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University M icrofilms International 300 N. ZEEB ROAD, ANN ARBOR, Ml 48106 IB BEDFORD ROW. LONDON WC1R 4EJ, ENGLAND 7922527

HCNEAL, HORACE PITMAN * JR. A METHOD OF ANALYSIS BASED ON CONCEPTS AND PROCEDURES DEVELOPED BY ALLEN FORTE AND APPLIED TO SELECTED CANADIAN STRING QUARTETS, 1 9 5 3 — 1962 .

THE OHIO STATE UNIVERSITY, PH.D., 1979

University Microfilms In ternational 300 N ZEEB ROAD, ANN ARBOR, Ml 48106 A METHOD OF ANALYSIS BASED ON CONCEPTS AND PROCEDURES

DEVELOPED BY ALLEN FORTE AND APPLIED TO

SELECTED CANADIAN STRING QUARTETS, 1953-1962

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Horace Pitman McNeal, Jr., B.M., M.A.

* * * * *

The Ohio State University

1979

Reading Committee Approved By

Burdette L. Green

Norman Phelps

William Poland - Adviser School of Music ACKNOWLEDGMENTS

I wish to express my appreciation to Dr. Burdette Green for his thoughtful attention to this project. His interest and consistent encouragement have contributed immeasurably to the success of this in­ vestigation. The many painstaking hours he has contributed are deeply appreciated.

Invaluable literary evaluation has been received from Dr. Norman

Phelps. His attention to clarity and conciseness has sped the progress of this study. Astute observations relative to semantics have been contributed by Dr. William Poland, and the text communicates more clear­ ly than would have been possible without his probing comments.

Dr. George Pollcello has given his statistical expertise frequently during the course of the project and is thanked in particular for his interest.

Special thanks are due to The Canadian Music Centre for its gracious cooperation in supplying scores and cassettes for study. The friendliness and helpfulness of the people in this agency assisted

Immensely in my research.

I wish to thank my wife, Jennie, who has supported me in many ways, helping me to complete this study. Her kind help has given me the encouragement to stay with and finish this dissertation.

ii VITA

February 28, 1949...... Born - Cedartown, Georgia

1971 ...... B.M., Furman University, Greenville, South Carolina

1972-1974...... Teaching Assistant, Department of Music, The University of Georgia, Athens, Georgla

1974 ...... M.A., The University of Georgia, Athens, Georgia

1974-1976...... Teaching Associate, School of Music, The Ohio State University, Columbu s, Ohio

1976-1977...... University Fellow, The Ohio State University, Columbus, Ohio

FIELDS OF STUDY

Major Field: Music Theory

Studies in Contemporary Music Theory. Professor Burdette L. Green

Studies In History of Music Theory. Professor Norman F. Phelps

Studies in Stylistic Analysis. Professor B. William Poland

ill TABLE OF CONTENTS

Page ACKNOWLEDGMENTS...... i±

VITA ...... iii

LIST OF T A B L E S ...... vi

LIST OF FIGURES...... iac

LIST OF E X A M P L E S ...... xi

INTRODUCTION ...... 1

Chapter

I. THE COMPOSERS AND THEIR QUARTETS...... 16

Canadian Compositional Activities, 1920-1960 .... 16 Jean Coulthard ...... 23 Claude C h a m p a g n e ...... ,...... 29 Harry S o m e r s ...... 35 John W e i n z w e l g ...... 46

II. CLARIFICATION OF THE ANALYTIC M ETHO D...... 55

Relevant Concepts and Terms ...... 55

Pitch Class ...... 55 Interval Class ...... 58 Pitch Class Set ...... 63 Prime F o r m ...... 64 Interval Vector . . ' ...... 72 Similarity Relations...... 77

Supplemental Procedures Employed In This Study . . . 85

Hypothesis Testing ...... 85 Distributions ...... 87 Improved Procedure for Prime Form Identification...... 89 Statistical Comparisons ...... 92

Prime Form Comparison in the Quartets ...... 94

iv TABLE OF CONTENTS (Continued)

Chapter Page III. COMPARISON OF HORIZONTAL PRIME FORM S E T S ...... 98

Four Hypotheses...... 98 Procedures for Linear Segmentation ...... 101 Tabulation of Results...... 107 Interpretation of Results...... 117

IV. COMPARISON OF VERTICAL PRIME FORM SETS AND SUMMARY OF SIMILARITY R E L A T I O N S ...... 132

Four Hypotheses...... 132 Procedures for Vertical Segmentation ••••••.•• 133 Tabulation of Results ...... 138 Interpretation of Results 146 Similarity Relations Between the Two Planes ...... 163

V. EVALUATION AND CONCLUSIONS...... 177

APPENDIXES

A. Prime Forms and Inversions Notated Conventionally and in Integer Notation: 3-Sets and 4-Sets ...... 184

B. 3—Sets and 4-Sets in Integer Notation with Interval Vectors ...... 190

C. All Prime Forms Identified by Their Integer Intervals . • 192

SELECT BIBLIOGRAPHY ...... 203

v LIST OF TABLES

Table Page

1. Numerical Representation of Pitch Classes ...... 57

2. Subtraction of Each Integer Interval From a Constant of Twelve ...... 62

3. Numbers of Prime Forms for Pitch Class S e t s ...... 70

4. Numerical Relationships Between Pitch Class Sets and Interval Vectors ...... 74

5. The Format for Entries in Appendix B ...... 75

6. Basic Descriptions of Similarity Relations ...... 78

7. Rp— Maximum Similarity of TWO Pitch Class S e t s ...... 79

8. The Rp Relation Not Evident in Prime F o r m s...... 79

9. Rp Relation Revealed Through Transposition...... 80

10. Ri““Maximum Similarity of Interval Class Content with Interchange Feature ...... 81

11. R«— Maximum Similarity of Interval Class Content Without Interchange F e a t u r e ...... 82

12. Rq—Minimum Similarity of Interval Class Content...... 83

13. The R^, Rp R e l a t i o n ...... 84

14. The R2, Rp Relation ...... 84

15. The Rg, Rp Relation 85

16. Successive Reorderings and Differences ...... 90

17. Horizontal Plane - Prime Form Counts, 3-Sets ...... 107

18. Horizontal Plane - Prime Form Counts, 4-Sets 108

vi LIST OF TABLES (Continued)

Table Page

19. Horizontal Plane - Prime Form Percentages, 3-Sets...... 109

20. Horizontal Plane - Prime Form Percentages, 4-Sets ..... 110

21. Prime Form Totals: 3-, 4-, 5-, 6 - S e t s ...... 112

22. Frequent Horizontal 3-Set Forms, Coulthard ...... 113

23. Frequent Horizontal 4-Set Forms, Coulthard ...... 119

24. Frequent Horizontal 3-Set Forms, Ch a m p a g n e...... 119

25. Frequent Horizontal 4-Set Forms, Champagne ...... 120

26. Frequent Horizontal 3-Set Forms, Somers 121

27. Frequent Horizontal 4-Set Forms, Somers ...... 122

28. Frequent Horizontal 3-Set Forms, We i n z w e i g ...... 123

29. Frequent Horizontal 4-Set Forms, Weinzwelg ...... 124

30. Horizontal T-Square Data for Musically Significant Differences...... 129

31. Degree of Confirmation for Hypotheses Relative to Horizontal Plane ...... 130

32. Vertical Prime Form Counts, 3-Set F o r m s ...... 138

33. Vertical Prime Form Counts, 4-Set Forms 139

34. Vertical Prime Form Percentages, 3-Set Forms ...... 140

35. Vertical Prime Form Percentages, 4-Set Forms ...... 141

36. Frequent Vertical 3—Set Forms, Coulthard 147

37. Frequent Vertical 4-Set Forms, Coulthard ...... 148

38. Frequent Vertical 3-Set Forms, Champagne ...... 149

39. Frequent Vertical 4-Set Forms, Ch a m p a g n e ...... 151

40. Frequent Vertical 3-Set Forms, Somers ...... 152

vii LIST OF TABLES (Continued)

Table Page

41. Frequent Vertical 4-Set Forms, Somers ...... 153

42. Frequent Vertical 3-Set Forms, We i n z w e i g ...... 154

43. Frequent Vertical 4-Set Forms, We i n z w e i g ...... 155

44. Vertical Y-Square Data for Musically Significant Differences 160

45. Degree of Confirmation for Hypotheses Relative to Vertical Plane ...... 162

46. Degree of Confirmation for Hypothesis Relative to Horizontal and Vertical Planes ...... 176

viii LIST OF FIGURES

Figure Page

1. Inversion-Related Pairs of Integer Intervals ...... 63

2. Successive Reorderings of a 3-Set ...... 66

3. Differences in Orderings for a 3 - S e t ...... 66

4. 3-Set Transposed to Prime Form 67

5. Successive Reorderings of a 4-Set ...... 68

6. Differences in Orderings for a 4 - S e t ...... 68

7. 4-Set Transposed to Prime Form ...... 68

8. Procedure for Determining Interval V e c t o r ...... 73

9. Y—Square Model ...... 93

10. Graph for Coulthard, Horizontal 3-Sets ...... 113

11. Graph for Champagne, Horizontal 3-Sets 113

12. Graph for Somers, Horizontal 3-Sets ...... 114

13. Graph for Weinzweig, Horizontal 3-Sets ...... 114

14. Graph for Coulthard, Horizontal 4—Sets ...... 115

15. Graph for Champagne, Horizontal 4-Sets ...... 115

16. Graph for Somers, Horizontal 4—Sets 116

17. Graph for Weinzweig, Horizontal 4-Sets...... 116

18. Graph for Coulthard, Vertical 3-Set3 ...... 142

19. Graph for Champagne, Vertical 3-Sets 142

20. Graph for Somers, Vertical 3-Sets ...... 143

21. Graph for Weinzweig, Vertical 3-Sets ...... 143

ix LIST OF FIGURES (Continued)

Figure Page

22. Graph for Coulthard, Vertical 4-Sets ...... 144

23. Graph for Champagne, Vertical 4-Sets ...... 144

24. Graph for Somers, Vertical 4-Sets ...... 145

25. Graph for Weinzweig, Vertical 4-Sets .....•••••• 145

26. Comparative Graph for Coulthard, Horizontal 3-Sets .... 167

27. Comparative Graph for Coulthard, Vertical 3-Sets ...... 167

28. Comparative Graph for Coulthard, Horizontal 4-Sets .... 168

29. Comapratlve Graph for Coulthard, Vertical 4-Sets ...... 168

30. Comparative Graph for Champagne, Horizontal 3-Sets .... 169

31. Comparative Graph for Champagne, Vertical 3-Sets . • • • • 169

32. Comparative Graph for Champagne, Horizontal 4-Sets .... 170

33. Comparative Graph for Champagne, Vertical 4-Sets ..... 170

34. Comparative Graph for Somers, Horizontal 3—Sets ...... 171

35. Comparative Graph for Somers, Vertical 3-Sets • 171

36. Comparative Graph for Somers, Horizontal 4-Sets ...... 172

37. Comparative Graph for Somers, Vertical 4-Sets ...... 172

38. Comparative Graph for Weinzweig, Horizontal 3-Sets .... 173

39. Comparative Graph for Weinzweig, Vertical 3-Sets ...... 173

40. Comparative Graph for Weinzweig, Horizontal 4-Sets .... 174

41. Comparative Graph for Weinzweig, Vertical 4-Sets ..... 174 LIST OF EXAMPLES

Example Page

1. Coulthard, String Quartet No, 2. First Movement, mm* 1-3, first violin...... 28

2. Coulthard, String Quartet No. 2. Third Movement, mm. 8-9, first violin...... 28

3. Coulthard, String Quartet Ho. 2, Second Movement, mm. 2-3, first violin...... 29

4. Champagne, Quatuor £ Cordes. First Movement, mm. 1—2, all parts ...... 34

5. Champagne, Quatuor £ Cordes. First Movement, mm. 89—90, all p a r t s ...... 34

6. Champagne, Quatuor 5 Cordes, Second Movement, mm. 124-27, first violin and c e l l o ...... 35

7. Condensed Row from Somers, String Quartet No. 3. mm. 1-9, c e l l o ...... 42

8. Somers, String Quartet No. 3. mm. 92-93, all parts .... 43

9. Somers, String Quartet No. 3. mm. 61-64, all parts .... 44

10. Somers, String Quartet No. 3. mm. 243-46, all parts .... 45

11. Row from Weinzweig, String Quartet No. 3 ...... 52

12. Weinzweig, String Quartet No. 3. Second Movement, mm. 111-17, all par t s...... 53

13. Weinzweig, String Quartet No. 3. Fourth Movement, mm. 14-18, cello ...... 54

14. Instances of Pitch Class c_ ...... 56

15. Triads Notated Conventionally and Expressed in Pitch Class Numbers...... 58

xl LIST OF EXAMPLES (Continued)

