thesis by portillo

SEQUENTIAL APPROACHES TO POST-TONAL

HARMONIC DICTATION

by

JESSICA A. PORTILLO, B.M.

A THESIS

IN

MUSIC THEORY

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF MUSIC

Approved

Matthew Santa Chairperson of the Committee

Michael Berry

Peter Martens

Accepted

John Borrelli Dean of the Graduate School

August, 2006

ii

ACKNOWLEDGEMENTS

I owe many thanks to and am extremely grateful for the following: God; Cristina

Portillo, my mom, who has always supported my education; my thesis committee:

Matthew Santa, chairperson and advisor, who is always such a great inspiration and has

seen me through this entire process; Michael Berry and Peter Martens who graciously

agreed to serve on my committee and have offered numerous valuable insights; my good

friend, Miguel Ochoa, who spent many frustrating hours formatting the appendices;

Renee Salandy, Wade Lair and Kenneth Metz for reading this thesis and offering editorial

remarks. Thank you to all who helped me accomplish this task.

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

LIST OF FIGURES vi

LIST OF TABLES ix

CHAPTER

I. INTRODUCTION 1

Ear Training for Twentieth-Century Music 1

Harmonic Dictation 2

Trichordal/Tetrachordal Set Classes 3

Sequencing Trichordal/Tetrachordal Set Classes 4

Similarity Relations 6

II. MICHAEL FRIEDMANN AND JOSEPH STRAUS: APPROACHES TO SEQUENCING TRICHORDAL AND TETRACHORDAL SET CLASSES FOR HARMONIC DICTATION 8

Introduction 8

Friedmann 1990 8

Trichordal Set Classes 8

Tetrachordal Set Classes 11

Straus 2005 15



Friedmann and Straus 17

Sequencing Set Classes Using Similarity Relations 18

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III. ROBERT MORRISS SIM RELATION 20

SIM 20


Similar Trichordal Set Classes 22

Dissimilar Trichordal Set Classes 26

Morriss SIM Applied to Tetrachordal Set Classes 27

IV. JOHN RAHNS MEMB RELATION 34

MEMB: The Embedding Function 34



Sequencing Trichordal Set Classes Using the MEMB Relation 42

Chains of Similar Set Classes 43

Chains of Dissimilar Set Classes 44

V. DAVID LEWINS REL RELATION 46

RELT 46


Sequencing Trichordal Set Classes 50

Sequencing Tetrachordal Set Classes 54

VI. ERIC ISAACSONS ICVSIM RELATION 59

Calculating IcVSIM 59

Sequencing Chains of Trichordal Set Classes Using IcVSIM 61

Sequencing Chains of Tetrachordal Set Classes Using IcVSIM 66

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Averages of SIM, MEMB, RELt, and IcVSIM 69

Conclusion 70

BIBLIOGRAPHY 73

APPENDIX A 76

TABLES OF TRICHORDAL SET CLASSES 76

APPENDIX B 80

LINE GRAPHS OF TRICHRORDAL SET CLASSES 80

APPENDIX C 92

TABLES OF TETRACHORDAL SET CLASSES 93

APPENDIX D 105

LINE GRAPHS OF TETRACHORDAL SET CLASSES 105

APPENDIX E 115

TABLES OF AVERAGESSIM, MEMB, RELT, ICVSIM 115

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LIST OF FIGURES

1.1 Trichordal Set Classes/Friedmanns Common Tetrachordal Set Classes 4

2.1 Friedmanns Interval Families 9

2.2 Friedmanns Common Tetrachordal Set Classes 12

3.1 Similar Pairs of Trichordal Set Classes Based on the SIM Relation 22

3.2 Paths of Similar Trichordal Set Classes Based on the Initial Thread (012) - (013) - (014) (015) - (025) - (027) 25

3.3 Dissimilar Pairs of Trichordal Set Classes Based on the SIM Relation 26

4.1 MEMBn[X,(012),(013)] 35

4.2 Line Graph of Morriss SIM and Rahns MEMB 4-1 (0123) 42

4.3 Chains of Trichordal Set Classes Derived from the MEMB Relation 43

4.4 Chains of Similar Trichordal Set Classes Using SIM and MEMB Beginning on (012) 44

4.5 Chains of Dissimilar Tetrachordal Set Classes Using SIM and MEMB Beginning on (0123) 45

5.1 RELt(4-1,4-2) 47

5.2 Line Graph Comparing SIM, MEMB, and RELt as Applied to 3-1 (012) 50



5.5 Similar Trichordal Set Class Chains Beginning With (012) Based on SIM, MEMB, and RELt 53

5.6 Line Graph of (0123) According to SIM, MEMB, and RELt 56

5.7 Line Graph of (0369) According to SIM, MEMB, and RELt 57

5.8 Dissimilar Chains of Friedmanns Common Tetrachordal Set Classes Beginning With (0123) Based on SIM, MEMB, and RELt 57

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6.1 IcVSIM of (012) and (013) 60

6.2 Line Graph of SIM, MEMB, RELt, and IcVSIM Applied to (012) 63

6.3 Line Graph of SIM, MEMB, RELt, and IcVSIM Applied to (013) 64

6.4 Chains of Similar Trichordal Set Classes Beginning on (012) According to SIM, MEMB, RELt and IcVSIM 65

6.5 Line Graph of (0123) According to SIM, MEMB, RELt and IcVSIM 67

6.6 Line Graph of (0369) According to SIM, MEMB, RELt and IcVSIM 68

6.7 Chains of Similar Trichordal Set Classes Beginning With (012) and Based on the Averages of SIM, MEMB, RELt, and IcVSIM 70

B.1 Table and Line Graph of (012) 80












D.1 Table and Line Graph of (0123) 105


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D.5 Table and Line Graph of (0158 109






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LIST OF TABLES

2.1 Strauss Sequence of Trichordal Set Classes According to Friedmanns Interval Families 16

3.1 SIM Relation Applied to Trichordal Set Classes 21

3.2 SIM Relation Applied to Tetrachordal Set Classes 28

3.3 SIM Relation Applied to Friedmanns 10 Common Tetrachordal Set Classes 31 3.4 Pairs of Friedmanns 10 Common Tetrachordal Set Classes Arranged from Most Similar to Most Disparate According to Morriss SIM Relation 32 4.1 MEMB Relation Applied to Trichordal Set Classes 36

4.2 Comparison of Morriss SIM and Rahns MEMB 37

4.3 Comparison of SIM and MEMB with Converted MEMB Values 38

4.4 MEMB Applied to Friedmanns Common Tetrachordal Set Classes 40

4.5 Morriss SIM Relation and Rahns MEMB Relation Applied to Friedmanns Common Tetrachordal Set Classes 41

5.1 Trichordal Set Classes According to RELt 49

5.2 Converted RELt Values Applied to Friedmanns Common Tetrachordal Set Classes 55

6.1 Table of Trichordal Set Classes According to IcVSIM 61

6.2 Table of Friedmanns Common Tetrachordal Set Classes According to IcVSIM 66

6.3 Averages of Trichordal Set Classes According to SIM, MEMB, RELt and IcVSIM 69

A.1 Robert Morriss SIM Relation 76

A.2 John Rahns MEMB Relation 77

A.3 John Rahns MEMB Converted 77

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A.4 David Lewins RELt Relation 78

A.5 David Lewins RELt Converted 78

A.6 Eric Isaacsons IcVSIM Relation 79

A.7 Eric Issacsons IcVSIM Converted 79

C.1 Robert Morriss SIM Relation 93

C.2 Robert Morriss SIM Applied to Friedmanns Common Tetrachordal Set Classes 95

C.3 Robert Morriss SIM Applied to Friedmanns Common Tetrachordal Set Classes Converted 95 C.4 John Rahns MEMB Relation 96

C.5 John Rahns MEMB Relation Applied to Friedmanns Common Tetrachordal Set Classes 98

C.6 John Rahns MEMB Relation Applied to Friedmanns Common Tetrachordal Set Classes Converted 98

C.7 David Lewins RELt Relation 99

C.8 David Lewins RELt Relation Applied to Friedmanns Common Tetrachordal Set Classes 101

C.9 David Lewins RELt Relation Applied to Friedmanns Common Tetrachordal Set Classes Converted 101

C.10 Eric Isaacsons IcVSIM Relation 102

C.11 Eric Isaacsons IcVSIM Relation Applied to Friedmanns Common Tetrachordal Set Classes 104 C.12 Eric Isaacsons IcVSIM Relation Applied to Friedmanns Common Tetrachordal Set Classes Converted 104

E.1 Table of the Averages of SIM, MEMB, RELt, IcVSIM Trichordal Set Classes 115

xi

E.2 Table of the Averages of SIM, MEMB, RELt, IcVSIM Trichordal Set Classes 116

1

CHAPTER I

INTRODUCTION

Many music theorists have researched and written on the subject of

relating set classes in post-tonal music (Buchler, 2000; Chrisman, 1977; Demske, 1995;

Everett, 1997; Forte, 1973; Friedmann, 1990; Herder, 1973; Isaacson, 1990; Lewin, 1977,

1997, 1998, 2001; Lord, 1981; Morris, 1995, 1994, 1979; Perle, 1981; Quinn, 2001; Rahn

1979/1980; Silberman, 2003; Straus, 2005; Teitelbaum, 1965). Few, however, have

delved into research concerning post-tonal harmonic dictation, specifically concerning

methods and sequential approaches to presenting trichordal and tetrachordal set classes in

the undergraduate-level music theory classroom (Everett, 1997; Friedmann, 1990;

Herder, 1973; Mead, 1997; Morris, 1994; Uno, 1997; Quinn, 2001, Silberman, 2003).

