval

is then manipulated by the addition of another (unstated) interval. The resulting sum of the two

intervals dictates the size of the interval immediately following the stated interval. In Figure 2.1,

[2] is added to the stated interval [3] producing [5]. As can clearly be seen, the interval

immediately following E4G4 is G4C5, which is [5]. This process is repeated, employing the

same unstated interval, to produce the next interval. Here, [2] is added to [5]: (G4C5) resulting

in [7]: (C5G5). This type of expansion is essentially a chain of transformations that extends

each resulting interval by a constant.5For the present example, the transformation could be

named ADDTWO since it is defined as a procedure that adds two semitones to each ensuing

interval.

In the following analyses, I will present examples of each type of expansion as individual

processes. Although the sections in the first quartet are texturally distinct and display relatively

little surface-level commonality, I will demonstrate that, not only is it possible, but it is quite

apparent that the expansion processes are indeed functioning as organizing principles for both

pitch and rhythm.



6Because of her interest in experimental sounds, Gubaidulina often composes works with microtonal

elements. The first and fourth quartets employ these microtones and the notations require some explanation. Since

the two quartets have different notational procedures, I will present each one in the context of that quartet (for the

fourth quartet, see chapter 5). In this work, the half-sharp is notated as a sharp missing one vertical and one

horizontal line while the one-and-a-half-sharp simply adds an extra vertical line to the half-sharp sign. The half-flatis notated with a flag (similar to an eighth-note flag) added to the flat sign and the one-and-a-half-flat simply

includes a horizontal line through the regular flat sign.

7It should be noted that the interval sizes involve fractions corresponding to the microtones. Also, [0]

indicates a unison.

8Throughout this study, rehearsal numbers are abbreviated with a capital R followed by the number.

27

Wedge Expansion

Figure 2.2 displays the solo viola motive from the opening measure of the piece. The G#3

is first established as a dotted half-note, ornamented with a gradually strengthening vibrato,

notated as a curved zig-zag line above the staff. Following a triplet A3-G#3-A3 figure, the G#3

descends through two quarter-tones to G3.6A3 is similarly inflected by a quarter-tone descent,

after which a two-quarter-tone ascent begins with G3. The return to G#3 is finally completed

with the final quarter-tone descent from A-half-flat. Displayed beneath each interval in the score

is the relative size in semitones (Figure 2.2).7It can be seen that the unison [0] increases in size

to a whole-tone [2] before collapsing back into a unison. This process is repeated in turn at the

same tonal level (G#3) in each of the remaining voices in the ensemble. Thus, the opening

motive establishes the model by which many of the processes operative throughout the work can

be understood.

In the first four measures of R1 (Figure 2.3),8the second violin presents a small wedge

expansion beginning with the center pitch C-half-#5. The first segment gradually expands

outward in quarter-tone intervals until reaching D-half-#5 in the ascending direction and B4 in

the descending direction. The second segment, separated from the first by a dotted eighth-rest,



9It is of interest to note how this passage invokes serial techniques since no pitch is repeated until the entire

aggregate has been established. Gubaidulina, however, rarely made use of this technique as a means of structuring a

composition.

28

discards the use of quarter steps and continues the asymmetrical expansion via semitones before

ending with a final upward ascent (G#4-D#5-B-half-#5). It is interesting that not only is the

initial expansion continued, but the expansion interval itself is affected by its own expansion

(quarter-step expanded to half-step).

A similar process occurs in mm. 310 of R8 (Figure 2.4) between three members of the

quartet. The first violin discontinues its previous highly chromatic motion (not shown in Figure

2.4), focusing on two pitches: F#6G6. The second violin maintains the F#6 as the first violin

descends to F6. The procedure occurs once more by the second violin and viola while the first

violin continues to expand in both directions to E6 and G#6. In m. 8, the expansion process

reverses by collapsing back into F#6. However, rather than retracing the exact steps utilized in

expanding, the contraction process employs quarter-tones. For instance, the first-violin E6

ascends to E-half-sharp6 while G#6 simultaneously descend a quarter step to G-half-#6. The

process continues until all three voices intersect at a unison F#6. Thus, in a similar way to the

second-violin line, shown in Figure 2.3, this passage exemplifies how multiple intervals can be

utilized in the wedge expansion process.

Figure 2.5 presents a passage in which a single wedge expansion is simultaneously

applied to three different voices. Beginning in m. 4 of R6, the two violins and cello lines are

canonic in their melodic structure while the rhythmic durations of individual pitches are

contrasted throughout. They each commence by asserting C#4 and gradually expand in pitch

space in both directions until all twelve chromatic pitches are stated.9



29

The second violin in mm. 49 of R7 (Figure 2.6) employs the identical method of wedge

expansion demonstrated in the semi-canonic passage described above. Beginning with an

asserted E5, the melodic material expands outward in pitch space until the entire twelve-tone

aggregate is established, ending with A#5 in m. 6. This passage is differentiated from the one

previously discussed by the inclusion of repeated pitches. In Figure 2.6 the circled pitches

represent those involved in the wedge expansion, while those not circled are repetitions of some

previously established pitch, always retaining the exact pitch level. In this way the expansion

process continues to function as the primary means of organization.