Example Page

16. An Interval Notated Conventionally and Expressed in Interval Class Numbers...... 59

17. The Twelve Integer Intervals Notated Conventionally and Expressed in Integer Interval Numbers ...... 61

18. Simultaneous and Successive Pitch Class Sets 64

19. Prime Form and Inversion Notated Conventionally and in Integer Notation ...... 71

20. A Comparison of Z-Related Prime F o r m s ...... 76

21. Phrase Marks as Segmenting Criterion, Coulthard, String Quartet No. 2. First Movement, m. 143, first violin...... 102

22. Articulation as Segmenting Criterion, Champagne, Quatuor & Cordes. Second Movement, mm. 200-01, viola ...... 103

23. Rhythm as Segmenting Criterion, Somers, String Quartet No. 3. m. 17, first violin...... 103

24. Rest as Segmenting Criterion, Weinzweig, String Quartet No. 3. First Movement, mm. 1-2, c e l l o ...... 104

25. Melodic Direction as Segmenting Criterion, Somers, String Quartet No. 3. mm. 449-50, second v i o l i n ...... 104

26. Isolated Pitches, Weinzweig, String Quartet No. 3 . Third Movement, mm. 8-9, first v i o l i n ...... 105

27. Conflicting Segmentation Criteria, Champagne, Quatuor A Cordes. Second Movement, mm. 344-45, second violin ...... 106

28. Horizontal Forms 3-2, 3-3, Coulthard, String Quartet No. 2. First Movement, mm. 6-7, first violin . . . 118

29. Horizontal Forms 4-3, 4-1, Coulthard, String Quartet No. 2 . First Movement, mm. 73-75, first violin...... 118

xii LIST OF EXAMPLES (Continued)

Example Page

30. Horizontal Form 3-2, Champagne, Quatuor 5 Cordes. Second Movement, mm. 255—56, second violin ...... 120

31. Horizontal Forms 4-11, 4-3, Champagne, Quatuor £ Cordes. First Movement, mm. 103-05, cello ...... 121

32. Horizontal Forms 3—2, 3-1, Somers, String Quartet No. 3. m. Ill, viola 122

33, Horizontal Forms 4—1, 4—2. 4—3, Somers, String Quartet No. 3, mm. 407-09, second violin...... 123

34. Horizontal Forms 3—1, 3-2, Weinzweig, String Quartet No. 3, Second Movement, mm. 114-15, first violin...... 124

35. Horizontal Forms 4-2, 4-1, Weinzweig, String Quartet No. 3. First Movement, mm. 73-74, cello ..... 125

36. Beat Defined for Set Selection, Coulthard, String Quartet No. 2. First Movement, mm. 9-10, all parts .... 134

37. Notes Articulated Prior to Beat as Set, Somers, String Quartet No. 3. m. 23, all p a r t s ...... 135

38. Set Selection in Sparse Texture, Weinzweig, String Quartet No. 3. First'Movement, mm. 20-21, all p a r t s ...... 136

39. Tremolos in Set Selection. Coulthard, String Quartet No. 2. First Movement, mm. 86-87, all parts . . . 137

40. Vertical Form 3-11, Coulthard, String Quartet No. 2 . First Movement, mm. 140-41, all parts ...... 147

41. Vertical Form 3-11, Coulthard, String Quartet No. 2 . Second Movement, mm. 15-16, all parts .....••••• 147

42. Vertical Form 4-27,'Coulthard, String Quartet No. 2 . First Movement, mm. 38-39, all parts ...... 148

43. Vertical Form 4-19, Coulthard, String Quartet No. 2, Third Movement, m. 95, all p a r t s...... 149

xiii LIST OF EXAMPLES (Continued)

Example Page

44. Vertical Forms 3-7, 3—3, 3—5, Champagne, Quatuor & Cordes, First Movement, mm. 20-21, all parts ...... 150

A5. Vertical Forms 3-3, 3-4, Champagne, Quatuor 3, Cordes, Second Movement, mm. 262-63, all parts ...... 130

46. Vertical Form 4-19, Champagne, Quatuor h Cordes. Second Movement, mm. 216-17, all parts...... 131

47. Vertical Forms 3-7, 3-4, Somers, String Quartet No. 3, m. 134, all p a r t s ...... 132

48. Vertical Form 4-15, Somers, String Quartet No. 3. m. 53, all parts ...... 153

49. Vertical Form 3-3, Weinzweig, String Quartet No. 3. Fourth Movement, mm. 31-32, all parts ...... 154

50. Vertical Form 4-2, Weinzweig, String Quartet No. 3. First Movement, mm. 14-15, all parts ...... 155

51. Vertical Form 4-6, Weinzweig, String Quartet No. 3. Second Movement, mm. 152—53, all parts • ...... 156

xiv INTRODUCTION

The analytic method used in this study was developed and tested

as a means for explaining contemporary music In more exact and relevant

terms than those employed for explaining traditional music. Contempo­ rary music contains complex melodic and harmonic structures that have no

satisfactory designations in traditional analytic terminology. Devel­ opments in free atonality, serialism, neoclasslc tonality, indetermi­ nacy, and blends of tonal and atonal elements (called here tonal/non- tonal music) have pushed far ahead of the terminology needed to deal with these techniques or combinations of techniques.

Allen Forte has shown that pitch class sets may be related to each other on the bases of either pitch class content or interval class content. In The Structure of Atonal Music^ he presented a set theory model that elucidates Interval class relationships in music written be­ tween 1908 and 1920 by Stravinsky, Schoenberg, Berg, and Webern. The

Interrelations of pitch class sets on the basis of interval class con­ tent is further pursued in this study.

While Forte developed his system to explain the atonal music written in the early part of the twentieth century, this investigation applies his system to works written in the years shortly after mid­ century. From this period four quartets by Canadian composers were

^■Allen Forte, The Structure of Atonal Music (New Haven: Yale University Press, 1973).

1 selected for an unordered set analysis. These works include Jean

Coulthard's String Quartet No. 2 (Threnody. 1953), Claude Champagne's

Quatuor h Corde3 (1956), Harry Somers' String Quartet No. 3 (1959), and

John Weinzweig's String Quartet No. 3 (1962).

The primary purpose of thi3 study is to test the validity of unordered set analysis through its practical application to these quar­ tets. I hypothesized that testing this method upon stylistically di­ verse contemporary quartets would yield relevant information in terms appropriate for stylistic comparisons— that is, that relevant stylistic comparisons might be made on the basis of the results obtained. A rational basis for formulating these stylistic observations provides objectivity in the method, as the study will show.

Certain delimiting factors were necessary to give a proper focus to the investigation. String quartets were selected for study because their relative homogeneity is ideally suited to the methodology.

The texture of quartets adapts well to comparisons of pitch class sets on the basis of interval class structure, because linear writing is usually emphasized in all parts. Hence, there are many horizontally conceived sets that may be described and related to one another. In the vertical plane sets of three and four pitch classes occur ordinar­ ily, permitting a uniform basis for relating simultaneities to each other.

Canadian string quartets proved to be well-suited to the study because they are plentiful, eclectic in style, and excellent in qual­ ity. Through the auspices of the Canadian Music Centre, fifty quartets were readily available, from which it was possible to select a represen­ tative sample of works written in a tonal/non-tonal idiom.

Limiting the chronological span during which the quartets were composed was necessary in order to achieve reasonable depth of examina­ tion of the works. The period 1953-1962 permitted the selection of four string quartets exhibiting a variety of compositional styles. One of the four selected (Coulthard*s) exhibits both tonal and non-tonal characteristics, while the remaining three are non-tonal. Of the three non-tonal works, two are serially based (Somers' and Weinzweig*s) and the third is non-tonal but not serial (Champagne's).

There are limitations to analytical techniques, such as Forte's, and also to the supplemental procedures used in this study. The prin­ cipal limitation is that real pitches and real Intervals are displaced by abstractions— pitch classes and Interval classes. These abstractions are further removed from tangible elements in that the pitch classes are both transposed and Inverted to obtain basic forms (called prime forms) that begin on c as a reference pitch class.

The relation of these abstractions to human perception is an

Important issue here— one that is considered but not pursued in this study. The* study of human perception is an entirely different area for investigation. The issue may be stated as follows: do we hear only the actual pitches and Intervals (and their combinations) in the spe­ cific context In which they are presented, or do we hear the broader, abstract patterns and relate them by means of memory over a span of time? 4

If we can perceive, perhaps unconsciously, abstract patterns, or basic sound types, then the results of this study are a step toward de­ fining explicitly that which we hear implicitly. The limitation in question is not a matter of negating traditional perceptual concepts, but is instead a matter of viewing musical sounds from a different per­ spective. Inasmuch as contemporary theory has not evolved a system to

satisfactorily account for exploitation of all pitch class set struc­ tures, the establishment of alternate perspectives is necessary. This work is but one in a series of steps towards an explanation of newer music that older theoretical systems cannot explain.

For the sake of this study, the assumption that we can hear ab­ stract patterns related by memory is accepted, and the existence and importance of basic sound types (prime forms) Is asserted. This in­

vestigation consists of the quantification of those sound types and an examination of the relationships between the sound types. Indeed, it may be that an understanding of the principles that underlie the basic

sound types of tonal/non-tonal music will lead us to explain real pitches and Intervals with greater effectiveness. We have not as yet reached this point. It is first necessary to search for whatever

basic principles or patterns may exist before attempting to explain the details of specific pitches and Intervals.

A final limitation of this approach is that time sequence is not considered. A large number of sets occurring over a period of time is decoded, or reduced, into prime forms and summarized, but the actual sequence is ignored. The previously discussed idea of demarcating patterns first also applies here. Time sequence is important in 5

relation to actual pitches and intervals, but the measurement of basic

sounds, or prime forms, is paramount to this study. The question of

real time is similar to the question of real pitch and is therefore

secondary.

Having defined the boundaries of the study, it will clarify matters to review the major texts associated with the area of inquiry.

As indicated earlier, Forte's The Structure of Atonal Music is the

primary source for the methodological concepts in this investigation.

Forte presents a uniform system for designating and describing pitch class sets as prime forms related by Interval class content. As the title of the text indicates, his area of investigation is atonal music. However, this title is not as inclusive as one might assume.

Forte does approach a significant area of atonal music, but the area of concentration i3 relatively small. His field of focus is

in fact limited to notable atonal works written between 1908 and 1920.

His reasons for this focus are made reasonably clear by the following statements:

In 1908 a profound change in music was initiated when Arnold Schoenberg began composing his "George Lieder" Op. 15. In this work he deliberately relinquished the traditional sys­ tem of tonality, which had been the basis of musical syntax for the previous two hundred and fifty years. Subsequently, Schoenberg, Anton Webern, Alban Berg, and a number of other composers created the large repertory known as atonal music.