This thesis will explore approaches to sequencing trichordal/tetrachordal set classes in

harmonic dictation based on similarities and dissimilarities formulated using various

measures of similarity.

Ear Training for Twentieth-Century Music

The subject of post-tonal music theory pedagogy is pertinent to aural skills and

musicianship courses in the university setting. As more universities include the study of

set theory in the undergraduate theory sequence, the demand for various approaches to

teaching this subject becomes greater. Problems arise, however, when constructing a

curriculum that teaches ear training for twentieth-century music. Few scholars have

written on the subject (i.e. Friedmann, 1990; Morris, 1994; Straus, 2005; Herder, 1973)

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and most instructors choose to base the curriculum around various aspects such as

musical literature, written theory texts, and personal teaching styles. While literature,

correlation with the written theory textbooks, and personal teaching styles are all

important when forming a post-tonal aural skills curriculum, systematically sequencing

material is also important. This study will attempt to do so using various similarity

relations.

Harmonic Dictation

Ear training includes a wide variety of exercises and activities. These may

include sightsinging, melodic dictation, harmonic dictation, and improvisation.

Sightsinging, melodic dictation, and improvisation are all vital to understanding and

hearing post-tonal music. Although a discussion of these elements is beyond the scope of

this study, they should be included in any curriculum covering post-tonal music.

The focus of this study is harmonic dictation, more specifically the harmonic

dictation of trichordal and tetrachordal set classes in isolation. Harmonic dictation

encompasses a variety of exercises such as dictation of chords in isolation, dictation of

intervals, and dictation of harmonic progressions which may include Roman numeral

analysis and the notation of two or more melodic lines. Harmonic dictation may also

include the dictation of set classes in inversion and transposition. To discuss all of these

subjects would be beyond the scope of this study. This study will sequence trichordal

and tetrachordal set classes to be presented for harmonic dictation and will require the

student to identify the set class when played in isolation. Identifying set classes in

inversion and transposition is important to the comprehension of post-tonal music.

3

Though they will not be discussed in this study, these transformations could easily be

integrated with the chains of similar and dissimilar set classes that will be proposed.

These chains can also be used to form various exercises for the dictation of harmonic

progressions, even though this study will not explore them. Also the question of whether

it is best to present dictation on the piano, using various instruments, or through

recordings is not addressed in this study. This thesis will offer different approaches to

sequencing material for harmonic dictation, leaving approaches to teaching harmonic

dictation in general to the preference of the instructor.

Trichordal/Tetrachordal Set Classes

In post-tonal music set classes range from the unad to the chromatic scale. Not

only would discussing all set classes be beyond the scope of this study, it would be

impractical. That is, an instructor would not have the time necessary to discuss all set

classes and present them in dictation in a one-hour, one-semester aural skills course. In

order to narrow the possibilities this study has focused on trichordal and tetrachordal set

classes. The twelve trichordal set classes are manageable in the time frame allotted for

most aural skills courses. The twenty-nine tetrachordal set classes, however, could be a

problem. The examples in this study, therefore, focus on the twelve trichordal set classes

and ten tetrachordal set classes as identified in Friedmanns Ear Training for Twentieth

Century Music.1 These ten, he labels common tetrachordal set classes identifiable by

1 Michael Friedmann, Ear Training for Twentieth Century Music. New Haven: Yale University Press, 1990; pp. 80-81.

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eye and ear.2 Figure 1.1 is a list of the twelve trichordal set classes and Friedmanns ten

common tetrachordal set classes.

Trichordal Set Classes Friedmanns Common Tetrachordal Set Classes 3-1 (012) 4-1 (0123) 3-2 (013) 4-3 (0134) 3-3 (014) 4-10 (0235) 3-4 (015) 4-11 (0135) 3-5 (016) 4-20 (0158) 3-6 (024) 4-21 (0246) 3-7 (025) 4-23 (0257) 3-8 (026) 4-26 (0358) 3-9 (027) 4-27 (0258) 3-10 (036) 4-28 (0369) 3-11 (037) 3-12 (048) Figure 1.1. Trichordal Set Classes/Friedmanns Common Tetrachordal Set Classes

Although this study has been narrowed to focus on Friedmanns common

tetrachordal set classes, all approaches to sequencing introduced in this thesis may be

used to sequence all twenty-nine tetrachordal set classes, as well as any of the remaining

set classes.

Sequencing Trichordal/Tetrachordal Set Classes

There are various approaches to sequencing set classes for harmonic dictation.

Friedmann (1990) and Straus (2005) have their own unique ways for introducing set

classes. For example, Friedmann uses interval families. Each interval family contains a

2 Ibid.

5

group of set classes. These interval families can be used to sequence set classes that are

most similar or most disparate.3

Straus suggests a sequencing of trichordal set classes which follows closely to

Friedmanns interval families. Both Friedmanns and Strauss approaches are further

discussed in Chapter II.

Ronald Herder (1973) uses a different approach to teaching ear training for both

tonal and atonal music. Herder organizes his chapters based on a sequence of intervals

and combines both tonal and atonal music for all exercises. He includes examples of

sightsinging, melodic dictation, and harmonic dictation exercises. He does not require

students to identify set classes in dictation.

The problem with this approach is that it would probably take more than four

semesters to cover all the chapters because of the combination of two large masses of

music. A longer sequence would be essential in securing a developed intellectual aural

understanding of both types of music. Another problem is that, by organizing the

chapters by intervals, the student may not acquire a holistic view of the music. Herders

approach may lead the student to an interval-to-interval approach to sightsinging and

dictation rather than to the ability to comprehend entire phrases or in this case to identify

set classes based on the overall sonority and not just based on the intervals from which

they are built. Herder does not offer an approach to sequencing set classes for dictation.

3 See Chapter II.

6

Similarity Relations

In order to systematically sequence set classes one must first identify the

relationship between set classes. Knowing this relationship will allow the instructor to

sequence set classes based on presenting a set of similar or dissimilar set classes. One

way of determining the relation between set classes is by calculating similarity relations.

Similarity relations use various methods such as interval vectors, common embedded

interval classes and difference vectors to determine the level of similarity between set

classes. This study will focus on four measures of similarity: Morris SIM relation,

Rahns MEMB relation, Lewins RELt relation and Isaacsons IcVSIM relation.4

This study examines sequences that present trichordal and tetrachordal set classes

separately and avoids combining set classes of different cardinalities. It systematically

orders set classes of the same cardinality into two kinds of chains: those that juxtapose

the most similar set classes first and move gradually to the least similar relations, and

those that juxtapose the least similar set classes first and move gradually toward higher

degrees of similarity. All chains are based on the four similarity relations mentioned

above.5

It is critical to include the topic of post-tonal ear training in todays undergraduate

music theory sequences. When a syllabus is designed systematically, based on concrete

methods, class time is used more efficiently and the students hopefully leave with a more

fundamental understanding of the subject. This thesis will offer not only alternative 4 Robert Morriss SIM relation is addressed in Chapter III, John Rahns MEMB relation in Chapter IV, David Lewins RELt relation in Chapter V, and Eric Isaacsons IcVSIM relation in Chapter VI. 5 Chains in this study refer to links of trichordal or tetrachordal set classes arranged as a similar or dissimilar set class network.

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approaches to the literature that is currently being used, but will also serve as a guide to

constructing sequences of trichordal/tetrachordal set-classes in a systematic way, thereby

offering music educators choices in the presentation of material that can be tailored to fit

different teaching styles and curricular designs.

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CHAPTER II

MICHAEL FRIEDMANN AND JOSEPH STRAUS: APPROACHES TO SEQUENCING TRICHORDAL AND TETRACHORDAL

SET CLASSES FOR HARMONIC DICTATION

Introduction

Michael Friedmann (1990) and Joseph Straus (2005) have two distinct approaches

to ear training for twentieth-century music. More specifically, they offer various methods

of arranging trichordal and tetrachordal set classes for harmonic dictation. While their

approaches may differ, elements of each can be combined to form yet another method of

sequencing trichordal and tetrachordal set classes for harmonic dictation. Before

discussing this approach, it is necessary to understand the differences between Friedmann

and Straus.

Friedmann 1990

Michael Friedmann is one of the few who has suggested a sequencing of post-

tonal harmonies (dyads to hexachords) for the purpose of harmonic dictation in a

classroom setting (Friedman, 1990). The Friedmann approach is currently a commonly

used method of teaching ear training for post-tonal music, including teaching the

dictation of trichordal and tetrachordal set classes.

Trichordal Set Classes

Friedmann organizes trichordal and tetrachordal set classes into three interval

families. Each interval family is based on the interval classes found in each set class.

Friedmanns first interval family is comprised of all set classes containing interval class

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1. The second interval family is comprised of all set classes containing interval class 2,

but not interval class 1. The third interval family is made up of all set classes containing

neither interval class 1 nor interval class 2. Figure 2.1 is a chart of all trichordal and

tetrachordal set classes organized into Friedmanns interval families.