The remainder of the passage following the aggregate completion continues to be

structured by the expansion begun in m. 4. Following the A#5, C#6 is asserted as a break in the

ascent, which, until now, has expanded by semitones. The descending portion follows suit by

stepping a whole-tone from the previous B4 to A4 and continuing to descend chromatically.

However, the Bb4 is immediately stated, re-establishing the possibility of a continued semitone

expansion. Following the Eb5-A5-Ab4 segment (all repeated pitches), the line ends with another

semitonal descent from F4 to B3. Although the descending portion of the wedge receives heavier

emphasis, it can be viewed as an acutely unbalanced continuation of the initially balanced wedge

expansion process.

A particularly salient example of wedge expansion occurs at R21 (Figure 2.7). Beginning

on D4, the first violin gradually ascends by semitones (circled in Figure 2.7), alternating each

new pitch with a restatement of the initial D4. The remaining voices in the ensemble support the

expansion through the reiteration of the newly established pitches. For example, in m. 2 the first-

violin line expands from D4 to E4, passing through Eb4. Once the first violin reaches E4



10This passage also exemplifies how the wedge expansion principle is not bound by temporal placement.

Either component may occur first or they may occur simultaneously. Just as the passage in Figure 2.6 was

imbalanced by the emphasis on the descending portion of the wedge, the passage in Figure 2.7 is heavily imbalanced

toward the ascending component. This is partially based on the fact that the descending component begins over half-

way through the passage.

30

(doubled by the viola), the second-violin and cello lines repeat the Eb4. Thus, as the first-violin

line ascends, the remaining quartet members chromatically saturate the created interval with

repetition. As the interval becomes larger, the literal repetition of every asserted pitch must cease

for a practical reason (i.e., there are too few instruments). Interestingly, the first instance of

losing a previously stated pitch (m. 7), in which the pitch segment lacks

Eb4, occurs just prior to the commencement of the descending component of the wedge in the

cello and viola lines (m. 9). Throughout this passage, the voices retain the D4 as a continuous

drone that functions as both a stable point of repose and a catalyst for an expansion

simultaneous motivation for continuity andchange.10

A complex example of wedge expansion can be found at R25 (Figure 2.8a). The cello

and second violin provide an accompaniment to the improvisational first-violin melody (omitted

from Figure 2.8a). These accompanying voices are structurally based on the wedge-expansion

process outlined above. D5 is asserted in the cello for four beats before sliding down through

Bb4 and returning upward to D-half-b5, one quarter-tone below the initial pitch. Another slide

to Bb4 is followed by another ascent to Db5. This process continues, always ascending to a pitch

one quarter-tone below the previous ascent, until C5 is attained. At this point the slide is

followed by an ascent to a half-note Eb5, and the process recommences from D5. As is

demonstrated in the analytic voice-leading graph of the cello line (Figure 2.8b), each descent

from D5 terminates on a pitch expanded downward by one quarter-tone from the previous



31

termination. Each of these pitches is also immediately followed by an ascent to a pitch expanded

upward by one semitone in relation to the previous ascent. In other words, each descent and

ascent expands away from D5 in both directions; however, the pitch space below D5 is

organized by quarter-tone expansions while the space above D5 is expanded by semitones. The

process culminates on the final pitch B-half-b4, one quarter-tone from the slide pitch Bb4.

The D5 for the cello also acts as the center for a wedge expansion in the second-violin

line, although the method of expansion is less consistent than for the cello. As a counter-balance

to the cello drone and the rhythmically fluctuating melodic material for the first violin, the

second-violin line is decisively rhythmic. The initial interval [4]: (C#5F5), defined by the

pitches in the first measure, has already expanded the pitch space in both directions around D5.

Gb5 and G5 are asserted in mm. 67, creating an unbalanced ascent compared to the descending

component, which has not exceeded C#5. Balance is achieved in mm. 1011 when the wedge

expands downward to Bb4, the lower boundary of the cello line. Thus, the wedge expansion

process not only structures each of the individual voices of the duet, it also organizes the pitches

betweenthe voices through the emphasis on D5 and B4. The emphasis on descent by the cello is

balanced by the emphasis on ascent in the second violin, even though the voices as individual

lines are imbalanced.

In sum, wedge expansion organizes the pitches within a passage of music around a

previously asserted central pitch. Not only can this process occur among multiple voices, it can

also structure a single line. Most importantly, wedge expansion does not consistently organize

the pitches symmetrically or even in any balanced manner. However, because of the nature of the

wedge expansion, some form of symmetry does often arise.



32

Additive Expansion

Whereas wedge expansion is specifically bound to a particular pitch in pitch space, the

second type of expansion process, additive expansion, uses a constant to create a chain of events.

Each event is a transformation of the preceding event rather than a transformation based on a

single pitch. In this way, the expansion is bound to a temporal process as well as a spatial

organization.