The above quote 3tates by implication that Schoenberg, Webern, and Berg are the primary composers to be considered in an indefinite time period beginning with the year 1908. He later supplies a

■^Forte, The Structure of Atonal Music, p. ix. 6 justification and, by implication, an end point for his area of concen­

tration :

Although a good deal of attention has been paid to the icono­ clastic nature of atonal music, there has been a tendency to overlook its significance within the art form. This circum­ stance is unfortunate and should be corrected. One need only remark that among the major works in this repertory are Schoenberg's Five Pieces for Orchestra Op. 16 (1909), Webern's Six Pieces for Large Orchestra Op. 6 (1910), Stravinsky's The Rite of Spring (1913), and Berg's Wozzeck (1920).3

The inclusion of Stravinsky does broaden the scope consider­ ably, and this widening of the area of concern is acknowledged by

Forte:

The Inclusion of Stravinsky's name in the list above suggests that atonal music was not the exclusive province of Schoenberg and his circle, and that is indeed the case. Many other gifted composers contributed to the repertory: Alexander Scriabin, Charles Ives, Carl Buggies, Ferruccio Busoni, and Karol Szymanowski— to cite only the more familiar names.^

Nevertheless, Forte is primarily concerned with the inception of atonality and its development prior to the advent of serlalism in the 1920*s. Because this transitional period has not been studied and understood with the same degree of thoroughness as have other periods, both before and after, Forte's attempt to shed light upon early atonal music is a meritorious endeavor. Although he does admittedly confine himself to a narrow focus, he extends the applicability of the term atonal to music written after 1920. However, his conditions for apply­ ing the term to a work are quite restrictive:

The present study draws upon the music of many of the com­ posers mentioned above. It does not, however, deal with 12-tone music, or with what might be described as paratonal

3Forte, The Structure of Atonal Music, p. ix.

4Ibid. music, or with more recent music which is rooted in the atonal tradition. This is not to say that the range of applicability is narrow, however. Any composition that exhibits the structural chracteristics that are discussed, and that exhibits them throughout, may be regarded as atonal.^

The requirement that a piece exhibit the characteristics found in early atonal works and that it exhibit them throughout is possibly overly restrictive. Atonallty has been achieved through serlalism (although a serial piece is not necessarily atonal) and there has also been music composed since 1920 that is atonal but not serial.

Although Forte’s approach to the term atonal is perhaps limited, the limitation is one that is offset by the positive value of his con­ tribution. He has formulated an analytic model that uniformly defines pitch class sets and their interval class structure. Previous analytic conventions such as chord symbols or serial row forms have required that a pitch belong to a recognizable chord or preconceived row form before that pitch could be properly labeled and understood. Forte’s model places no such restrictions upon a pitch and only stipulates that a given pitch be considered In relation to other pitches in the context of a pitch class set. The initial step that Forte has taken, then, is to describe groups of pitches equally and systematically, within the context of early atonal music.

Two other authors who, like Forte, have attempted to explain pitch class sets in twentieth-century music not explicable by tradition­ al means are Howard Hanson and George Perle. Hanson’s Harmonic

^Forte, The Structure of Atonal Music, p. lx. Materials of Modern Music** focuses on "resources of the tempered scale"

(its subtitle), and Perle's Serial Composition and Atonality^ centers

on set usage of Schoenberg, Berg, and Webern (subtitled: "An Intro­

duction to the Music of Schoenberg, Berg, and Webern"). The works of

both authors deal with basic pitch class sets (similar to Forte's prime

forms) available in twentieth-century music, although their approaches

are quite different from Forte's.

Hanson's Harmonic Materials of Modern Music is essentially a

dictionary of pitch class sets designed to assist a composer in his

awareness of all sets. In contrast to Forte, Hanson explains the deri­ vation of many sets by processes such as intervallic projection (build­

ing by Interval), involution (Inversion), and complementary scales (the

remaining sets when certain sets are. subtracted from the twelve pitch

classes). An extensive chart showing the structures of a large number

of pitch class sets accompanies the text and provides in summary form

the basic materials that the text presents.

Although the sets classified by Hanson are like Forte's sets,

a codification of available distinct types, an essential difference in

perspective is found between the two authors. Forte presents a concise

and tightknit logical system of pitch class relations that deals with

elements in his chosen body of music. Hanson, in contrast, presents a

^Howard Hanson, Harmonic Materials of Modem Music: Resources of the Tempered Scale (New York: Appleton-Century-Crofts, Inc., 1960).

^George Perle, Serial Composition and Atonallty: An Introduc­ tion to the Music of Schoenberg. Berg, and Webern, Fourth Edition, revised (Berkeley: University of California Press, 1977). 9

diverse Inventory of pitch class sets. Rls preface clearly states his

purpose:

X hope that this volume may serve the composer in much the same way that a dictionary or thesaurus serves the author. It Is not possible to bring to the definition of musical sound the same exactness which one may expect In the defi­ nition of a word. It is possible to explain the derivation of a sonority, to analyze its component parts, and describe its position in the tonal cosmos. In this way the young, composer may be made more aware of the whole tonal vocabu­ lary; he may be made more sensitive to the subtleties of tone fusion; more conscious of the tonal alchemy by which a master may, with the addition of one note, transform and illuminate an entire passage. At the same time, it should give to the young composer a greater confidence, a surer grasp of his material and a valid means of self-criticism of the logic and consistency of his expression.®

Because of the pedagogic intent of the work, much explanation

of the structure and the sound of pitch class sets is presented. State­ ments regarding the "soun^,f of pitch class sets tend to be more subjec­

tive than logically clear (Forte’s goal) but may aid in the explanation

of perception (Hanson’s goal). One could say that The Structure of

Atonal Music is written primarily for the music theorist, while

Harmonic Materials of M o d e m Music is intended mainly for the composer.

Forte presents pitch class set possibilities in mathematical terms un­

clouded by perception, and Hanson presents the same possibilities

through musical characterizations.

The musical characterization is also emphasized in Perle’s

Serial Composition and Atonality, but to a lesser degree than in the

Hanson text. Perle's book "ties together the compositional method and

the listener's experience of hearing what has been composed."^ In

O Hanson, Harmonic Materials of M o d e m Music, p. viii. q Perle, Serial Composition and Atonality, dust jacket. 10 terms of logical analytic outlook and the listener's perception, his text falls between the two perspectives offered by Forte and Hanson.

The emphasis in the greater part of his text is on specific sets as they occur in the music. However, basic sets are also presented in symbolic notation in accompanying tables. Perle's tables are struc­ tured more according to logic than to perception and, in this respect, are closely related to Forte's tables. Perle's area of concern, both the non-serial and serial music of Schoenberg, Berg, and Webern from roughly 1908 to 1950, is broader than Forte's. The use of serialism as a means of solving problems raised by composing in a non-tonal style is placed in perspective by Perle:

In 1923 Schoenberg first published a composition employing the "method of composing with twelve notes." This "method" soon proved to have some general relevance to the special problems of atonal composition. It is consistent with both the positive and negative premises of atonality, affirming the availability of twelve notes while denying a priori functional precedence to any one of them. ®

Although both free atonal works and serial works are used for examples of sets, he places the emphasis on specific sets of three, four, five, and six pitches, regardless of the classification of the composition as atonal, twelve—tone, or serial. He focuses on the explanation of the compositional manipulation of selected sets in specific musical pas­ sages, so that one may reasonably expect to perceive In sound that which is explained. Unlike either Forte or Hanson, he frequently men­ tions temporal features in relation to the sets.

A rationale for selecting Forte's approach as the basis for this study concludes the review of the literature. His system has not

lOperle, Serial Composition and Atonality, p. 1. 11 been chosen because his text is better than the others— all three works are Important. His logical and systematic approach is simply the one most appropriate to the task at hand. Suitability of purpose and in­ herent flexibility have dictated its selection for use in this disser­ tation.

Any discussion of atonal music calls for an agreement on a definition of tonality, because atonality is generally taken to mean the negation of tonality, or the state of being "not tonal." To say that a musical work, or a portion of that work, is "not tonal" requires a clear concept of "tonal."

A piece may be said to be tonal if any single pitch class is emphasized enough to become predominant as a center. By extension, more than one pitch class may receive significant emphasis in relation to the remaining pitch classes so that pitch classes which are emphasized may be said to constitute tonal centers. Critical judgment of the evidence is essential for determining whether tonality exists in a piece accord­ ing to either the initial definition or the extended definition above.

For example, if an analyst views a piece such as Stravinsky's

Threni, the decision as to whether the term tonality applies to this work is a matter of perspective and judgment. In the sense that the work is serial and that dissonance is integral to it, Threni may be said to be atonal. However, a careful analysis of the initial and con­ cluding pitches of the row forms at structural points in the piece will show that there is indeed an emphasis upon the following pitch classes, 12 arranged In two pairs: dff-fff and a-c. ^ These pairs represent a polarity of black keys versus white keys (on a keyboard) and establish complementary tonal areas, that is, d#-f# and a-c. A tritone relation­ ship exists between the two pairs: d£ to a^ is one tritone, and f t o c^ is another. In the sense of formal arrangement of row forms, then,

Stravinsky's Threni may be said to be tonal, given a liberal definition of the term.

Tonality established through selection of key structural pitch classes is the method employed by Stravinsky in Threni. Other means of establishing tonality include the standard harmonic progressions of the common practice era and, more recently, the borrowing of certain inter- vallic relationships placed within a context of contrapuntal texture.

An excellent example of the latter can be found in the first movement of Bartok's Music for Strings. Percussion, and Celeste. The tonal center of a_ is established by means of an orderly progression of pitch classes In perfect fifths above and below £i, culminating in a beauti­ fully climactic passage emphasizing e^, the meeting point of the two streams of fifths originating from the central pitch a.

In relation to tonality, then, the term atonal may be applied to a work if that work does not establish "loyalty” to one or more pitch class centers or areas. The limits of our traditional analytic approaches do not absolutely preclude the presence of tonality. We can only say that our analytic methods have not proven the presence of tonality.

^Horace McNeal, "An Analytical Investigation of Igor Stravinsky's Threni," M.A. thesis, University of Georgia, 1974. 13

In the tonal/non-tonal works selected for this study, the set of

terms and techniques chosen to accomplish the measurement of intervallic properties of pitch class sets is derived from Forte's text* nis ter­ minology, which is explained in Chapter II (Clarificatien of the Anal­ ytic Method), is adopted here for three reasons. First, his terminology

is complete with respect to labeling of pitch class sets. Given at least three pitch classes and excluding sets of ten or more pitch classes, there are none that cannot be expressed using Forte's terms.

Although Perle provides a system for labeling sets with up to twelve pitch classes, such mathematical thoroughness is not necessary in this study. Sets of less than three pitch classes are excluded here because their Intervallic content is Insufficient for making comparisons with more complex sets. And, sets of five or more pitch classes are gener­ ally excluded due to the prevailing use of three and four parts in the simultaneous structures of the string quartets.

Second, Forte's terms are exact. Every set of pitch classes to be examined may be accurately described by one and only one term in

Forte's language. Thus, the measurements permit no "shades" or judg­ mental decisions, although judgment does enter into the interpretation of the resulting data. The exactness of measurement Insures a rigorous basis for subsequent interpretative observations.

The relevance of his techniques is a third reason for using only Forte's terminology. This is not meant to imply that his terms are the only suitable ones to describe his techniques. They are, how­ ever, concise and workable. 14

The present Investigation expands Forte’s techniques in order

to apply them to music written much later than the period upon which he

focused. The thorough application of his techniques to four large works

required some refinements. For example, a much needed refinement of the procedure for determining the appropriate name of any given pitch class set resulted in the rapid identification of prime forms. Also, the entire Investigation was placed within the context of a scientific

sequence (hypothesls-testing-results—Interpretation), which was employed

to give empirical control to the investigation.

The emphasis on examining large groups of prime forms led to the employment of statistical measures. A means for controlled interpreta­ tion of various differences In thp content of prime forms among the quartets was essential, and statistical comparisons provided this measure of control. Because of the large number of comparisons, the computer proved to be a useful tool for the statistical calculations.

In this respect the use of the computer In this study differs from

Forte’s approach. Whereas Forte used the computer to establish possi­ bilities In a logically complete system, I have employed it to test differences among quartets in the frequency of certain prime forms.

The content and order of the following chapters delineate the testing of Forte’s concepts and provide background information on the string quartets. Chapter I presents a description of Canadian musical activity from 1920 to 1960 and gives biographical Information on the four composers whose quartets have been selected. Chapter II provides a detailed explanation of the concepts and procedures employed in this study. Many of Forte’s ideas are clarified in a condensed review. Chapter III presents a study of the melodically generated pitch class sets occurring in the four quartets. In parallel fashion Chapter IV provides data for the vertical, or simultaneous sets and concludes with a comparison of the data for the two planes. Chapter V evaluates the research, draws conclusions, and points the way toward further research. CHAPTER I

THE COMPOSERS AND THEIR QUARTETS

Canadian Compositional Activities, 1920-1960

To place In proper perspective the composers and their selected

string quartets, it is necessary to consider the state of Canadian music from 1920 to 1960. As one might expect in a country so geneolog-

ically indebted to England and France, the influences of those two countries dominated musical traditions in Canada through the eighteenth, nineteenth, and early twentieth centuries. By the 1920*s French-

Canadian composers had achieved eminence in Canada. French impression­

ism was a dominant force in their styles. In the 1930fs Canadian com­ posers, trained at home and in America, became prominent. Helmut

Kallmann provides specific names relating to these two decades:

Among the first of our native composers to express themselves in a contemporary idiom were a number of French-Canadians (bom between 1880 and 1900) who were often strongly influ­ enced by French impressionism: CHAMPAGNE, J. J. GAGNIER, GRATT0N, LALIBERTE, LAPIERRE, R. MATHIEU, MORIN, ROY, TANGUAY and others.