Interval Family 1: Set Classes Containing Interval Class 1

Trichordal Set Classes:

3-1 (012), 3-2 (013), 3-3 (014), 3-4 (015), 3-5 (016)

Tetrachordal Set Classes:

4-1 (0123) 4-2 (0124) 4-3 (0134) 4-4 (0125) 4-5 (0126) 4-6 (0127) 4-7 (0145) 4-8 (0156) 4-9 (0167) 4-10 (0235) 4-11 (0135) 4-12 (0236) 4-13 (0136) 4-14 (0237) 4-Z15 (0146) 4-16 (0157) 4-17 (0347) 4-18 (0147) 4-19 (0148) 4-20 (0158) 4-Z29 (0137)

Interval Family 2: Set Classes Containing Interval Class 2, but not Interval Class 1 Trichordal Set Classes: 3-6 (024), 3-7 (025), 3-8 (026), 3-9 (027) Tetrachordal Set Classes: 4-21 (0246), 4-22 (0247), 4-23 (0257), 4-24 (0248), 4-25 (0268), 4-26 (0358) 4-27 (0258)

Interval Family 3: Set Classes Containing Neither Interval Class 1 nor Interval Class 2 Trichordal Set Classes: 3-10 (036), 3-11 (037), 3-12 (048)

Tetrachordal Set Classes: 4-28 (0369) Figure 2.1. Friedmanns Interval Families Ear Training for Twentieth-Century Music (1990) pp. 51, 73

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Friedmann sets forth guidelines for harmonic dictation using his interval families.

These guidelines form a process in which students learn to take harmonic dictation. The

first exercise in this process is to identify set classes according to the interval family to

which they belong. Friedmanns Exercise 4.10 (the first exercise following the

introduction of set classes according to their interval families) states: Given the interval

family of a played trichord, specify which set class it is.1 While this exercise may seem

straightforward, it leaves room for interpretation. Friedmann does not limit the instructor

to playing trichordal set classes in closed position, but allows other realizations of set

classes to be included as well. The only requirement of the instructor is to advise the

student as to which interval family the trichord belongs. Friedmanns next dictation

exercise (Exercise 4.12) states:

Three-note chords and three-note melodic figures are played. Identify the set class. At first the exercise can be limited to each of the three interval families so as to limit the field of choices, but eventually it should be possible to recognize the twelve trichordal set class types. There are two versions of this exercise: the first uses only the most compact pitch representatives of the set classes; the second uses scattered pitch representatives of the set classes.2

This exercise requires students to identify a trichordal set class from one

particular interval family before moving on to the next. It allows for two presentations of

set classes, one in closed voicing and another in open voicing. Friedmann suggests to

begin with the dictation of set classes in closed voicing before moving to the dictation of

set classes in open voicing.

1 Exercise 4.10 can be found in Michael Friedmann, 1990. Ear Training for Twentieth-Century Music. New Haven: Yale University Press, pg. 51. 2 Ibid.

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The Exercises 4.10 and 4.12 can be interpreted as contrasting approaches to

presenting trichordal set classes for harmonic dictation. Exercise 4.10 can be used to

present trichordal set classes from most similar to most disparate or from most disparate

to most similar. Depending on the interpretation, an instructor may arrange trichordal set

classes from most similar to most disparate using all set classes from interval family 1,

then all set classes from interval family 2, followed by all set classes from interval family

3; or from most disparate to most similar using a mix of set classes from all three interval

families.3

Exercise 4.12 arranges set classes from most similar to most disparate.

Friedmann states that at first the exercise should be limited to only set classes in each

interval family. As the student becomes comfortable, then set classes may be mixed

between interval families. The objective here is for the student, based on the three

interval families, to distinguish first between set classes and then to identify all twelve

trichordal set class types.

Tetrachordal Set Classes

Friedmann also organizes tetrachordal set classes according to the three interval

families. He then discusses methods with which to identify the interval classes that make

up the interval families as a stepping stone to identifying tetrachordal set class types. For

example, when presenting a tetrachordal set classes from interval family 1, an instructor

may play 4-1 (0123) and then ask the student to identify the interval class that would

place it in its interval family, in this case ic1, interval family 1.

3 The measure of similarity in this case is the common interval class.

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Friedmann also organizes tetrachordal set classes according to supersets as

another method of grouping. This grouping is meant to provide the student with another

measure of similarity. Placing tetrachordal set classes into supersets is another way for

the student to distinguish between set classes. He does not, however, ask the student to

identify the set class according to its superset.

Friedmann does not use the same approach to presenting tetrachordal set classes

as he did when presenting trichordal set classes. He requires the student to identify only

10 of the 29 tetrachordal set classes. These ten he describes as common tetrachords that

are easy to identify by eye or ear (Friedmann, 1990). Figure 2.2 is a list of the ten

common tetrachordal set classes along with the descriptions Friedmann assigns to them.

4-1 (0123) chromatic A tetrachord that could be presented as four

consecutive members of a chromatic scale.

4-21 (0246) whole-tone A tetrachord that could be presented as four consecutive members of a whole-tone scale.

4-28 (0369) diminished A tetrachord that could be presented as four consecutive i representatives of the diminished seventh chord4.

4-23 (0257) fourth-chord A tetrachord that could be presented as four consecutive members of the circle of fifths.

4-27 (0258) Familiar from tonal contexts as the dominant

seventh chord (inversionally equivalent to the half-diminished seventh chord).

4-26 (0358) Familiar from tonal contexts as the minor seventh chord. Figure 2.2. Friedmanns Common Tetrachordal Set Classes Ear Training for Twentieth-Century Music (1990), pg. 81.

4 I is the same as interval class 3 (ic3).

13

4-3 (0134) octatonic A tetrachord that could be presented as four tetrachord 1 consecutive members of an octatonic scale starting with a half-step.

4-10 (0235) octatonic A tetrachord that could be presented as four

tetrachord 2 consecutive members of an octatonic scale starting with a whole-step.

4-11 (0135) major tetrachord 1 A tetrachord that could be presented as the first four degrees of a major scale (inversionally equivalent to Phrygian 1).

4-20 (0158) Familiar from the tonal contexts as the major seventh chord (this tetrachord is inversionally symmetrical, although this is not evident from the normal order name). Figure 2.2 Continued.

After introducing these tetrachordal set classes, Friedmann organizes them into

three groups. Exercise 5.6 states:5

a) To practice aural identification of the preceding set classes, divide the ten set classes into the three groups indicated below. The pianist plays the chords selected from each group in a variety of spacings; identify each set class Group 1 Group 2 Group 3 4-1 (0123) 4-10 (0235) 4-3 (0134) 4-11 (0135) 4-21 (0246) 4-27 (0258) 4-20 (0158) 4-23 (0257) 4-28 (0369) 4-26 (0358)

Friedmann does not specify why he grouped certain tetrachordal set classes together.

This is surprising because these groups do not follow the three interval families he

introduces before. While Friedmann organized trichordal/tetrachordal set classes

according to his three interval families, these ten common tetrachordal set classes do not

follow suit. One might speculate that each group contains tetrachordal set classes with

5 Exercise 5.6 from Friedmann (1990), pg. 81.

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distinctly contrasting sonorities. For example, Group 1 is comprised of the chromatic

tetrachord, the major tetrachord 1, the major seventh chord, and the minor seventh

chord. Friedmann uses this as a method of narrowing down the choices between

tetrachordal set classes and therefore makes it easier for the student to dictate these set

classes. When examining the descriptions of the tetrachordal set classes in each group,

one can say that each is made up of disparate tetrachords (as opposed to similar

tetrachords). Group 2 contains the octatonic tetrachord 2, the whole tone tetrachord,

and the fourth chord. Group 3 consists of the octatonic tetrachord 1, the dominant

seventh chord, and the diminished tetrachord. When played, the tetrachordal set

classes in each group will result in contrasting sonorities, as opposed to if all tetrachordal

set classes in interval family 1 were played.

Once the student is able to identify tetrachordal set classes from each group,

Friedmann then constructs two larger groups of tetrachordal set classes for dictation.

Exercise 5.6 (b) states: Perform the same exercise [as in 5.6 (a)] using the two following

larger groups:

Group 1 Group 2

4-1 (0123) chromatic 4-10 (0235) octatonic 2 4-3 (0134) octatonic 1 4-20 (0158) major seventh chord 4-11 (0135) major 1 4-21 (0246) whole-tone 4-23 (0257) fourth-chord 4-26 (0358) minor seventh chord 4-27 (0258) dominant seventh 4-28 (0369) diminished

After these groups have been presented for dictation, Friedmann instructs the

dictation of all ten tetrachordal set classes. This step by step approach is essential to the

15

mastery of the harmonic dictation of tetrachordal set classes. Once the student is asked to

identify all ten set classes, he/she is already familiar with the various sonorities.

Straus 2005

Joseph Strauss Introduction to Post-Tonal Theory is a textbook intended to cover

both the theoretical concepts and the aural skills necessary to apply set theory and twelve-

tone theory musically in analysis. When speaking of trichordal set classes, Straus

specifies a sequencing for harmonic dictation. This is the only ear training exercise in

this text which requires students to identify the twelve trichordal set classes in dictation.


Strauss Chapter 2 ear training exercise VI states:

Learn to identify the twelve different trichordal setclasses when they are played by your instructor. It may be easier if you learn them in the following order, adding each new one as the previous ones are mastered: 1. 3-1 (012) chromatic trichord 2. 3-9 (027) stack of perfect fourths or fifths 3. 3-11 (037) major or minor triad 4. 3-3 (014) major and minor third combined 5. 3-7 (025) diatonic trichord 6. 3-12 (048) augmented triad 7. 3-5 (016) semitone and tritone 8. 3-8 (026) whole-tone and tritone 9. 3-10 (036) diminished triad 10. 3-2 (013) nearly chromatic 11. 3-6 (024) two whole-tones 12. 3-4 (015) semitone and perfect fourth

This sequence may seem quite similar to Friedmanns sequencing of the ten common

tetrachordal set classes. For the most part, this ordering of trichordal set classes moves

16

through a path of disparate trichords. (012) and (027), for example, can be described as

having distinct sonorities. (012) is comprised of small intervals, ic1, which would create

a very dissonant sound while (027) is comprised of perfect fourths and fifths, which

would create a more open sound. It seems that Friedmann and Straus agree that ordering

trichordal and tetrachordal set classes for dictation may be easier for the student when

choosing to present set classes with contrasting sonorities. In fact, the first three

trichordal set classes in Strauss sequencing belong to the three Friedmann interval

families. This pattern continues until the last two trichords. Table 2.1 outlines Strauss

sequence of trichordal set classes arranged into the three Friedmann interval families.