For example, the viola line in mm. 23 of R1 (Figure 2.9) consists of three ascending

intervals, each starting on a different pitch. The first interval [9]: (Eb4C5) is expanded by [1] in

the next interval [10]: (A4G5). This process is repeated as the next interval is expanded once

more ([11]: (D5C#6)). Thus, a single operative transformation can be defined as add [1] to the

previous interval. Throughout the remainder of this chapter, I will refer to this particular

transformation as ADDONE.

Figure 2.10 displays mm. 14 of R4, in which the second violin presents a descending

melody based on the ADDONE additive transformation. The initial interval [1]: (F#7F7) is

expanded by [1], resulting in [2]: (F7Eb7). As can be seen by the labels above the score in

Figure 2.10, each interval in this descent is the result of ADDONE applied to the immediately

preceding interval. However, the line produced in this passage is fundamentally different from

that in the previous discussion. In Figure 2.9, the first interval [9]: (Eb4C5) is separated from

[10]: (A4G5) because they do not share any pitches. In the present example, each new interval

formed as a result of ADDONE is built on the terminating pitch of the previous interval. Thus,

[2]: (F7Eb7) shares its first pitch with the previous interval [1]: (F#7F7). This transformational

expansion continues until the pitches are out of the second-violin range, at which time they are



11

This idea was inspired by the notions of projection and deferral as presented in Christopher Hasty,Meter and Rhythm(Oxford: Oxford University Press, 1997) and by the work of David Lewin: On Formal Intervals

Between Time-Spans,Music Perception1, no. 4 (1984): 41423 and Generalized Musical Intervals, 6081. The

concept of constantly-undermined expectations of rhythmic continuity is especially prominent in Gubaidulinas

experimentation with the stream-of-consciousness character of time perception. By avoiding any constancy of

metric division, countless manners of temporal perception within a composition become possible. In the particular

context of this analysis, an additive character lends a unique quality to the piece as a whole, in which time seems to

be constantly expanding.

33

continued by the cello line. The pattern terminates only after the cello has reached its registral

nadir.

The possibility of rhythmic organization utilizing a wedge formation is difficult to

imagine since rhythms are limited to temporal space. Not only must temporal aspects occur in

sequence whereas pitches can occur simultaneously, pitches have an added benefit of being

perceived on vertical levels in space corresponding to a listeners impression of high and low.

Thus, it is possible for pitches to move outward and expand in pitch space in two different

directions. Sequential events are incapable of spatial movement and are cognized as occurring in

chains, which necessarily do not allow for bidirectional perception. Based on this understanding,

wedge expansion, because of its dependence on multiple directionality, is only capable of

structuring pitch material. However, the additive expansion process is perfectly able to formally

structure rhythmic material.

Additive expansion applied to rhythmic space follows the same basic procedure as it does

when applied to pitch space. Rhythmic values are expanded in duration by some constant

rhythmic value. As with intervals, the transformation acts upon each resulting duration in order

to produce the next duration.11For example, the cello melody in mm. 24 of R1 (Figure 2.11) is

made up of four (012) melodic trichords: , , , and

. Each trichord is rhythmically articulated as a short-short-long segment; however,



12To clarify, although the example under discussion involves transformations acting upon durations of

pitches, the ADD16th function can also (and will) apply to durations of rests as well.

34

though the short is consistently presented as a sixteenth-note, the long is systematically

lengthened in each consecutive statement. The first statement employs a long duration of three

sixteenth-notes, and the second statement includes an expanded long duration of four

sixteenth-notes. The third and fourth statements continue the process of expanding the previous

long duration by one more sixteenth-note. This additive expansion could be seen as the

rhythmic analogue of expanding an interval by [1]. Therefore, I will define this particular

transformation as add one sixteenth-note duration to the previous duration. For the remainder

of this chapter, I will refer to this defined transformation as ADD16th.12

Returning to Figure 2.10, the identical additive process ADD16th structures the rhythmic

properties of the descending melody. Although the pitch statements retain a constant sixteenth-

note duration, the duration of the rests between these pitches fluctuates. Between the first two

(F#7F7) exists a zero duration of silence. The transformation ADD16th is applied to this

zero duration resulting in a one-sixteenth-rest between the second and third pitches (F7Eb7).

Applying ADD16th once more results in a two-sixteenth-rest between the third and fourth

pitches (Eb7C7). This additive expansion continues unhindered, even as the melody passes into

the cello line, resulting in a duration of nine sixteenth-rests between the final two pitches

(B2C2). The process is graphically demonstrated with labels below the score in Figure 2.10.

Thus, the additive expansion process effectively organizes both the pitch material and rhythmic

durations of this passage.

Another example of this interaction of pitch and rhythm via additive expansion is

demonstrated in mm. 415 of R48 (Figure 2.12). B4 is asserted by the first violin, viola, and



35

cello in unison. As the viola sustains B4, the first violin ascends by a quarter-tone, and the cello

descends the same distance. This apparent wedge expansion is immediately subsumed by an

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