A generation with even more modernistic tendencies matured in the 1930's. Unlike that of most of the earlier composers, their background is usually secular and their training chiefly American or Canadian. Among those whose work is most often heard are ARCHER, BLACKBURN, BROTT, COULTHAED,

16 17

DELA, GEORGE, PAPINEAU-COUTURE, PENTLAND, RIDOUT, VALLERAND and WEINZWEIG.1

This latter group of composers secured a professional status

for the Canadian composer, with the help of the Canadian Broadcasting

Corporation:

But It was a younger generation whose determination and talent made a place in Canada for the professional com­ poser. . . .Aided especially by the Canadian Broadcasting Corporation (established 1936), these composers employ various media of composition. Those b o m In the 1920* s and later became the first to receive training In contem­ porary techniques. Harry Somers (b. 1925) and Pierre Mercure (b. 1927) are typical of this generation.^

International trends entered with Increasing force by 1930.

’*Murray Adaskin (b. 1906) relates to Milhaud in his early music, Jean

Papineau-Couture (b. 1916) to Stravinsky, Barbara Pentland (b. 1912) to 3 Hindemith and Copland." Two of the composers whose works are studied here, John Welnzwelg and his pupil Harry Somers, were Influenced by

international trends:

. . . the violin and cello sonatas of John Welnzwelg (b. 1913) draw on the Jewish expressive vein that was also explored by Bloch. . . .Welnzwelg was the first Canadian to adopt the 12-tone method (in a short piano piece of 1939, composed, according to the composer, in response to a first hearing of Berg*s Lyric Suite). Welnzwelg*s serial structures make use of a slow ex­ posure technique by which, for example, notes 1—3 are set In motion for a few pars, then note 4, then 5

^■Helmut Kallmann, ed., Catalogue of Canadian Composers, reprint of revised and enlarged ed. (St. Clair Shores, Michigan: Scholarly Press, 1972), p. 18. 2 Helmut Kallmann, "Canada," Harvard Dictionary of Music. 2nd ed., p. 123. 3 John Beckwith, "Canada," Dictionary of Contemporary Music, ed. by John Vinton (New York: Dutton, 1974), p. 122. 18

and 6, etc., until the whole repretory is established; the economic but kinetic character of his rhythmic style has been described by one critic as "pokes in the mid­ riff." Similar features can be seen also in the music of such pupils of his as Harry Somers (b. 1925) and Harry Freedman (b. 1922).4

In both quantity and quality, Canadian musical composition be­ came significant in its own right in the 1940's. Further, public rec­ ognition of Canadian composers increased:

In the last decade [1940's] Canada has become more self- sufficient in all musical matters but a great deal of pioneering remains to be done. It may, however, be said that later than painting and literature, musical composi­ tion in Canada has finally come of age. . . .But what is more encouraging than the mere quantity of our composers is the fact that so much of their music is good. At last the Canadian public is becoming aware of the existence of the Canadian composer and audiences in other American and in European countries are demanding to hear our music.

In 1952 the Canadian League of Composers was created. Welnzwelg and Somers were Instrumental in its formation:

In 1952 such composers as John Welnzwelg, Jean Papineau— Couture, John Beckwith, Murray Adaskin, Violet Archer, Barbara Pentland, Pierre Mercure, Harry Freedman, Clermont Pepin, and Harry Somers formed the Canadian League of Com­ posers to press for more recognition; a decade later, the League numbered some 60 members, accepted only on the basis of established professional competence and serious intent.®

^Beckwith, "Canada," p. 122.

^Kallmann, Catalogue of Canadian Composers, p. 19.

^Keith MacMillan, "Canada," The Encyclopedia Americana (International Edition; New York: Americana Corporation, 1971), II, p. 437. Despite the solidifying effect of the Canadian League of Com­ posers, there was a scarcity of support for Canadian compositional activity:

Still, the situation for our composers is not ideal. Radio and film as well as certain other functional types of music provide an outlet for a number of them . . . but commissions and publications are not very frequent and few of our com­ posers are able to devote sufficient time to composition. Performances and publications of Canadian works need to be multiplied and public curiosity stimulated.^

The need for support of Canada's native composers was met by the establishment of the Canada Council In 1957. Considerable finan­ cial support was given to music and to the other fine arts by the

Canada Council, resulting in an Increasingly prominent Canadian cul­ tural life. The commissioning activities of the Canada Council are described by Beckwith:

The Canada Council has given scholarships to young musi­ cians and leave—fellowships and short-term work grants to established older ones. It has also given grants to performing organizations, sometimes for the specific purpose of commissioning and/or performing new works.

The Canadian Music Centre was established In 1959 by the

Canada Council. This agency serves as a library of Canadian music and provides scores for study, aiding In the dissemination of Canadian music:

The Canadian Music Centre was established to promote Canadian music and to provide a reference and loan col­ lection of scores and recordings; by 1970 the collection included some 4,000 works by about 130 c o m p o s e r s . 9

7 'Kallmann, Catalogue of Canadian Composers, p. 19.

8Beckwith, "Canada," p. 120.

9Ibld. 20

Central to the national exposure of Canadian music Is the

Canadian Broadcasting Corporation. This agency has supported composers

mainly by giving national broadcasts of their works. The practical

effect of the Canadian Broadcasting Corporation's support is described

below:

. . . Canadian works of merit seldom go long without at least one performance. Furthermore composers have access to a nationwide audience, not just a local one, and they can tape performances of their works for private study and use. Since 1960 the CBC has been commissioning music at the rate of six to 20 or more works per season. In­ cluded are folk music arrangements, chamber works, orch­ estral pieces, musicals, and operas.

The experimentation by Canadian composers with international

trends has to some degree overshadowed any Canadian identity or school

of composition. Although there does not seem to be any distinct

Canadian style in the twentieth century, Canada's position with regard

to both cultural influences and geographical position may be said to be unique. The influence of both British and French culture has shaped

Canada, but due to its geographical proximity to the United States,

that country too may be said to exert a third shaping effect. The in­

fluence of the United States is mentioned by MacMillan, although he points out that its role has not prevented independent Canadian musical growth:

The development of music in Canada has clearly been shaped by historical and geographical circumstances, principally its size and its proximity to the overpowering United States. Except for the Quebec "chansonniers," virtually all popular music is foreign; the serious budding artist

^■^Beckwith, "Canada," p. 121 21

must make "the debut" In New York. However, such Inter­ national stars as Glenn Gould, pianist, Lois Marshall, soprano, Maureen Forrester, contralto, Louis Qulllco, baritone, Jon Vickers, tenor, Theresa Stratas, soprano, and Mario Bemardl and Wilfrid Pelletier, opera conductors (all trained in Canada), help to belie the myth of non­ musical Canada.

A strong desire for a Canadian identity is examined by Beckwith as follows:

A leitmotiv in 20th-century Canadian culture is the long­ ing for separate artistic identity (or identities)• This is no belated quest for nationalism. Rather it corresponds to the impulse that drives Japanese composers of today to reexamine their dying classical traditions and their In­ volvement with post-Weberaism, electronics, etc. Canadians, habitually sceptical about their own artistic achievements, have been slow to define what music threads might lend a local touch to their handling of global movements. On the one hand, of course, such Indigenous accents are implied in the character of the best works by the country’s best composers. On the other, a renewed Interest In ethnomusic- ology may be on the brink of revealing hitherto little un­ derstood facets of Canadian musical culture.^

A gradual shift away from outside domination and toward a

Canadian consciousness (although not a Canadian school or style) has occurred In this century. Canadian folklore has become at least a competing force with external Influences, a point made by Kallmann:

Up to the 1920's most of our composers were trained abroad; and they were composers who happened to live in Canada, rather than "Canadian composers" by style or conviction. Partly through the revival of our folklore, partly through the growing national consciousness, composers began to strive for a specifically Canadian content in composition, or at least to ponder the possibility of doing so. Musical nationalism has not up to now developed Into a great force, and our composers continue to be influenced by a variety of international currents. It is therefore hardly possible to observe distinct schools although there certainly are some

^MacMillan, "Canada," p. 437

12Beckwith, "Canada," p. 124. 22

composers whose works bear unique characteristics of style and expression sometimes distinctly Canadian, and often drawing inspiration from Indian, Eskimo and French- Canadian themes, or from our l a n d s c a p e . ^3

The variety and experimentation in contemporary Canadian music have produced a large body of string quartets that are available for study on a loan basis from the Canadian Music Centre in Toronto. I examined fifty quartets and found many that were worthy of consider­ ation. Those that were chosen for this study represent distinct gradu­ ated differences in nontraditional writing. In my judgment the

Coulthard work is similar In texture and rhythm to late nineteenth century works. Mildly dissonant in pitch content, it is fairly con­ servative for a modem string quartet. The Champagne quartet shows greater experimentation with pitch content and rhythm, although the texture is somewhat traditional. The Somers quartet shows striking use of dissonant pitch combinations with much rhythmic experimentation.

The Welnzwelg quartet shows the most extreme use of "pointillistic" textures. It includes much dissonance and several passages without perceptible rhythmic beat. It Is the most modem of the four works.

These skillfully written compositions are examples of various trends in

Canadian writing. The biographical Information following should assist the reader to place the selected works in perspective.

The sources used for the information on the composers are

Thirty—Four Biographies of Canadian Composers. ^ Contemporary Canadian

^Kallmann, Catalogue of Canadian Composers, p. 18.

-^Canadian Broadcasting Corporation, Thirty-Four Biographies of Canadian Composers (Canada: International Service of the Canadian Broadcasting Corporation, 1964). 23 15 16 Composers. and the biography Harry Somers. Brief remarks about the

quartets conclude the biographical sketches of each composer.

Jean Coulthard

Jean Coulthard, a Canadian of Scottish descent, has had a long

and distinguished career in Canadian musical circles. Born In

Vancouver on February 10, 1908, she received her earliest training from

her mother, a singer, pianist, and organist who "is said to have intro­

duced the music of Debussy to Western Canadian audiences."^ Also en­

couraged to study music by her father, a physician, Coulthard started writing songs and short piano pieces when she was nine years old. Her later teachers were Frederick Chubb (theory) and Jan Chemiavsky

(piano). Early successes as a pianist and composer followed for

Coulthard:

Soon she started to make a name for herself as a pianist, appearing in many local concerts and also, while still in her teens, as a composer, winning the prize for composi­ tion at the first Competitive Music Festival in British Columbia.18

Her compositions enabled her to win a scholarship In 1928 from the Vancouver Women’s Musical Club for training at the Royal College of Music in London. There she studied piano with Kathleen Long and composition with Ralph Vaughan Williams, both of whom she feels, had a

^Keith MacMillan and John Beckwith, eds., Contemporary Canadian Composers (Toronto: Oxford University Press, 1975.

18Brian Cheraey, Harry Somers (Toronto: University of Toronto Press, 1975).

^Thirty-Four Biographies, p. 28.

18Ibid. 24

"decisive influence on her life, spiritually as well as musically."^

Returning to Vancouver in 1930 she continued her work in piano and com­ position, studying with Arthur Benjamin, an Australian composer living in Vancouver at that time.