Table 2.1. Strauss Sequence of Trichordal Set Classes According to Friedmanns Interval Families Interval Family 1 (ic1) Interval Family 2 (ic2, not ic1) Interval Family 3

(neither ic1 nor ic2) 1. 3-1 (012) 2. 3-9 (027) 3. 3-11 (037) 4. 3-3 (014) 5. 3-7 (025) 6. 3-12 (048) 7. 3-5 (016) 8. 3-8 (026) 9. 3-10 (036) 10. 3-2 (013) 11. 3-6 (024) 12. 3-4 (015) Tetrachordal Set Classes Straus (2005) does not mention the dictation of tetrachordal set classes. Based on

his approach to sequencing trichordal set classes, one might assume that he would take

the same approach as Friedmann in sequencing tetrachordal set classes for dictation.

That is, one might assume Straus would use a disparate to similar approach to sequencing

set classes as he has done with trichords. Strauss arrangement and descriptions of

17

trichordal set classes are quite reminiscent of Friedmanns ten common tetrachordal set

classes. Friedmann, however, does not sequence tetrachordal set classes according to his

interval families as both Friedmann and Straus do when sequencing trichordal set classes.

Each group in Friedmanns exercise 5.6 is a mix of tetrachordal set classes that belong to

different interval families.

Friedmann and Straus

Strauss ordering of trichordal set classes can be viewed as one way to realize

Friedmanns exercise 4.10. While Straus does not refer to Friedmanns interval families,

it may benefit the student to combine both approaches. Asking a student to first identify

the interval family is a much more manageable task than identifying all twelve trichordal

set classes. After this is mastered, the Friedmann approach asks for students to identify

all trichordal set classes in one interval family before moving to the next. Instead of this

method, Straus chooses to introduce a set class from interval family 1, then a set class

from interval family 2, followed by a set class from interval family 3. This is repeated

until all trichordal set classes have been presented. Identifying set classes with distinct

sonorities may also be a much more manageable task for a student than identifying all set

classes in one interval family before moving to the next. A combined approach might

include introducing the various interval families, and then asking students to identify

disparate sounding trichordal set classes in dictation (assuming the measure of similarity

is the interval class common in each interval family).

18

Sequencing Set Classes Using Similarity Relations

Strauss sequencing of trichordal set classes fits very nicely into Friedmanns

exercise 4.10, as mentioned before. Yet, it is only one way of realizing this exercise.

Straus, for example, moves, within interval family 1, from 3-1 (012) to 3-3 (014), then to

3-5 (016) followed by 3-2 (013) and 3-4 (015). Why would he not start at 3-1 (012) and

then move to 3-2 (013), then 3-3 (014), and so on? That is, why would he not choose a

path that started by comparing those set classes that were most similar to each other and

then move towards diversity? Any arrangement of set classes that moves from one

interval family to the next, continuously, will result in a sequence of disparate sounding

set classes. For example, beginning with 3-1 (012) from interval family 1, then moving

to 3-6 (024) in interval family 2, followed by 3-10 (036) from interval family 3 will result

in a sequence of trichordal set classes which relies on disparate sonorities, if the measure

of similarity is the common interval class in each interval family. The problem is that

there are no restrictions, using the Straus and Friedmann method, as to which trichordal

set classes should be presented first. Straus says that his sequence may be easier for the

student to master, but does not specify why. Friedmanns approach is much more in

depth and expanded yet still leaves room for interpretation. An instructor still needs to

decide which set classes he/she will introduce first from each interval family. While one

may assume that an instructor would start with the first set class in interval family 1, then

move to the first in interval family 2 and so on, this is not a well-founded assumption.

Straus is a great example. He chooses not to follow this path; rather, he chooses a

different sequence when moving through the interval families.

19

One way to answer these questions is to calculate the measure of similarity

between set classes. Knowing which trichordal and tetrachordal set classes are most

similar within an interval family will aid the instructor in building a sequence of

trichordal and tetrachordal set classes that uses a disparate-to-similar approach, or a

similar-to-disparate approach, depending on the instructors preference. This measure of

similarity can be formulated in many ways. Chapter III will explore Robert Morriss

SIM relation, which could be used to sequence more systematically trichordal and

tetrachordal set classes for harmonic dictation.

20

CHAPTER III

ROBERT MORRISS SIM RELATION

One of the most straight-forward ways to measure similarity between set class

types is by the Similarity Index (SIM) relation (Morris, 1979/1980), which compares

interval vectors. Morriss SIM relation calculates the number of interval classes (ics) that

are different between pairs of set classes. A larger numerical value indicates set classes

with a greater amount of dissimilar ics and therefore yields dissimilar pairs, while a

smaller numerical value indicates set classes with a lesser number of dissimilar ics and

therefore yields a more similar pair. Morriss SIM relation can be applied to sequencing

paths of trichordal and tetrachordal set classes that move from similarity to dissimilarity

and from dissimilarity to similarity, and thus might aid the aural skills instructor wishing

to introduce post-tonal harmonies in a systematic order that fits their own teaching

philosophy.

SIM

To calculate the measure of similarity between two set classes using Morriss SIM

relation, one compares their interval vectors. Morriss SIM relation tallies the number of

ics that are different between set classes, for example (012) and (013). To calculate the

measure of similarity, take the sum of the absolute value of the difference of the vectors:

6

SIM = | an bn| n = 1

21

In this instanceto find the measure of similarity between (012) and (013)

one would subtract the corresponding ics, take the absolute value, then add:

| 2-1 | + | 1-1 | + | 1-0 | + | 0-0 | + | 0-0 | + | 0-0 |. This equals 1 + 0 + 1 + 0 + 0 + 0 = 2.

SIM (012, 013) = 2. The numeric result reflects the number of differences between the

two sets; (013) has one less instance of ic1 than (012), and 1 more instance of ic3. The

lower the final number is, the greater the degree of similarity.


This measure of similarity can be applied to all possible pairs of set classes in

order to identify pairs that are either most similar or most disparate. A SIM relation of 0

indicates the greatest degree of similarity between set classes. A SIM relation of 2 yields

a greater measure of similarity than 4, which yields a greater measure of similarity than 6.

The following table (Table 3.1) charts the SIM relation as applied to the twelve trichordal

set classes.

Table 3.1. SIM Relation Applied to Trichordal Set Classes1

1 Data formulated by using the Isaacson PCSet Similarity Relation Calculator < http://theory.music.indiana.edu/isaacso/research.html>

(012) (013) (014) (015) (016) (024) (025) (026) (027) (036) (037) (048) (012) 0 2 4 4 4 4 4 4 4 6 6 6 (013) 2 0 2 4 4 4 2 4 4 4 4 6 (014) 4 2 0 2 4 4 4 4 6 4 2 4 (015) 4 4 2 0 2 4 4 4 4 6 2 4 (016) 4 4 4 2 0 6 4 4 4 4 4 6 (024) 4 4 4 4 6 0 4 2 4 6 4 4 (025) 4 2 4 4 4 4 0 4 2 4 2 6 (026) 4 4 4 4 4 2 4 0 4 4 4 4 (027) 4 4 6 4 4 4 2 4 0 6 4 6 (036) 6 4 4 6 4 6 4 4 6 0 4 6 (037) 6 4 2 2 4 4 2 4 4 4 0 4 (048) 6 6 4 4 6 4 6 4 6 6 4 0

22

Similar Trichordal Set Classes

This method identifies ten pairs of similar trichordal set classes and two

individual trichordal set classes that could not be paired with any of the other set classes,

at least not with such a strong degree of similarity. Figure 3.1 illustrates all similar pairs

of trichordal set classes. Pairs with a SIM relation of 0 have been excluded as 0 indicates

two identical set classes. All of the following pairs yield a SIM relation of 2.

(012) (013) (013) (014) (013) (014) (025) (015) (014) (015) (015) (024) (037) (016) (037) (026) (025) (025) (027) (037) Figure 3.1. Similar Pairs of Trichordal Set Classes Based on the SIM Relation

The two trichordal set classes that could not be paired with another set class that would

yield a SIM of 2 are (036) , and (048) .

After these pairs have been identified, they may be used to construct a network of

possible sequences2:

(012) (013) (014) (015) (016) | | (025) (037) (024) (036) | | (027) (026) (048)

2 For a similar diagram refer to Robert Morris, 1979/1980. A Similarity Index for Pitch-Class Sets. Perspectives of New Music 18(1-2): 455.

23

This network can be used in developing paths of trichordal set classes which move from

most similar to most disparate. Using the diagram above as a guide, one may begin a

sequence of similar trichordal set classes by first presenting (012), then (013). Once

(013) has been presented, 2 similar set classes remain that can be presented after (013):

(014) and (025). The set class that is chosen to follow (013) will at least in part

determine the path in one particular sequence. For example, (012) - (013) - (014) (015)

- (025) - (027) (016) (024) (026) (036) (048) is one possible path that moves

from similarity to dissimilarity. This thread begins at (012) and moves through the

network from similar pair to similar pair, and ends with the two trichordal set classes that

cannot be paired. When the thread reaches (027) the pattern reaches a predicament: (027)

is not as similar to any of the remaining trichordal set classes as those bound together by

a similarity value of 2. The possibilities then grow in number as three of the five set

classes in bold have a SIM of 4 with (027), and two of them have a SIM value of 6 with

(027).

This sequencing method used to generate the example requires that pairs of

trichordal set classes be linked together whenever possible. For instance, in the example

above, although a crisis has been reached at (027), there is still one pair of similar

trichordal set classes which have not been introduced, (024) and (026). If (024) is

introduced then the set class that follows it must be (026) and vice versa, no matter where

in the path these set classes occur. This way every possible link to a similar trichordal set

class will be used.

24

The problem with linking similar pairs of set classes is that in certain places along

the chain a link between dissimilar set classes will occur. For example, in the chain

above, (027) is not similar to (016), (016) is not similar to (024), (026) is not similar to

(036) and (036) is not similar to (048), that is, if the definition of similarity is a SIM

value of 2. This is not a chain of similar set classes; rather, it is a chain that moves from

similar set classes to dissimilar set classes. A chain that links set classes from most

similar to most disparate can be confusing when presenting in a classroom for harmonic

dictation. An instructor would have to make clear the distinction between set classes that

are similar and those that are disparate. That is, if an instructor were to begin by

presenting similar set classes, the student may be confused when then asked to identify

dissimilar set classes.

Another problem that arises is that there are multiple paths in using this method to

sequence trichordal set classes, all of which seem equally justifiable. Figure 3.2 is an

example of all the possible paths that could be formulated using the initial thread (012) -

(013) - (014) (015) - (025) - (027).

25

(012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (024) - (026) - (036) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (024) - (026) - (048) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (026) - (024) - (036) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (026) - (024) - (048) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (036) - (024) - (026) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (036) - (026) - (024) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (036) - (048) - (024) - (026) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (036) - (048) - (026) - (024) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (048) - (024) - (026) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (048) - (026) - (024) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (048) - (036) - (024) - (026) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (016) - (048) - (036) - (026) - (024) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (024) - (026) - (016) - (036) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (024) - (026) - (016) - (048) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (024) - (026) - (036) - (016) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (024) - (026) - (036) - (048) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (024) - (026) - (048) - (016) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (024) - (026) - (048) - (036) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (026) - (024) - (016) - (036) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (026) - (024) - (016) - (048) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (026) - (024) - (036) - (016) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (026) - (024) - (036) - (048) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (026) - (024) - (048) - (016) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (026) - (024) - (048) - (036) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (016) - (024) - (026) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (016) - (026) - (024) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (016) - (048) - (024) - (026) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (016) - (048) - (026) - (024) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (024) - (026) - (016) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (024) - (026) - (048) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (026) - (024) - (016) - (048) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (026) - (024) - (048) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (048) - (024) - (026) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (048) - (026) - (024) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (048) - (016) - (024) - (026) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (036) - (048) - (016) - (026) - (024) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (016) - (024) - (026) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (016) - (026) - (024) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (016) - (036) - (024) - (026) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (016) - (036) - (026) - (024) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (024) - (026) - (016) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (024) - (026) - (036) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (026) - (024) - (016) - (036) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (026) - (024) - (036) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (036) - (016) - (024) - (026) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (036) - (016) - (026) - (024) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (036) - (024) - (026) - (016) (012) - (013) - (014) - (015) - (037) - (025) - (027) - (048) - (036) - (026) - (024) - (016) Figure 3.2. Paths of Similar Trichordal Set Classes Based on the Initial Thread (012) - (013) - (014) (015) - (025) - (027)

26

It is quite difficult to systematically choose one path with all the choices that are

available.

Dissimilar Trichordal Set Classes

Dissimilar pairs of trichordal set classes can also be identified using the SIM

relation. Figure 3.3 illustrates dissimilar pairs of trichordal set classes that have no

interval classes in commontrichordal set classes with a SIM relation 6 have no ics in

common:

(012) (012) (012) (013) (036) (037) (048) (048) (014) (015) (016) (016) (027) (036) (024) (048) (024) (025) (027) (027) (036) (048) (036) (048) (036) (048) Figure 3.3. Dissimilar Pairs of Trichordal Set Classes Based on the SIM Relation

(026) is the only trichordal set class that does not have a difference of 6 when comparing

ics with the other set classes. A network of dissimilar set classes can be realized as

follows:

(015) (036) (024) (016) | (027) (048) (012) | | | (014) (013) (037) (025) (026)

27

The same problems arise when realizing a path of dissimilar trichordal set classes

for harmonic dictation that arose when sequencing similar pairs of trichordal set classes.

Morriss SIM relation, however, does present a starting point in thinking about

systematically sequencing trichordal set classes using similarity relations.

Morris's SIM Relation Applied to Tetrachordal Set Classes

Morriss SIM relation can also be applied to find the measure of similarity

between pairs of tetrachordal set classes. Table 3.2 is a table of the SIM relation applied

to the 29 tetrachordal set class types. When applied to tetrachordal set classes a SIM

relation of 10 indicates set classes that are most dissimilar. Set classes have a greater

degree of similarity as the SIM relation decreases from 10 to 0 (0 indicating identical set

classes and 10 indicating dissimilar set classes).

28

Table 3.2. SIM Relation Applied to Tetrachordal Set Classes

(0123) (0124) (0134) (0125) (0126) (0127) (0145) (0156) (0167)

(0123) 0 2 4 4 6 6 6 8 8

(0124) 2 0 2 2 4 6 4 6 8

(0134) 4 2 0 2 4 6 4 6 8

(0125) 4 2 2 0 2 4 2 4 6

(0126) 6 4 4 2 0 2 4 2 4

(0127) 6 6 6 4 2 0 6 2 2

(0145) 6 4 4 2 4 6 0 4 6

(0156) 8 6 6 4 2 2 4 0 2

(0167) 8 8 8 6 4 2 6 2 0

(0235) 4 4 4 4 6 6 6 8 8

(0135) 4 2 4 2 4 6 4 6 8

(0236) 6 4 2 4 4 6 6 6 8

(0136) 6 6 4 4 4 4 6 6 6

(0237) 6 4 4 2 4 4 4 4 6

(0146) 6 4 4 2 2 4 4 4 6

(0157) 8 6 6 4 2 2 6 2 4

(0347) 8 6 4 4 6 8 2 6 8

(0147) 8 6 4 4 4 6 4 4 6

(0148) 8 6 6 4 6 8 2 6 8

(0158) 8 6 6 4 6 6 2 4 6

(0246) 8 6 8 8 6 8 8 8 10

(0247) 6 4 6 4 6 6 6 6 8

(0257) 6 6 8 6 8 6 8 8 8

(0248) 8 6 8 8 6 8 8 8 10

(0268) 8 6 8 8 6 8 8 8 8

(0358) 8 6 4 4 6 6 6 6 8

(0258) 8 6 4 4 4 8 6 6 8

(0369) 10 10 8 10 10 10 10 10 8

(0137) 6 4 4 2 2 4 4 4 6

29

Table 3.2 Continued.

(0235) (0135) (0236) (0136) (0237) (0146) (0157) (0347) (0147) (0148)

(0123) 4 4 6 6 6 6 8 8 8 8

(0124) 4 2 4 6 4 4 6 6 6 6

(0134) 4 4 2 4 4 4 6 4 4 6

(0125) 4 2 4 4 2 2 4 4 4 4

(0126) 6 4 4 4 4 2 2 6 4 6

(0127) 6 6 6 4 4 4 2 8 6 8

(0145) 6 4 6 6 4 4 6 2 4 2

(0156) 8 6 6 6 4 4 2 6 4 6

(0167) 8 8 8 6 6 6 4 8 6 8

(0235) 0 2 4 2 4 4 6 4 4 6

(0135) 2 0 4 4 2 2 4 4 4 4

(0236) 4 4 0 2 4 2 4 4 2 6

(0136) 2 4 2 0 4 2 4 4 2 6

(0237) 4 2 4 4 0 2 2 4 4 4

(0146) 4 2 2 2 2 0 2 4 2 4

(0157) 6 4 4 4 2 2 0 6 4 6

(0347) 4 4 4 4 4 4 6 0 2 2

(0147) 4 4 2 2 4 2 4 2 0 4

(0148) 6 4 6 6 4 4 6 2 4 0

(0158) 6 4 6 6 2 4 4 2 4 2

(0246) 8 6 6 8 8 6 6 8 8 8

(0247) 4 2 6 6 2 4 4 6 6 6

(0257) 4 4 8 6 4 6 6 8 8 8

(0248) 8 6 6 8 8 6 6 8 8 6

(0268) 8 6 6 8 8 6 6 8 8 8

(0358) 4 4 4 4 2 4 4 4 4 6

(0258) 4 4 2 2 4 2 4 4 2 6

(0369) 8 10 6 6 10 8 10 8 6 10

(0137) 4 2 2 2 2 0 2 4 2 4

30

Table 3.2 Continued.