Other figures from whom she obtained critical guidance were

Aaron Copland (1939), Arnold Schoenberg (1941), Darius Milhaud (1942),

Bela Bartok (1944), Bernard Wagenaar (1945), Nadia Boulanger (1955), and Gordon Jacob (1965-66). The values that Coulthard espoused are evident in her following statement;

In this great age of scientific development, I feel that human values remain the same, and that unless music is able to reach the heart in some way, it loses its com­ pelling power to minister to human welfare. I also think that a composer's musical language should be instructive, personal and natural to him, and not forced in any way as to specific style or technique of the moment. For if one becomes over-involved in the mechanics of one's thoughts, inspiration is easily lost.2®

Summing up her experience as a composer, Coulthard has said:

I have written many kinds of musical compositions, from quite simple forms and combinations of instruments, to large forms of full orchestra, and in it all, my aim is simply to write music that is good.2^

She began her collegiate teaching career as the head of the 22 music departments at St. Anthony's College and Queen's Hall School.

She became lecturer in music at the University of British Columbia in

1947 and ten years later attained a rank of senior instructor. In 1955

19 Thirty-Four Biographies. p. 28.

20Ibid.. p. 29.

^MacMillan and Beckwith, "Jean Coulthard," in Contemporary Canadian Composers (Toronto: Oxford University Press, 1975), p. 54.

22Ibid 25

she lived in France for a year under a Royal Society of Canada Fellow­ ship. While in France Nadia Boulanger examined her scores and offered critical advice.

In addition to the Royal Society Fellowship, Coulthard has earned a long list of awards, grants, and commissions. On the occasion of Queen Elizabeth's coronation, the Canadian Broadcasting Corporation commissioned A Prayer for Elizabeth (1953), a work mentioned by Lazare

Saminsky:

Her "Prayer for Elizabeth," a specially commissioned work for String Orchestra, premiered following the Queen's first broadcast address to the Commonwealth, was rated by a Toronto critic as a "little masterpiece in its moving sincerity and one of the musical highlights of the Coro­ nation season."22

Spring Rhapsody was written on commission in 1958 for the Vancouver

International Festival. A Violin Concerto (1959) and a choral cantata,

This Land (1967) were commissioned by the Vancouver Symphony Orchestra.

Stylistically, Coulthard1s works may be described as contempo­ rary with frequent diatonic tonal emphasis. Saminsky describes her partial reliance on tonality in the Two Sonatinas for Violin and Piano:

The first of these sonatinas is music of a kind of post- impressionist style, but there is a subtle emotional sub­ stance in it and tonal charm. Its salient trait is an interplay and imaginative balancing of modal shades with the atonal in her harmony. Her climaxes are not without force, and the laconic form is attractive. These,quali­ ties are better stressed in the second sonatina.

The broad range of her compositional style is mentioned in Contemporary

Canadian Composers:

22Thirty-Four Biographies, p. 29.

24Ibld. 26

On hearing Miss Coulthard*s work, one is immediately aware of the influence of her teacher, Vaughan Williams. Her music is generally tonal with only occasional digressions into chromaticism or dissonant polyphony (String Quartet No. 2, 1953). She often chooses picturesque titles—A Quiet Afternoon (1964), The Frisky Pony (1964), Legend of The Snows (1970), Song to the Sea (1942), Rider on the Sands (1953)— descriptive of a style that involves tone- painting to a great extent, again a characteristic of English music of the early part of this century.

Her tonal style is well-suited to vocal music, Coulthard*s forte, "where perhaps her particular diatonic style is most at home."^

The evocative tonality characteristic of Coulthard is appropriate for her many songs, as pointed out by Eric McLean, a music critic (Montreal

Star):

As in many of the Coulthard songs, the vocal part is eminently singable--a direct and tasteful translation of works into sustained sound. It is in the accompani­ ment, rather, that we find her harmonic and rhythmical deviations from the nineteenth century models. They are full of strongly evocative colour, best illustrated in the first of the group "Now Great Orion."^^

Notwithstanding the influence of various male teachers and com­ posers In the male—dominated field of composition, Coulthard has main­ tained a feminine touch in her music, according to Chester Duncan in the Canadian Music Journal:

It is hard to be a woman composer at any time. Either listeners expect you to be nicer and weaker than other people and your music goes all watery with impression­ ism, or you overcompensate by developing a will like a ramrod and music like iron fillings in a steel

^MacMillan ^ Beckwith, "Jean Coulthard," p. 54.

26Ibid. 27 Thirty-Four Biographies, p. 29. 27

kaleidoscope. Jean Coulthard's Preludes [for piano], however, retain their feminity without either embar­ rassment. °

CoulthardTs successes and recognition are by no means limited

to Canada. Her music Is known In the United States, Europe, England, and Israel, and many of her piano and vocal pieces have been published.

The reputation that Coulthard has built for herself Is supported by

the quantity of her output. She has written a large number of works for stage, orchestra, voices, instrumental ensembles, and piano. Al­

though Coulthard's primary compositional interest may be said to be In the area of vocal music, she has written a significant number of In­ strumental chamber works. More formally titled works such as String

Quartet No. 1 (1948) and String Quartet No. 2 (1953), a Piano Quintet

(1932), Sonata for Oboe and Piano (1947), and Divertimento for Five

Winds and Piano (1968) stand in contrast to more freely titled pieces such as Day Dream (violin and piano, 1964), Lyric Trio (violin, cello, piano, 1968), and The Birds of Lansdowne (violin, cello, piano, and electronic tape, 1972).

One of Coulthard's most experimental works is the one selected for study in this dissertation: the String Quartet No. 2 (1953). As she says, it reflects a different approach to writing "music that is good" and that Is "not forced in any way as to specific style or tech­ nique of the moment." The insertion of atonal passages and the special emphasis on dissonant style in this work are not typical for Coulthard.

However, the melodic writing is lyrical and frequently oriented towards

^Thirty-Four Biographies, p. 29. 28 motion In seconds and thirds. Throughout the work there are distinct

themes and motives that give a melodic emphasis to the work and alleviate the adventuresome dissonant harmonies to a large degree. The quartet has three movements, each divided into distinct sections based on tempo change and thematic or motivic content. The principal theme of the first movement, in quasi-sonata form, is also used in severely abbreviated form as the primary motive of the third movement. In mod­ erate tempo, the first movement presents this theme at the beginning:

Fbeo mosso J- * 72, Ccirca) ContiMe esprtes.

J*)- j l can+iblc.

Example 1. Coulthard. String Quartet No. 2. First Movement, mm. 1—3, first violin

The third movement contains an alteration of Lento and Allegro sections, and the abbreviated theme, now a motive, occurs in the fast sections:

"V Allegro con brio k -- i SS r-\ r s 1', T. __ 1 ----- — 1**“ ■' - r -- y- - — ^ 9~ Aljegre con brio ^ area

Example 2. Coulthard, String Quartet No. 2 . Third Movement, mm. 8-9, first violin

The middle movement is slow and does not employ the principal theme material. Here more slowly moving melodies with gently winding motion 29 are emphasized for contrast, as found in the first violin in the open­ ing measures:

Example 3. Coulthard, String Quartet No. 2, Second Movement, mm. 2-3, first violin

Thus, although the work is experimental in harmonic structure, it is stylistically typical of Coulthard in melodic structure. A mature work, the Quartet No. 2 reflects Coulthard*s training, composi­ tional experience, and inspiration in a way that transcends the mechan­ ical aspects of the melodic and harmonic structures that are analyzed in this study.

Claude Champagne

For Canadian composers Claude Champagne served as *'a bridge between old and new, between the nineteenth and twentieth centuries."^

B o m Adonal Desparois Champagne in Montreal on May 27, 1891, his early experience enabled him to hear and perform Quebec folk music. He par­ ticularly admired his grandfather, a violinist quite popular in the region of Repentlgny and this admiration excited his Interest in music:

However, it was the violin, the instrument of his beloved grandfather, that the young Claude wanted to play. His school teacher, himself a violinist, helped him to realize

^Louise Bail-Milot, ,rClaude Champagne," in Contemporary Canadian Composers (Toronto: Oxford University Press, 1975), p. 41. 30

his wish. On the boy's 15th birthday, his schoolmates, headed by the teacher, came to his house and presented him with a violin which has been his favoured instrument ever since.30

In this early period he studied violin with Albert Chamberland

and piano with RomaIn Octave Pelletier and at the age of fifteen earn­

ed the diploma of the Dominion College of Music. Three years later he

graduated from the Montreal Conservatoire National de Musique.

The "rather raw and joyous modal sonorities of Quebec folk

music"3*- made an early impression upon Champagne that was never entire­

ly lost in his later musical development as a composer. In his later

compositions he employed both modality and folk tunes with a "limpid

and touching simplicity."

Champagne also came in contact with the sophisticated elitism

of Canadian composers who were studying and Imitating European trends

around 1900-1920. Through the influence of Alfred Lallbert£, and

others, Champagne was exposed to music that went far beyond the region­

al folk melodies of Quebec. In spite of international trends, however,

this Indigenous music would later provide a part of the impetus toward

a greater Canadian consciousness already implicit at this stage: "This was a time of great divergencies between ideas of universalism on the no one hand, and those of regionalism, 'the soil,* on the other."

Given the various regional factors and universal currents that

Imprinted themselves upon young Champagne's consciousness, it should

^Thirty-Four Biographies, p. 24.

3*-Bail-Mllot, "Claude Champagne," p. 41.

32Ibid. 31 not be surprising that be became a dilettante. The following additional quotes from Bail-Milot clarify the above statement:

In short, the years 1900-20 were particularly fruitful in cultural manifestations and Champagne disovered in that period an essential element that he developed throughout his life— dilettantism.

He was a dilettante from every point of view, as were most men to some degree in that generation. Champagne never finished his advanced education; his daughter speaks of him as always having a book in his hand——books that dealt with a vide variety of subjects, since Champagne was interested in virtually everything. His subtle spirit, linked to a gentle but firm character, never became partisan to any given system; to his students he was a master who knew how to develop their individual personalities.^

In 1921 a scholarship provided by the Government of the

Province of Quebec enabled Champagne to travel to Paris for study.

Encouraged by LallbertS and by Serge Rachmaninoff during a visit to

Montreal, Champagne spent seven years in Paris, studying composition with Andr£ Gedalge and Raoul Laparra and violin with Jules Conus. His

Suite canadienne (1927) was performed in Paris by the Pasdeloup Orch­ estra in 1928.

When Champagne returned from Europe in 1928 he applied his energies to the development of Canadian musical life in Quebec. He supervised solf&ge training in the elementary schools of the Montreal

Catholic School Commission, and taught composition at the McGill Uni­ versity. His pupils included Canadian composers such as Marvin Duchow,

Violet Archer, Maurice Dela, Pierre Mercure, Roger Matton, Clermont

Pepin, and Franqois Morel.^

^Bail-Milot, "Claude Champagne," p. 41. 34 Thirty—Four Biographies, p. 25. 32

Champagne's compositional style is considered to be a blend of neo-Romanticism and clarity of expression:

One notes in his music a clarity and a consciseness that are often astonishing. His output Is actually quite small; in all only some fifteen works attest to a reaction against the excesses of romanticism, such as extended developments and complication of style . . . it is his later mature works that bear the mark of conciseness.^

A songlike quality and the subordination of harmony and rhythm are alluded to by Bail-Milot:

One finds with Champagne a sort of intimate union of melody and rhythm that serves to eliminate the rigidity of bar lines and creates a kind of unending song, where all char­ acteristic rhythmic formulas are abandoned.

. . . it is perhaps the intimate union of harmony and rhythm, both subordinate to the melodic line, that imparts to Champagne's music such a limpid and touching simplicity.3**

The "doyen" of Canadian composers,37 Champagne consistently built his reputation. He spent some time in Brazil teaching and con­ ducting after his highly successful works such as Suite canadienne.

Danse villageoise. and Symphonie gaspSsienne.

In the years following 1945 his writing became somewhat less

"regional:"

After the "Symphonic Gaspesienne" Champagne moved into what music writers characterized as his "post-impressionist style." It Is represented by such works as the Concerto for Piano and Orchestra, and by the String Quartet.

3^Bail-Milot, "Claude Champagne," p. 42.

36Ibid., p. 43.

37Ibid.. p. 42.