(0158) (0246) (0247) (0257) (0248) (0268) (0358) (0258) (0369) (0137)

(0123) 8 8 6 6 8 8 8 8 10 6

(0124) 6 6 4 6 6 6 6 6 10 4

(0134) 6 8 6 8 8 8 4 4 8 4

(0125) 4 8 4 6 8 8 4 4 10 2

(0126) 6 6 6 8 6 6 6 4 10 2

(0127) 6 8 6 6 8 8 6 8 10 4

(0145) 2 8 6 8 8 8 6 6 10 4

(0156) 4 8 6 8 8 8 6 6 10 4

(0167) 6 10 8 8 10 8 8 8 8 6

(0235) 6 8 4 4 8 8 4 4 8 4

(0135) 4 6 2 4 6 6 4 4 10 2

(0236) 6 6 6 8 6 6 4 2 6 2

(0136) 6 8 6 6 8 8 4 2 6 2

(0237) 2 8 2 4 8 8 2 4 10 2

(0146) 4 6 4 6 6 6 4 2 8 0

(0157) 4 6 4 6 6 6 4 4 10 2

(0347) 2 8 6 8 8 8 4 4 8 4

(0147) 4 8 6 8 8 8 4 2 6 2

(0148) 2 8 6 8 6 8 6 6 10 4

(0158) 0 8 4 6 8 8 4 6 10 4

(0246) 8 0 6 8 2 2 8 6 10 6

(0247) 4 6 0 2 6 6 2 4 10 4

(0257) 6 8 2 0 8 8 4 6 10 6

(0248) 8 2 6 8 0 2 8 6 10 6

(0268) 8 2 6 8 2 0 8 6 8 6

(0358) 4 8 2 4 8 8 0 2 8 4

(0258) 6 6 4 6 6 6 2 0 6 2

(0369) 10 10 10 10 10 8 8 6 0 8

(0137) 4 6 4 6 6 6 4 2 8 0

31

One can see that formulating a path of similar and dissimilar tetrachordal set

classes would be more cumbersome than the attempt to sequence trichordal set classes.

In order to purvey an example of a possible sequencing of set classes, let the focus be on

Friedmanns 10 common tetrachordal set classes.3 Table 3.3 is a table of the SIM

relation as applied to Friedmanns 10 common tetrachordal set classes. Friedmanns 10

common tetrachordal set class types have been paired, beginning with the most similar

pairs and ending with the most disparate. Table 3.4 is a listing of these pairs from most

similar to most disparate.

Table 3.3. SIM Relation Applied to Friedmanns 10 Common Tetrachordal Set Classes

(0123) (0134) (0235) (0135) (0158) (0246) (0257) (0358) (0258) (0369)

4-1 (0123) 0 4 4 4 8 8 6 8 8 10

4-3 (0134) 4 0 4 4 6 8 8 4 4 8

4-10 (0235) 4 4 0 2 6 8 4 4 4 8

4-11 (0135) 4 4 2 0 4 6 4 4 4 10

4-20 (0158) 8 6 6 4 0 8 6 4 6 10

4-21 (0246) 8 8 8 6 8 0 8 8 6 10

4-23 (0257) 6 8 4 4 6 8 0 4 6 10

4-26 (0358) 8 4 4 4 4 8 4 0 2 8

4-27 (0258) 8 4 4 4 6 6 6 2 0 6

4-28 (0369) 10 8 8 10 10 10 10 8 6 0

3 For information on Friedmanns 10 common tetrachordal set class types refer to Chapter II, or Friedmann (1990), Ear Training for Twentieth-Century Music. This will allow for an example to be provided of a possible sequencing of tetrachordal set classes without having to sequence all 29 set classes. While Morris (1979/80) does not refer or limit the application of the SIM relation to tetrachordal set classes by Friedmanns 10 common tetrachordal set classes, this limitation will be used for the purposes of this study. Any of the methods used in this study to sequence Friedmanns 10 common tetrachordal set classes can be applied to a sequencing of all 29 tetrachordal set classes.

32

Table 3.4. Pairs of Friedmanns 10 Common Tetrachordal Set Classes Arranged from Most Similar to Most Disparate According to Morriss SIM Relation. 4

SIM 2 4 6 8 10 1. (0135) (0235)

(0123) (0134)

(0123) (0257)

(0123) (0158)

(0123) (0369)

2. (0258) (0358) (0123) (0235)

(0134) (0158)

(0123) (0258)

(0135) (0369)

3. (0123) (0135) (0235) (0158)

(0134) (0246)

(0158) (0369)

4. (0134) (0235) (0135) (0246)

(0134) (0257)

(0246) (0369)

5. (0134) (0135) (0158) (0257)

(0134) (0369)

(0257) (0369)

6. (0134) (0358) (0158) (0258)

(0235) (0246)

7. (0134) (0258) (0246) (0258)

(0235) (0369)

8. (0235) (0257) (0257) (0258)

(0158) (0246)

9. (0235) (0358) (0258) (0369)

(0246) (0257)

10. (0235) (0258) (0246) (0358)

11. (0135) (0158) (0358) (0369)

12. (0135) (0257)

13. (0135) (0258)

14. (0135) (0358)

15. (0158) (0358)

16. (0257) (0358)

4 Pairs with a SIM relation of 0 have been left out of this chart as 0 indicates two identical set classes. The numerical values in the first column were added in order to easily count the number of pairs in each column.

33

There are only two tetrachordal pairs which yield a SIM relation of 2. One could

present these two pairs and then proceed to present pairs under the SIM column 4,

followed by those in 6, 8 and 10. This however, would not be a true chain of similar

tetrachordal set classes; rather, it would be a chain that moves from pairs most similar to

most disparate. This process in reverse could be used to form a chain of tetrachordal set

classes that moves from most disparate to most similar.

Morriss SIM relation is a great introduction in calculating the measure of

similarity between various pairs of trichordal and tetrachordal set classes. However,

there are other similarity relations that suggest finer gradations of similarity and thus

would narrow down the choices, making it easier for one to choose a sequence of

trichordal and tetrachordal set classes for harmonic dictation. The next few chapters will

explore similarity relations suggested by John Rahn, David Lewin, and Eric Isaacson.

34

CHAPTER IV

JOHN RAHNS MEMB RELATION

MEMB: The Embedding Function

John Rahn (1979/80) has formulated a similarity relation that suggests a finer

gradation of similarity between trichordal and tetrachordal set classes as a response to

Robert Morriss SIM relation. Rahns MEMB relation is an embedding function that

calculates the number of common interval classes embedded in two set classes. The

larger the number of embedded interval classes, the more similar the pair. Thus

MEMB(A, B) counts all subsets of a specific size embedded mutually in set A and B.1

Rahn (1979/80) offers the following formula when calculating the measure of

similarity between two sets:2

MEMBn(X,A,B) = EMB(X,A) + EMB(X,B) for all X such that #X= n and EMB (X,A) >0 and EMB(X,B)>0 It is most important to realize that, in the definition of MEMBn(X,A,B), X must be

embedded at least once in both sets A and B to be counted; then all instances of X in

either set are counted. Subsets that appear in one set but not the other are not counted at

all.3 In other words, MEMBn(X,A,B) first identifies the interval classes that are

1Rahn (1979/80) distinguishes between EMB and MEMB in that EMB is a count of how many times a set is embedded in another and MEMB is a count of how many subsets are mutually embedded in two sets. MEMB is therefore more discriminatory as any two sets of the same size under the EMB function will result in a similarity of 0. MEMB measures subsets embedded in two sets of the same cardinality and offers a way too calculate the measure of similarity between sets of the same cardinality. 2Rahn (1979/80) Relating Sets, pg. 492. #X refers to the cardinality of the subset X. Rahn (1979/80) mentions that there is no point in counting embedded subsets of size zeroevery set has exactly one(pg. 493). 3Rahn (1979/80) Relating Sets, pg. 492.

35

common in both set A and set B. These common interval classesdyadsmake up the

subset X. Then the subsets labeled X are counted as they appear in set A and added to

the number of times they appear in set B. The sum of the two equals the measure of

similarity, or the number of times X is embedded in both set A and set B. Figure 4.1 is a

realization of this formula using sets 3-1 (012) and 3-2 (013).

(012) (013) Using the interval vectors one can clearly identify the two common interval classes, ic1 and ic2. X = (01) and X = (02) First calculate how many times (01) or ic1 is embedded in (012) and (013). MEMB2[(01),(012),(013)] = EMB [(01), (012)] + EMB [(01), (013)]

Using the interval vectors above as a reference one can see that there are 2 instances of (01) in (012) and 1 instance of (01) in (013).

MEMB2[(01),(012),(013)] = 2 + 1 MEMB2[(01),(012),(013)] = 3; there are 3 instances of (01) embedded in (012) and (013). This same formula is then applied to the other common interval class; ic2 or (02)

Looking back at the interval vectors of (012) and (013) one can see that there is 1 instance of (02) in (012) and 1 instance of (02) in (013).

MEMB2[(02),(012),(013)] = 1 + 1

MEMB2[(02),(012),(013)] = 2; there are two instances of (02) embedded (012) and (013). Therefore MEMB2[X,(012),(013)] = 5. Figure 4.1. MEMBn[X,(012),(013)].

36


Rahns embedding function may be able to distinguish finer levels of similarity

between set classes than Morriss SIM relation. Table 4.1 is a table of Rahns MEMB

relation applied to the 12 trichordal set classes.

Table 4.1. MEMB Relation Applied to Trichordal Set Classes.4 Rahn, John. 1979/1980. Relating Sets. Perspectives of New Music 18 (1-2): 483-98.

(012) (013) (014) (015) (016) (024) (025) (026) (027) (036) (037) (048) (012) 6 5 3 3 3 3 2 2 2 0 0 0 (013) 5 6 4 2 2 3 4 2 2 3 2 0 (014) 3 4 6 4 2 2 2 2 0 3 4 4 (015) 3 2 4 6 4 2 2 2 3 0 4 4 (016) 3 2 2 4 6 0 2 2 3 2 2 0 (024) 3 3 2 2 0 6 3 5 3 0 2 4 (025) 2 4 2 2 2 3 6 2 5 3 4 0 (026) 2 2 2 2 2 5 2 6 2 2 2 4 (027) 2 2 0 3 3 3 5 2 6 0 3 0 (036) 0 3 3 0 2 0 3 2 0 6 3 0 (037) 0 2 4 4 2 2 4 2 3 3 6 4 (048) 0 0 4 4 0 4 0 4 0 0 4 6

In order to confirm that Rahns MEMB relation really does offer a finer gradation

of similarity than Morriss SIM, results from both must be compared. Table 4.2 is a

comparison of Morris SIM relation and Rahns MEMB relation as applied to trichordal

set class (012).


37

Table 4.2. Comparison of Morriss SIM and Rahns MEMB.

It is important to note that when using Morriss SIM relation, the larger the

numerical value, the less similar the set classes. When using Rahns MEMB relation, the

larger the numerical value, the more similar the set classes. This may present a problem

when attempting to compare across both similarity relations. Rahns values, therefore,

will be converted to match the range of Morriss SIM relation. The range for all tables

MORRIS SIM RAHN MEMB

(012) (012)

(012) 0 6

(013) 2 5

(014) 4 3

(015) 4 3

(016) 4 3

(024) 4 3

(025) 4 2

(026) 4 2

(027) 4 2

(036) 6 0

(037) 6 0

(048) 6 0

38

and graphs for trichordal set classes will be 0-6, 0 indicating two identical set classes and

6 indicating pairs of greatest dissimilarity.5 Table 4.3 is the same comparison between

SIM and MEMB where MEMB values have been converted to fit the range of SIM.

Table 4.3. Comparison of SIM and MEMB with Converted MEMB Values.

It is evident in the table above that the MEMB relation does offer finer gradations

in measuring similarity than SIM does. The chart shows, using Morriss SIM relation,

that (012) is just as similar to (014), (015), (016), (024), (025), (026), and (027) while

5 In order to convert Rahns values to fit the range 0-6, one must subtract the existing value from 6. Thus, when referring to the MEMB relation, 6 becomes 0, 5 becomes 1, 4 becomes 2 and so on.

MORRIS SIM RAHN MEMB

(012) (012)

(012) 0 0

(013) 2 1

(014) 4 3

(015) 4 3

(016) 4 3

(024) 4 3

(025) 4 4

(026) 4 4

(027) 4 4

(036) 6 6

(037) 6 6

(048) 6 6

39

Rahns MEMB relation shows (012) is less similar to (025) than (014). MEMB will

narrow the possible sequences of trichordal set classes. MEMB also identifies finer

levels of similarity when applied to tetrachordal set classes.


The problem with Morriss SIM relation is even more evident when speaking

about tetrachordal set classes. It is quite apparent that many of the set classes under

Morriss SIM relation are similar to each other, making it difficult to systematically

sequence them for harmonic dictation.

It may be easier to focus on Friedmanns 10 common tetrachordal set classes

(Friedmann, 1990) as presenting all 29 tetrachordal set classes may be too time

consuming for a one semester aural skills course. Table 4.4 is a table of MEMB as

applied to Friedmanns 10 common tetrachordal set classes. The MEMB range has been

converted to match 0-12. The range 0-12 will remain for the remainder of tetrachordal

tables and graphs, where 0 indicates identical set classes and 12 indicates the greatest

dissimilarity.

40

Table 4.4. MEMB Applied to Friedmanns Common Tetrachordal Set Classes

(0123) (0134) (0235) (0135) (0158) (0246) (0257) (0358) (0258) (0369) (0123) 0 1 1 2 6 7 6 6 6 7 (0134) 1 0 2 1 3 5 6 4 4 6 (0235) 1 2 0 1 4 7 1 2 3 6 (0135) 2 1 1 0 2 4 2 1 2 7 (0158) 6 3 4 2 0 8 5 2 3 7 (0246) 7 5 7 4 8 0 7 5 3 9 (0257) 6 6 1 2 5 7 0 1 2 7 (0358) 6 4 2 1 2 5 1 0 1 6 (0258) 6 4 3 2 3 3 2 1 0 3 (0369) 7 6 6 7 7 9 7 6 3 0

Table 4.5 is a comparison of SIM and MEMB applied to (0123) and restricted to

Friedmanns common tetrachordal set classes. This table makes clear the distinctions

between SIM and MEMB. In some instances MEMB offers finer levels of similarity,

while in some cases SIM offers a finer gradation. Both of these relations can be used in

conjunction to sequence chains of tetrachordal set classes.

41

Table 4.5. Morriss SIM Relation and Rahns MEMB Relation Applied to Friedmanns Common Tetrachordal Set Classes

Morris's SIM Rahn's MEMB

(0123) (0123)

(0123) 0 0

(0134) 4.8 1

(0235) 4.8 1

(0135) 4.8 2

(0158) 9.6 6

(0246) 9.6 7

(0257) 7.2 6

(0358) 9.6 6

(0258) 9.6 6

(0369) 12 7

Figure 4.2 is a line graph of the SIM and MEMB relations as applied to set class

4-1 (0123). This graph makes clear the differences between the SIM and the MEMB

relation. While in the SIM relation, (0123) is just as similar to (0235) as it is to (0135), in

the MEMB relation (0123) is more similar to (0235) than it is to (0135). This occurs

again when comparing (0158) and (0246) to (0123). In the SIM relation they are just as

similar to (0123) while in the MEMB relation (0158) is more similar to (0123) than

(0246).

42

Figure 4.2. Line Graph of Morriss SIM and Rahns MEMB 4-1 (0123)

Sequencing Trichordal Set Classes Using the MEMB Relation

One approach to sequencing trichordal set classes is to arrange them in a

manner that moves from similarity to dissimilarity. That is, a starting point would be

chosen, such as (012). The set class following (012) should be one that yields the

greatest similarity, in this instance (013). This sequence has progressed from (012)

(013). The MEMB value of (012) and (013) is 1. The next link in this sequence should

have a MEMB value of 2. As the numerical values increase so does the measure of

dissimilarity. Essentially what would result is a progression of pairs that are similar, then

less similar and so on, eventually forming a chain of trichordal set classes that moves

from most similar to most disparate. Figure 4.3 is a list of several possible realizations of

this approach.

SIM, MEMB (0123)

0123456789

101112

(0123) (0134) (0235) (0135) (0158) (0246) (0257) (0358) (0258) (0369)


Mea

sure

of S

imila

rity

SIMMEMB

43

1 2 2 3 3 3 4 4 4 4 6 /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ (012) - (013) - (014) - (015) - (027) - (024) - (025) - (026) - (037) - (016) - (036) - (048)

1 2 2 2 2 3 3 4 4 4 4 /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ (027) - (025) - (013) - (014) - (048) - (024) - (012) - (015) - (026) - (037) - (016) - (036)

Figure 4.3. Chains of Trichordal Set Classes Derived from the MEMB Relation.6

This approach, however, may be quite difficult to use for the purposes of teaching

harmonic dictation. It would be challenging for the student to make the distinction

between the first part of the chain that presents similar trichordal set classes and the latter

part of the chain which links dissimilar trichordal set classes. Perhaps, rather than

forming a chain that moves from most similar to most disparate, a chain of similar or

dissimilar set classes may be more efficient. Then, an instructor may choose which

method to use.

Chains of Similar Set Classes

A chain of similar set classes can be derived using Rahns MEMB relation by

beginning with set class (012) and progressing to the next most similar set class, (013).7

Once a set class has been introduced it should not be repeatedif any part of the chain is

most similar to a set class that has already been introduced, then the next most similar set

class should follow. In this instance, according to MEMB, (013) is just as similar to

(014) and (025), excluding (012). This then divides the chain of similar set classes in

6(012) is simply an example of a starting point. This method can be applied to sequences starting on any of the twelve trichordal set classes.

44

two: (012) (013) (014) and (012) (013) (025). To continue, one would need to

find the most similar set class to (014), excluding (012) and (013); and the most similar

set class to (025), excluding (012) and (013). Figure 4.4 lists examples of possible

chains of similar set classes that can be derived using Rahns MEMB relation.8

(012) - (013) - (014) - (048) - (026) - (024) - (027) - (025) - (037) - (015) - (016) - (036) (012) - (013) - (014) - (015) - (016) - (027) - (025) - (037) - (048) - (026) - (024) - (036) (012) - (013) - (025) - (027) - (015) - (037) - (048) - (014) - (036) - (016) - (026) - (024)

Figure 4.4. Chains of Similar Trichordal Set Classes Using SIM and MEMB Beginning on (012) While the possibilities are numerous using this approach to sequencing, they are

not nearly as many as when attempting to sequence similar pairs. This approach may be

useful if first asking the student to identify the similar interval classes between two sets,

in an effort to help the student understand how the two sets are related.

Chains of Dissimilar Set Classes

The same idea can be used to form chains of dissimilar set classes. As an

example, let the focus be on Friedmanns common tetrachordal set classes. Set class

(0123) will be the proposed starting point, although the chain may begin on any of the 10

tetrachordal set classes. The next link in the chain would be the set class that is least

similar to (0123). According to MEMB the next set class would be (0246) or (0369).