•^Thirty-Four Biographies, p. 25. 33

His tone poem for choir and orchestra, Altitude, also represents

experimentation and absorption of modern trends. Ball-Mllot says of It:

Newer trends are reflected in two later compositions, the Quatuor h cordes (1951) [1956?] and Altitude (1959)— some­ what surprising in one of Champagne1s generation and demonstrating his absorption of Schoenberg and Messiaen. Altitude was Inspired by his first extended visit to the Canadian Rockies and employs forces both orchestral (in­ cluding ondes Martenot) and choral for musical description and religious comment.^9

After considerable acclaim both in composition and in teaching,

Champagne had a concert hall in the Ecole Vincent d'Indy named after

him while he was still living. He belonged to the Canadian Arts Council

(of which he was the honorary president In 1951), the Academle de

Musique du Quebec, and the Canadian League of Composers. He was editor-

in-chief of the publication division of BMI Canada, and he was awarded

the Canada Council Medal in 1962. "He died as quietly as he lived, on

Tuesday, December 21, 1965, at 11:30 a.m., while writing Christmas

greetings."^®

The dilettantism of Champagne is evident In the quartet anal­ yzed in this dissertation, Quatuor a cordes (195A). Turning away from

his intimate union of harmony and rhythm, subordinated to the melodic

line, he turns to a multi-layered linear approach in this work. These

layers are distinguished by varying rhythmic values throughout the

two movements of this work. For example, the first two measures of

oq Bail-Milot, "Claude Champagne," p. A3.

40ibtd.' the first movement ("Introduction") contain four levels of time values whole notes, half notes, quarter notes, and eighth notes.

A/SOAfSTlfSO JsiaO

Example 4. Champagne, Quatuor Cordes. First Movement, mm. 1-2, all parts

There are some sections, however, in which all parts move in one rhythmic "layer," as in the first movement:

J_LL6GrRO .M.ODEKJLJO J „,2

— .T - y - b l 3 rl -h ird-IjH-- fKT— * h i a r i J bJ J h J r --- 3--- 1 P: g 1------2 ------rj---2---" V U - - - ■ J- ■■ J.tfJ----- /.---

EZjpE-.. 1? V y d - L j — - — - — -—

Example 5. Champagne, Quatuor H Cordes, First Movement, mm. 89—90, all parts 35

The cello deviates slightly from the pattern in the other instruments, but this deviation is not sufficient to establish a new "layer."

An example of layers Involving simultaneous metrical signatures is found at the beginning of the second movement:

IT

«

Example 6. Champagne, Quatuor 5. Cordes, Second Movement, mm. 124-27, first violin and cello

The whole work is sectional with sections distinguished by tempo changes, as was the case with much of the Coulthard quartet.

Like Coulthard's, this quartet is linearly oriented, and, al­ though it is experimental it is, nevertheless, a mature work by a com­ poser with fully developed stylistic characteristics. These character­ istics are clearly defined moods, tuneful melodies, and subordination of harmony and rhythm to the melody.

Harry Somers

Harry Somers, "one of Canada's leading composers of the younger generation,was born in Toronto on September 11, 1925. He developed an Interest in the field of music quite suddenly and intensely at age

fourteen:

^^Thirty-Four Biographies, p. 91. 36

At school he was interested in sports and drawing and did not participate in musical activities. Recalling his school years, Somers once wrote: "Around the age of twelve or thirteen, I entered high school. As yet there were no musical associations which stirred me. . . .1 used my pocket money which might have admitted me to the children's concert, for the rugby game. . . .

That was his state of mind at thirteen, and it could hardly have been expected that only a year later Harry would decide that music would be his life's work. It was in 1939 that Somers became converted to music. Spending part of his summer vacation at a friend's country place he had an opportunity to listen to some music by Beethoven, Brahms, and Mozart played on an old piano by two amateur music enthusiasts. Perhaps it was not so much the music itself, but the sheer enjoyment and the obvious respect for music revealed by the two piano players that fascinated Somers most. This short music session and the following warm and unpretentious conversation about the art of music- making convinced young Somers that the newly revealed world of enchanted and mysterious sounds was going to become his own.^2

Following this sudden exposure to serious music, Somers studied piano with Dorothy Hornfelt. He successfully completed the examination requirements for Grade 8 in the Toronto Conservatory and began the study of composition, writing a string quartet by age seventeen. Somers regards his relative lack of formal schooling prior to this period as an advantage:

From the beginning I had been writing my own music. These early works went through various styles, but with a differ­ ence: they were not imitative because I had no knowledge of much music other than a few classics. I was fortunate In learning to speak for myself at the beginning.^

The person most influential in Somers' development as a com­ poser was John Welnzweig. Somers studied with him during the 1940*s;

^Thirty-Four Biographies, p. 91.

43Ibid. 37

The impact of Weinzweig on his new pupil can be measured to Somer's underexposure to music, and also to the 1942 musical diet in Toronto. Through Weinzweig, Somers had his intro­ duction to the brave new world of twelve-tone composition and to Webern, Schoenberg, Berg, the early Stravinsky, and later on, Bartok. Weinzweig, who regards Somers as having been his most talented pupil, tells of Harry’s initial colour-mindedness, his naivete in organizing musical ideas, and his lack of enthusiasm for eighteenth century craft. For his teacher, craft is almost creed. In this respect he is at one with Schoenberg, who regularly plunged eager but startled pupils into weeks of exercises in pure diatonic melodic construction. There is, however, one difference. With Somers, Weinzweig started at both levels, with strict exercises and with free composition.^4

Brian Cheroey provides more musical detail in his account of Weinzweig's influence by listing five specific traits that Somers absorbed from

Weinzweig:

Weinzweig was probably the most crucial single factor in Somers' development. At least five important aspects of his influences can be seen in Somers* work during the late 1940*s and early 1950*s: (i) a highly flexible use of serial technique (the series on the whole controlling linear rather than vertical events), for Instance repeat­ ing certain pitches that are rhythmically varied, as if examining them from a different viewpoint; (ii) an emphasis on a highly controlled and elegant long melodic line that bears the main weight of the musical argument; (iii) trans­ parent, clear textures (in orchestral works there are rarely more than three or four voices); (lv) an awareness of instrumental colour; and (y) a dry wit, manifested in a play of short rhythmic ideas.

In 1949 Somers' talent as a composer was recognized sufficiently to earn him a scholarship for study abroad from the Canadian Amateur

Hockey Association. Studying composition with Darius Milhaud in Paris

(1949-50), he applied himself seriously to compositional problems en­ countered in the organization of large pieces. Somers states, "It was

^Thirty-Four Biographies, p. 91.

^^Brlan Chemey, "Harry Somers," in Contemporary Canadian Com­ posers (Toronto: Oxford University Press, 1975), p. 208. a chance to study and write and I needed every minute. Milhaud forced

me to rethink and confirm my ideas and this was good."^^

During this period of Intense development (1948—50) a consistent

stylistic trait of Somers emerges:

a sustained sound (whether a vertical aggregate of two or more pitches, or a single pitch, Isolated or prolonged in a melodic line) that is Infused with a dynamic envelope or shape of its own, becoming as it were an active living organism.^

The "dynamic envelope" to which Chemey refers may assume the role of a

structural "motive." Specific works that contain this compositional device are North Country (1948, string orchestra), String Quartets

Nos. 2 and 3 (1950, 1959— analyzed in this study), The Fool (1953,

opera), and Five Concepts for Orchestra (1961), "where it becomes the basic compositional premise."^®

Somers returned to Toronto in 1950 and began a period of exper­

imentation with Baroque contrapuntal techniques synthesized with his own style. Somers says of this period of his development:

During the early fifties I was very involved with contra­ puntal technique, attempting to unify conceptions of the Baroque and earlier, which appealed to me enormously, with the high tensloned elements of our own time. A series of works were thus involved-—my First Symphony, a set of piano fugues, "12 by 12," two violin sonatas, a chamber opera, "The Fool," and the Passacaglia and Fugue.49

During the remainder of the 1950* s Somers explored the use of fugal textures and the juxtaposition of different musical styles:

^®Thirty-Four Biographies, p. 92. ^C h e m e y , "Harry Somers," p. 208. 48 Brian Chemey, "Harry Somers," p. 208.

^®Thlrty-Four Biographies, p. 93. 39

The Somers ’fugue1 . . . bears little resemblance to the academic notion of fugue. There are two general types of subject: CD fragmented and rhythmic (Passacaglia and Fugue, the first subject of the double fugue In the String Quartet No. 3) or (11) lyrical and sustained (the slow movement of the Symphony for Woodwinds, Brass and Per­ cussion) . The success of the fugal movements is due In part to the . . . characteristic of superimposing several layers moving at different speeds; this is ideally suited to a contrapuntal texture. But above all the fugue is a highly credible vehicle in Somers’ hands: he continually presents the subject in new perspectives by adding new strands of figuration in the surrounding voices by re­ casting the subject itself.50

Close juxtaposition of tonal and atonal styles was used by

Somers for dramatic effect. In the Suite for Harp and Chamber Orches­ tra (1949) he first realized this conception. Chemey describes the style juxtaposition found in works from the 1950's:

The device [style juxtaposition] can be seen In the largest works of the decade--The Fool. Piano Concerto No. 2, the ballet scores The Fisherman and his Soul (1956) and Ballad (1958)— as well as In the slow movement of the Violin Sonata No. 1. In all of these, it is used in the sense of the first aim mentioned above [see quote below], that is, to heighten the dramatic and emotional impact of a given situ­ ation by juxtaposing (or mixing) a non-tonal language employing serial methods of pitch organization with a rather synthetic tonal language (often using baroque tech­ niques and forms).51

Somers himself explains his reasons for exploring juxtaposed styles as follows:

My alms were at least twofold. One: to achieve maximum tension by this type of superimposition. Tonal and tonal- centre organizations create their "solar systems" so strongly that, for me, maximum tenxion is achieved only by fracturing them and joining them with non-tonal material. This, I believe, is not only aesthetic but also psychological, for tonality has such strong associations of order In most people, that It is a shock when it is broken, or challenged

50Cherney, "Harry Somers," p. 209.

^Chemey, Harry Somers, p. 71. 40

In the same composition. . . .So I was deliberately using memory and association as compositional elements. The second aim was to attempt to realize the super Imposition of planes of sound. (The clearest example of this is The Unanswered Question of Ives. • .).^

In the 1960*s Somers extended his compositional experimentation beyond the limits of standard musical notation. His orchestral works

"show Increasing experimentation with non-thematic textures (Five Con­ cepts. Stereophony) and with the spatial and visual aspects of the per­ forming situation (Movement for Orchestra and Stereophony, respective­ ly)."53

Chemey discusses the musical application of serlalism In

Somers* works. In the 1940*s he experimented with three-or-four-note interval cells In North Country (1948) and In Suite for Harp and Cham­ ber Orchestra (1949). Use of all twelve notes is quite free, as de­ scribed by Chemey:

In a few works of the 1940*s he deliberately used all twelve pitches In Isolated melodic statements; for ex­ ample, in the first movement of Testament of Youth and Moon Haze (1944). In slightly later works such as the First Piano Concerto (1947), the Rhapsody for violin and piano (1948), and the Woodwind Quintet (1948), a series is stated at the beginning as a theme, and certain raotivic ideas are extracted from it. In subsequent works the ex­ pressive character of certain intervals in a given series, rather than a systematic application of the series itself, determines the choice of pitch.

Following his early experimentation with serlalism, Somers* serial works show varying degrees of strictness in application of the series. Both pitches and row segments may be presented out of order.

52 Chemey, Harry Somers, p. 71. C "I Chemey, "Harry Somers," p. 210.

5AIbid., p. 209. 41

Also a given work, such as the Symphony for Woodwinds, Brass and

Percussion (1961) may contain more than one series. The use of tonal references in the series is discussed by Chemey:

It is important to point out that even in works after 1950 that apply the series more strictly there are tonal reference, either built into the melodic line (and Inherent in the series) or else as points of vertical repose before or after a more dissonant climactic episode (as in the first section of the Symphony No. I). Beginning with the String Quartet No. 3 (1959), tonal references sometimes found in the serially organized works of the earlier years are avoided in their experimental counterparts of the early 1960*3, as in Five Concepts. Stereophony, or Twelve Miniatures (1964).^5

The commissions that Somers has received are numerous and show the active role that he has played in Canadian musical life. He has written music on commission for a number of different organizations.

Included is the work studied in this dissertation, String Quartet No. 3

(1959), commissioned by the Vancouver Festival Society for the

Hungarian String Quartet.