They both yield a MEMB value of 7 on a scale of 0-12. It is interesting to note, however,

that SIM offers a finer level of similarity than MEMB in this instance. SIM finds (0369)

8 Refer to Table 4-2 for numerical values used to form chains based on Rahns MEMB relation.

45

to be less similar to (0123) than (0246). The chain should then progress (0123) (0369),

always using the finest gradation possible. The next link should be the set class most

disparate to (0369). According to Table 4.4, there are two possible chains of dissimilar

tetrachordal set classes beginning on (0123), restricted to Friedmanns common

tetrachordal set classes. Figure 4.5 lists the two chains of dissimilar tetrachordal set

classes.

(0123) (0369) (0246) (0158) (0257) (0134) (0358) (0235) (0258) (0135) (0123) (0369) (0246) (0158) (0257) (0134) (0258) (0235) (0358) (0135)

Figure 4.5. Chains of Dissimilar Tetrachordal Set Classes Using SIM and MEMB Beginning on (0123)9

There still remain numerous possibilities for chains of similar set classes. Rahns

MEMB relation, however, has helped to narrow down the choices. Linking pairs of

similar and dissimilar set classes has proven to be quite cumbersome. This approach,

when applied to other similarity relations, will identify a limited amount of possibilities

for sequencing set classes using similarity relations. The next chapter will explore how

David Lewins REL relation could be used to further narrow down choices in a sequential

approach to harmonic dictation that is based on similarity.

9 These chains were constructed using the same method as when constructing chains of similar set classesdescribed above. The objective is to have the lowest measure of similarity possible between set classes.

46

CHAPTER V

DAVID LEWINS REL RELATION

David Lewin (1979) has also developed a formula for determining the degree of

similarity between various set classes. This function, RELt, was formulated as a response

to John Rahns response to Robert Morriss SIM relation (1979/80) and David Lewins

REL2 relation (1977).1 This chapter will use RELt to calculate finer gradations of

similarity between set classes in an attempt to systematically sequence trichordal and

tetrachordal set classes for harmonic dictation.

RELt

RELt can be calculated using the formula offered by Lewin (1979): 2 RELt (A,B) = 1 [EMB(/X/,A)EMB(/X/,B)] [TOTAL(A)TOTAL(B)] The sum over all /X/ in test.

Where TOTAL(A) is the sum of all values EMB(/X/, A) as /X/ ranges over the members of TEST; and EMB (X,Y) is the number of distinct forms of X (distinct members of /X/) which are

1REL (relatedness) is the similarity relation developed by David Lewin in 1977. Robert Morriss SIM relation was developed in 1979/80. John Rahn (1979/1980) formulated the MEMB function, an embedding relation that offers finer gradations of similarity than REL and MEMB, as a response to David Lewin 1977 and Robert Morris 1979/80. David Lewin (1979) responded to Rahns response (1979/80) with another version of REL that offers further generalizations and alternatives to Rahns embedding functions and Morriss Similarity Index functions. The distinctions REL2 and RELt were applied to Lewins relations by Eric Issacson in order to distinguish between REL (1977), which calculates embedded subsets of only size 2, and REL (1979), which calculates subsets of all sizes embedded in two set classes. This chapter will focus on RELt, Lewins 1979 similarity relation. 2A Response to a Response: On PCSet Relatedness, David Lewin (1979), pg. 500. The subscript t has been added to this formula according to the labels set forth by Eric Isaacson to distinguish between REL(1977) and REL(1979).

47

subcollections of Y, and /X/ notates the chord-type of X in some group of canonical transformations.3

This formula can be expressed in simpler terms. First identify the common

embedded subsets in both set classes. Then, for each subset common to both set classes,

multiply the number of times the subset is embedded in the first set class by the number

of times the subset is embedded in the second set class and then take the square root of

that product. Next, take the results for each common subset and add them together. Then,

divide the result by the total number of possible subsets embedded in any two set classes.

Figure 5.1 is an example of RELt as applied to set classes 4-1 (0123) and 4-2 (0124).

First calculate the number of common embedded subsets in each set class:

4-1 (0123) 4-2 (0124)

3 instances of (01) 2 instances of (01) 2 instances of (02) 2 instances of (02) 1 instance of (03) 1 instance of (03) 2 instances of (012) 1 instance of (012) 2 instance of (013) 1 instance of (013)

Next, for each subset common to both set classes, multiply the number of times the subset is embedded in the first set class by the number of times the subset is embedded in the second set class, then take the square root of that product:

(01) = (3 x 2) = 6 = 2.45 (02) = (2 x 2) = 2 (03) = (1 x 1) = 1 (012) = (2 x 1) = 2 = 1.41 (013) = (2 x 1) = 2 = 1.41

Next take the sum of all the values above: 6 + 2 + 1 + 2 + 2 = 8.27791686

Next, tally the total number of subsets that can occur in any tetrachordal set class: Figure 5.1. RELt(4-1,4-2).

3Definitions of COMPARE, TEST, and TOTAL(A) are taken directly from Lewin (1979). Definition of EMB(X,Y) is taken from Rahns (1979) description of Lewins (1977) definition.

48

Tetrachordal set classes contain 1 unad, 6 dyads, and 4 trichords. The total number of subsets found in any tetrachordal set class is 11.4

Next take 8.27791686 (sum found above) and divide by the total number of possible subsets found in tetrachordal set classes (11).

RELt(4-1,4-2) = 0.7525379

Figure 5.1 Continued. Trichordal Set Classes The range of RELt is 1 0, 1 indicating the most similar set classes (identical set

classes) and 0 indicating the least similar set classes. Table 5.1 is a table of all trichordal

set classes according to RELt. Just as John Rahns (1979) MEMB values were converted

in Chapter IV, so will Lewins (1979) values be converted so that the range is 0 6, 0

indicating the most similar set classes (identical set classes) and 6 indicating the least

similar set classes.5 This will yield data that is easier to compare to the other similarity

relations discussed in this study. Lewins RELt values were converted by subtracting all

numerical values from 1 and then multiplying by 6 (to adjust the range).

4The numerator in this equation does not account for the unad embedded in both subsets, as all set classes have pc 0 in common. The denominator, which accounts for all possible subsets embedded in any two set classes, does account for the unad.

5The range has been set at 0-6 in order to match the range of Morriss SIM and Rahns MEMB when applied to trichordal set classes.

49

Table 5.1. Trichordal Set Classes According to RELt.6

(012) (013) (014) (015) (016) (024) (012) 0 2.367 3.879 3.879 3.879 3.879 (013) 2.367 0 3 4.5 4.5 3.879 (014) 3.879 3 0 3 4.5 4.5 (015) 3.879 4.5 3 0 3 4.5 (016) 3.879 4.5 4.5 3 0 6 (024) 3.879 3.879 4.5 4.5 6 0 (025) 4.5 3 4.5 4.5 4.5 3.879 (026) 4.5 4.5 4.5 4.5 4.5 2.379 (027) 4.5 4.5 6 3.879 3.879 3.879 (036) 6 3.879 3.879 6 4.5 6 (037) 6 4.5 3 3 4.5 4.5 (048) 6 6 3.402 3.402 6 3.402

(025) (026) (027) (036) (037) (048) (012) 4.5 4.5 4.5 6 6 6 (013) 3 4.5 4.5 3.879 4.5 6 (014) 4.5 4.5 6 3.879 3 3.402 (015) 4.5 4.5 3.879 6 3 3.402 (016) 4.5 4.5 3.879 4.5 4.5 6 (024) 3.879 2.379 3.879 6 4.5 3.402 (025) 0 4.5 2.377 3.879 3 6 (026) 4.5 0 4.5 4.5 4.5 3.402 (027) 2.377 4.5 0 6 3.879 6 (036) 3.879 4.5 6 0 3.879 6 (037) 3 4.5 3.879 3.879 0 3.402 (048) 6 3.402 6 6 3.402 0


50

Sequencing Trichordal Set Classes

Figure 5.2 is a line graph which compares Morriss SIM relation (1979), Rahns

MEMB relation (1979/80) and Lewins RELt Relation (1979) as applied to set class

(012). A similar line graph can be produced for each trichordal set class and can be

found in Appendix B. This will allow a comparison over all three similarity relations and

a method in which to combine data from SIM, MEMB, and RELt.

Figure 5.2. Line Graph Comparing SIM, MEMB, and RELt as Applied to 3-1 (012).7

If one were attempting to sequence trichordal set classes using the similar set

class approach and began at (012), one would need to introduce (013) next in the

sequence according to Figure 5.2. If one were attempting to sequence trichordal set

7 This graph uses the converted MEMB and RELt values. All similar graphs in this chapter will also use these converted values.

SIM, MEMB, RELt (012)

0

1

2

3

4

5

6

7

(012) (013) (014) (015) (016) (024) (025) (026) (027) (036) (037) (048)Trichordal Set Classes

Mea

sure

of S

imila

rity

SIM

MEMB

RELt

51

classes using the dissimilar set class approach and began at (012), one could present

(036), (037), or (048) next in the sequence according to this graph. Now that this graph

has provided the first links of a possible sequencing chain, other links will be added

according to similar graphs. Let the focus be on a chain of similar set classes, keeping in

mind that this same process can be applied to a chain of dissimilar set classes. Consider

Figure 5.3, which is a line graph of (013) according to SIM, MEMB, and RELt.

Figure 5.3. Line Graph Comparing SIM, MEMB, and RELt as Applied to 3-2 (013).

According to this graph the next link in a similar set class chain should be either

(014) or (025). (014) and (025) are more similar to (013) than any of th

thesis by portillo

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