Somers* String Quartet No. 3 (1959) is a mature work that re­ flects the highly flexible serial technique, superimposed layers of sound, avoidance of tonal references, and style juxtaposition that are his established traits in the late 1950*s. There Is no particularly new element for Somers in this quartet; rather, the compositional approach represents a continuing line of experimentation for a

55 Chemey, "Harry Somers," p. 209. composer who has systematically evolved his style by means of experi­ mentation.

The quartet consists of four movement-like sections (slow-fast slow-fast) based on one row, presented first by the cello as an Intro­ ductory solo (condensed below in non—metric form):

k o - A o - TT

T3~

Example 7. Condensed Row from Somers, String Quartet No. 3 , mm. 1-9, cello

As with the quartets of Coulthard and Champagne, this work is section­ al within "movements," tempo and thematic content determining section­ al divisions. 43

Sustained sounds occur with great frequency in the quartet, much of the time in conjunction with superimposed static layers, as shown below:

1

$

Example 8. Somers, String Quartet No. 3 , mm. 92—93, all parts

Rapid style juxtaposition through rhythmic contrast, as opposed to the tonal/non—tonal juxtaposition discussed earlier, occurs in the first "movement": 44

t j r J

1PICC

f * — if * s r !>}/ r s * _i------Ite--- M — JH-: i j ~ f f f

Example 9. Somers, String Quartet Mo. 3, mm. 61—64, all parts Another typical passage, Involving agitated motion contrasted succes­ sively with a gently undulating melody, occurs in the third "movement”

a s # * h y s s — ------pn»«fc. — r ftinpo 0 _ T JJ 7r m — - y *" » h tTrnpo ptitno 3 r i ! - rrw *'--- y *— y---

‘h r fTfnpa primd

b l w - . . ■1' ~ 9 I 1 L _ I_ ■

____

L U i

3 wfswt.

Example 10. Somers, String Quartet No. 3. mm. 243-246, all parts 46

As one listens to this quartet, the most striking feature is perhaps the rhythmic force that results from both sustained driving action and abrupt contrasts, Somers* individuality is apparent and im­ parts a high level of emotional force and intensity to this quartet.

John Weinzweig

John Weinzweig stands as one of the foremost composers, educa­ tors, and organizers in the field of contemporary Canadian music.

Born in Toronto, Ontario on March 11, 1913, his earliest experiences with music are recorded as follows:

His first interest in music was inspired by Jewish and Italian folk music which was often heard in his ethnically diverse community— the new home of his parents who came to Canada from Poland. His earliest instrument was the mando­ lin, on which he soon learned to improvise his favourite tunes with musically minded neighbors.

Formal studies for Weinzweig began at age fourteen with piano and theory lessons from Gertrude V. Anderson. Subsequently he learned to play tenor saxophone, tuba, and bass.

Weinzweig?s interest in composition began while playing in the student orchestra at the Toronto Harbord Collegiate Institute and

Central High School of Commerce. The orchestra leader, Brian McCool, inspired Weinzweig to plan a career in music:

This new experience under the guidance of an enthusiastic teacher, Brian McCool, who later became Director of Music in the Ontario Department of Education, opened to him the whole world of music. He became so fascinated by the un­ limited possibility of mixing the orchestral sounds that, at the age of nineteen, he decided that composition was to be his life work. To earn money for his advanced

^®”John Weinzweig,” in Thirty-Four Biographies of Canadian Composers, p. 104. 47

studies he went to work as a bookkeeper and typist. With his savings, and his father*s help, he entered the Faculty of Music of the University of Toronto in 1934.57

While at the University of Toronto he studied harmony with Leo

Smith, counterpoint with Healey Willan, orchestration with Sir Ernest

MacMillan, and conducting with Reginald Stewart. Going a step beyond

his academic studies, Weinzweig founded and became the first conductor

of the University of Toronto Symphony Orchestra. This notable accom­

plishment foreshadows the tremendous organizational talents of Weinzweig

revealed by his later impact on the exposure and establishment of con­

temporary Canadian music.

Weinzweig earned a Bachelor of Music degree (1937) and wrote

his String Quartet No. 1 as his thesis. He continued his education at

the Eastman School of Music in Rochester, New York and received a

Master of Music degree a year later (1938) In composition and conduc­

ting. His teachers there were Bernard Rogers (orchestration and com­

position) and Paul White (conducting).

The brief period at Eastman was significant to Weinzweig in

terms of his future direction as a composer as well as his influence upon a younger generation of Canadian composers who have now reached prominence. Weinzweig*s time at Eastman was fruitful for his later

stylistic development:

His Imagination was caught by the new horizons revealed to him in works by Stravinsky and Alban Berg, helping him to mould his own artistic style and musical concepts,-*®

^Thirty-Four B lographies , p . 3 04. 58 Ibid., p. 105. 48

Weinzweig's sCudy at Eastman led to his adoption of serlalism,

and he became "the first Canadian composer to espouse the twelve-tone

technique.In 1939, the year after his Eastman training, he wrote

his first twelve-tone work, Dirgeling (found in the Suite for Piano

No. 1). Departing from the Viennese approach, Weinzweig's serial works

use the row as "a source for melodic invention;"^®

In Weinzweig's early music particularly, the use of the row is frequently difficult to detect. Dirgeling (from the Suite for Piano No. 1. 1939), his first twelve-note composition, is not constructed in the systematic manner employed by the Viennese composers. Weinzweig's row is primarily a source for melodic invention, which is applied in such a way that it more resembles the baroque technique of "Fortspinnung."6l

Weinzweig's early works frequently use different rows for each

movement, and harmonies are not usually derived from the row. "For

him the twelve note series is not a unifying element for all movements

of the work. He rather emphasizes the contrast of individual movements

by basing them on different rows."^ In his later works he more close­

ly approaches traditional serial writing. The same row is employed in

all movements of a given work, and harmonies are derived from the row.

In 1939 Weinzweig began teaching at the Royal Conservatory of

Music in Toronto. His teaching career was interrupted by service in

the Royal Canadian Air Force from 1941 until 1943. During the war

^Richard Henninger, "John Weinzweig," in Contemporary Canadian Composers (Toronto: Oxford University Press, 1975), p. 230.

60Ibid., p. 232.

61Ibid.

^Thirty-Four Biographies, p. 105. 49 years Weinzweig directed much of his musical energies toward writing film music. Radio dramas also received his attention, and Weinzweig is "the first composer (1941) to write incidental music for CBC radio dramas."^

Many students who were to achieve later distinction received the benefit of his instruction when Weinzweig was appointed Professor of Composition at the University of Toronto in 1952. Prominent students of his have included Murray Adaskin, John Beckwith, Norma Beecroft,

Samuel Dolln, Harry Freedman, Irving Glick, Philip Nimmons, Murray

Schafer, and Harry Somers. The significance of Weinzweigfs Influence upon these composers is Indicated by Henninger:

It is difficult to imagine what contemporary Canadian composition would be like had it not been for John Weinzweig. As teacher of many of the English-speaking composers in this volume, he gave some of them their first glimpse of mainstream twentieth-century techniques and styles of composition. He fostered and encouraged individuality in his students while challenging them with the need for high standards and self-criticism. The diversity and quality of Canadian composition today attests to this country’s debt to him In the arts.

Weinzweig*s talents as an organizer have greatly aided the pro­ motion of contemporary Canadian music. He stands as a "primary moti­ vating force in the establishment and success of most of the organiz­ ations that have promoted contemporary Canadian music during the past 65 quarter of a century." His concern and efforts have led to the

^Henninger, "John Weinzweig," p. 230.

64Ibid.. p. 231.

65Ibid. 50 creation of both the Canadian League of Composers and the Canadian Music

Centre.

Clarity is an essential element of Weinzweig*s personal style.

Most of his works are "characterized by clear, often thin, textures and lucid form with strong motivic organization, usually serially de­ rived. Some of the works from 1943 to 1950 reveal "an almost 67 ascetic clarity and brittleness." Examples are the Second String

Quartet (1946) , Divertimenti Nos. 1 (1946) and (1948) , Piano Sonata

(1950), and Interlude in an Artist*s Life (1943, string orchestra).

Most of Weinzweig*s music is instrumental and is written for orchestral and chamber ensembles. A tendency toward moderate to thin instrumentation is consistent in these works:

. . . after the mid-1940*s . . . Weinzweig uses his instru­ ments sparingly so that a chamber texture prevails. Melodies are often presented for several measures with little or no accompaniment. Phrases often change timbre by being passed from one instrument to another. Melodic doubling becomes almost entirely absent. Ensemble textures and accompaniments are usually divided among several sections so that the over­ all fabric is kaleidoscopic.®®

Careful motivic organization contributes to the clarity of

Weinzweig*s works and is "another consistent feature throughout

Weinzweig*s melodies."®^ Serial derivation of motives is his usual procedure, although Henninger says that "his use of the twelve-note row has been overemphasized, particularly by those reviewers and performers

®®Henninger, "John Weinzweig," p. 231.

^Thirty-Four Biographies, p. 105.

^Henninger, "John Weinzweig," p. 231.

69Ibid. 51 who, in order to dispose of his music as ’cerebral*, use the row’s presence to excuse their own limited understanding of the more signifi­

cant features of his music."7® The motivic exposition of row segments

is stressed by Hennlnger in this context:

Weinzweig dwells on certain segments of the row to such an extent that the music almost becomes tonal. An entire section often consists of one exposition of a form of the row, as though it is the motif and not the row that is the primary concern of the composition. That the motif comes from a row serves to tie the work more neatly to­ gether. 71

Weinzweig*s commissioned works form a large part of his output.

The Canadian Broadcasting Corporation commissioned Our Canada (1943),

Wine of Peace (1957), Concerto for Piano and Orchestra (1965-66),

Dummiyah/Silence (1969), Around the Stage in Twenty-five Minutes during which a Variety of Instruments are Struck (1970), and Trialogue (1971), among others.

In the category of instrumental ensemble are three string quartets, each written for a particular occasion or group. The First

String Quartet was composed as,a graduation thesis at the University of

Toronto in 1937; String Quartet No. 2 (1946) was commissioned by the

Forest Hill Community Centre and performed by the Parlow String Quartet in 1947; and String Quartet No. 3 (1962), studied in this dissertation, was commissioned by the Canadian String Quartet under a Canada Council grant.

In the late 1960’s and early 1970’s Weinzweig received several honors and awards for his contributions to Canadian music. The Canada

7®Henninger, "John Weinzweig," pp. 231-232.

7lIbid.. p. 232. 52

Council granted him a Senior Arts Award in 1968, and he was made an

honorary Doctor of Music by the University of Ottawa in 1969. He be­

came an Officer of the Order of Canada in 1974. Also, Weinzweig is on

the Board of Directors of the Canadian Music Centre, the Canadian

League of Composers, and CAPAC (Composers, Authors and Publishers

Association of Canada, Limited).

Weinzweig*s String Quartet No. 3 (1962) displays his typical

style traits: 1) free use of serialism, 2) prominent melodic lines,

3) clarity of texture, 4) attention to instrumental color, and 5) hu­

morous rhythmic play. In the five movements sectioned by motivic con­

tent within movements, the quartet is based on one row throughout:

I) ......

/ n L L n r t H (( s U & 7 £ c? L n b q U is Tr7—1 L r f ...

Example 11. Row from Weinzweig, String Quartet No. 3

The row is frequently found "out of order" with regard to specific

pitches, and many row statements are incomplete. Rows are dwelled

upon with an unfolding exposure, so that the statement of all twelve notes is delayed by repetition of row segments. An example of this

treatment is found in the second movement: r ' I+r = l - y ----- —------‘•-J — J f &#x*Si\anai

^ ------■------'J \ if i ~rf—- [if-fTf »-.<• ""■ I f \ If

I

Example 12. Weinzweig, String Quartet No. 3. Second Movement, nun, 111-17, all parts 54

Rhythmic play on, a row segment is found throughout much of the fourth movement in the cello. The following excerpt shows the "dry wit" that combines with an underlying rhythmic vitality in this move­ ment : p in .

1 , I • ~1 - ,---- i j ^ ,------v Irrr-W-— !*•*— - H f =

..

Example 13. Weinzweig, String Quartet No. 3. Fourth Movement, mm. 14—18, cello

The economy of means and occasional sparseness of Weinzweig's writing are distinctive. There is a consistent use of silence in all parts, and almost never do all four Instruments play simultaneously in a sustained passage. Like Somers1 quartet, this is a mature work that displays compositional techniques already developed, rather than a work written primarily for experimentation as is the case in the Coulthard and Champagne quartets. CHAPTER II

CLARIFICATION OF THE ANALYTIC METHOD

Since relevant concepts and terms found In The Structure of

Atonal Music form the basis for the analytic method employed, precise definitions are provided in the first part of the chapter. These in­ clude: 1) Pitch Class, 2) Interval Class, 3) Pitch Class Set, 4) Prime

Form, 5) Interval Vector, and 6) Similarity Relations. Although this study depends heavily upon concepts presented in the first part of

Forte*s text, practical procedures for applying these concepts are not sufficiently explicit to test them in the quartets selected for this investigation. It was therefore necessary to develop additional pro­ cedures in order to apply Forte's ideas appropriately to the four

Canadian quartets. These supplemental procedures are discussed in the second part of the chapter under the headings: 1) Hypothesis Testing,

2) Distributions, 3) Improved Procedure for Prime Form Identification, and 4) Statistical Comparisons.

Relevant Concepts and Terms

Pitch Class

The term pitch class identifies a concept that is basic to the understanding of interval class and the related concepts, pitch class set, prime form, interval vector, and R-relations. set forth in Forte's text. Pitch class has been defined by John Vinton as follows:

55 56

Pitch Class, a pitch without reference to a specific octave register* The term, which gained currency through the writings and teaching of Milton Babbitt, occurs most fre­ quently In reference to 12—tone music and is used to sep­ arate the property of pitch from other properties of tones (timbre, volume, duration, etc.)*^

The Importance of the concept is underscored by Benjamin Boretz when he states that "Babbitt devised the terms pitch class and Interval class to represent the traditional notions of the functional equiva­ lence of octave-related pitches. . . ."^ Example 1 gives three In­ stances of pitch class £• Traditionally called c1, c2, and c3, these three distinct pitches belong to the same pitch class, as does any other pitch with the given generic name c (natural)* Because the pitches can be considered to be functionally equivalent, any one of them may be expressed simply as ^c. i - e -

Exauiple 14. Instances of Pitch Class .c

*

There are twelve separate pitch classes, corresponding to the twelve semitone ratio relations of the equal tempered octave. In contemporary analyses, these twelve pitch classes are represented by

^■"Pitch Class," Dictionary of Contemporary Music, ed. by John Vinton (New York: E. P. Dutton and Co., Inc., 1974), p. 577.

^Benjamin Boretz, "Babbitt, Milton," Dictionary of Contemporary Music, p. 46, n. 1. 57 the numbers 0 through 11. An advantage of numerical representation over traditional letter representation Is that the distinctions normally necessary concerning enharmonic semitones are set aside, and theoretical operations may then be applied directly to the numbers* Por the sake of uniformity, the pitch class £, the standard point of reference In both conventional and more recent theoretical models, Is assigned the number 0* Ascending order by pitch and number Is followed after _c * 0.

This numerical system of pitch representation Is given In Table 1:

TABLE 1

Numerical Representation of Pitch Classes o 1 0 6 - f# * H 1 0 7 - g

2 - d 8 - g#

3-d# 9 - a

4 - e 10 - a#

5 - f 11 - b

*Sharps are employed arbitrarily for the sake of consistency.

Traditional notational spelling Is Ignored In the numerical

Identification of pitch classes. Por example, the major triads c#—e#— g# and d**—f-a^, equivalent In equal temperament yet spelled different­ ly, are both represented by the numbers 1, 5, and 8 : Example 15. Triads Notated Conventionally and Ex­ pressed in Pitch Class Numbers

Enharmonic meanings and distinctions are set aside through the assump­

tion of enharmonic equivalence. Thus, a given key on a standard key­

board Is represented by one and only one number. For example, the key

for _c (as well as bp and d^b) belongs to pitch class 0 , regardless of

the notatlonal spelling In the music.

Forte covers this point In his Glossary:

Pitch class (pc). One of the 12 pitch-classes designated by the Integers 0 through 11. Pitch-class 0 refers to all notated pitches C, B-sharp, and D-double-flat. Pitch- class 1 refers to all notated pitches C-sharp, D-flat, B-double-sharp, and so on.^

This numerical system Is adopted here to facilitate the application of

Integer notation to the four string quartets.

Interval Class

By analogy with pitch class, Babbitt's term interval class en­

tails a grouping of actual intervals into general classes. Forte

applies integer notation also to interval classes when he states that

an interval class is "one of the seven Interval classes designated by

the Integers 0 through 6".^ In Integer notation, which also eliminates enharmonic distinctions, an interval is the arithmetic difference

^Forte, The Structure of Atonal Music, p. 210.

4Ibid. 59 between pcO and pc3 is 3, and "3" is the Interval, as shown In the ex­ ample below:

Conventional Interval: minor third

Integer Interval: pc3 - pcO *■ 3

-B-

Example 16* An Interval Notated Conventionally and Expressed in Interval Class Numbers

In the above example, the actual notational spelling of pcO and pc3 is of no consequence in Integer notation. Conventionally, however, if e^ were spelled as jjl#, the interval designation Itself would change to "augmented second," an altogether different interval from the minor third in spelling and function. Thus, the conversion of pitch classes to numbers already Ignores enharmonic meanings, and, as a logical con­ sequence, integer intervals recognize no enharmonic distinctions.

The number of possible intervals in integer notation and the question of a negative difference obtained by subtraction are treated in Forte's explanation of "interval":

The interval formed by two pc integers a and b is the arlthmentic difference a-b. In order not to have to specify that a is greater than b it is assumed that the absolute value (positive value) of the difference is always taken. Thus, for example, the interval formed by 0 and 1 is 0—1 * 1; the interval formed by 5 and 9 is 5-9 ■ 4. It is not difficult to show that if the inter­ vals between all pairs of pc Integers were formed and 60

duplicates were removed there would remain the set [0 ,1 , 2,...,11]. Thus, there are 12 Intervals, corresponding to the 12 pitch—classes.5

The twelve possible Integer intervals are shown In Example 17 with con­ ventional Interval designations and numerical designations:

perfect prime minor second major second

S- 2

minor third major third perfect fourth

7 \ 1 U " ' J _ IS IS LS a ~0~ ~Q~ s - 3 4 5

diminished fifth perfect fifth minor sixth t 1 ' ft J IS & LS aa C)~ -~0~ 6 7 8

^Forte, The Structure of Atonal Music, p. 14. 61

major sixth minor seventh major seventh /------1 / a 1/ u __ w/r\ __ - uit J -e- -o- -e- 9 10 ii

Example 17. The Twelve Integer Intervals Notated Conventionally and Expressed In Integer Interval Numbers

By grouping the twelve Integer Intervals Into inversion-related pairs (explained below), the Intervals reduce to only seven Interval classes designated by the Integers 0 through 6. The procedure for

g reduction is to subtract each Integer in turn from twelve. This pro­ cedure is shown In Table 2:

g Pitch class numbers of twelve or more reduce by subtraction of the number 12 that represents the octave. In reference to pitch classes Forte States (page 3) that "any pitch number greater than or equal to 12 can be reduced to one of the pitch-class integers by ob­ taining the remainder of that number divided by 12." The procedure is called "modulo 12" (page 210) and also applies to integer Intervals as a means of assuming Inversions! equivalence. 62

TABLE 2

Subtraction of Each Integer Interval From a Constant of Twelve

Interval Invers ion—Related Constant Subtracted Interval

12 0 tm 12

12 1 - 11 12 2 10

12 3 - 9

12 4 - 8

12 5 - 7

12 6 - 6

12 7 - 5

12 8 - 4

12 9 - 3 12 10 at 2 12 11 m 1

12 12 - 0

Disregarding the first and last Inversion-related intervals In

Table 2, 12 (or 0, the perfect prime in conventional terminology), the remaining Inversion—related Intervals 11 through 1 may be grouped in pairs whose sum is 12: 63

1-11

2-10

3-9

4-8

5-7

6-6

Figure 1. Inversion-Related Pairs of Integer Intervals

Each of the above Integer interval pairs represents "an equiv­ alence class.The number of equivalence classes, or Interval classes.

Is six, and the first number of each of the six pairs, numbers 1 through

6, is taken to represent that Interval class. Two pitch classes are required to form an Interval class, and a specific interval class is expressed by the smallest arithmetic difference between two different pitch classes.

Pitch Class Set

A pitch class set is a group of different pitch classes consid­ ered as a collection. Forte defines the term numerically as "a set of distinct Integers (i.e., no duplicates) representing pitch classes."8

The pitch classes may occur either successively, as an Instrumental or vocal line, or as a simultaneity. The label pitch class set attaches primary Importance to the total combination of the pitch classes as a

^Forte, The Structure of Atonal Music, p. 14.

8Ibid.. p. 3. 64

collection that has a distinct character, as opposed to the individual

pitch classes considered in isolation.

Two pitch class sets appear in Example 18. The first one is a

simultaneity of three pitches; the second one consists of a melodic

line of four pitches. The form for expressing a pitch class set, i.e., numbers separated by commas and enclosed in brackets, follows Forte's procedure:

bo

[7, 1, 6]

Ix . .. . 1 o U Lf 0 ------A ---- [0, 5, 10, 4]

Example 18. Simultaneous and Successive Pitch Class Sets

Prime Form

Distinct pitch class sets such as those shown above may be compared with other pitch class sets by means of reduction to uniformly ordered sets called prime forms. A prime form is essentially a pitch class set expressed in closest possible ascending numerical order

(normal order) with 0 as the first integer. In Forte's words, "the 65

form of a pc set such that it is In normal order (or best normal order)

and the first integer is 0 is called a prime form."^

Let us take the first of the two pitch class sets in Example 18*

pc set [7, lt 6], as an illustrative example. The first step In reduc­

ing this set to its prime form is to arrange the Integers in ascending

numerical order [1, 6* 7]. Next, this set must be examined to see if

it represents the "closest possible ascending numerical orderThere

are two other possible arrangements of the Integers in ascending numer­

ical order. These are obtained by placing the fir3t Integer last and

adding 12 to it. The addition of 12 preserves the pitch-class equiva­

lency of a given integer (if 0 “ c, then 12 - c)*® and allows any

Integer of a pitch class set to be considered as the "first" In order

to determine the closest possible ascending numerical order. These

successive reorderings**' are shown on the following page:

^Forte, The Structure of Atonal Music, pp. 4-5.

*®This is another Instance of "modulo 12" (see footnote 12). The addition of 12, Instead of its subtraction, is employed here.

**The reorderings are termed "circular permutations" by Forte (pages 3-4 In his text). 66

[1, 6, 7]

(1 + 12 - 13)

[6, 7, 13]

(6 + 12 - 18)

[7, 13, 18]

Figure 2. Successive Reorderings of a 3-Set

The next step Is to select the closest possible ascending

numerical order, determined by subtracting the first Integer from the

last in each of the three orderings:

Difference Between First Ordering and Last Integer

1, 6, 7 6

6. 7, 13 7

7, 13, 18 11

Figure 3. Differences In Orderings for a 3-Set

Because the original ordering [1, 6 , 7] shows the smallest difference between the first and last Integers, It represents the closest possible

ascending numerical order or, In Forte's words, "normal order."

It should be noted that in some cases it Is possible to have no

single "smallest difference" between the first and last Integers. In

such cases, the first and second Integers are used as a basis for de­

termining the smallest difference and, thus, the normal order. Forte

deals with this contingency in the following manner: 67

If the least difference of first and last Integers Is the same for any two permutations, select the permutation with the least difference between first and second Integers. If this is the same, select the permutation with the least difference between the first and third Integers, and so on, until the difference between the first and the next to last Integers has been checked. If the differences are the same each time, select one ordering arbitrarily as the normal order.

Returning now to our example, the final step in the reduction

of set [1, 6 , 7] to its "prime form" is to transpose it so that the

first Integer Is 0. Transposition Is accomplished by subtracting the

first Integer of the set (in this case 1) from every member of the set.

Figure 5 below shows the procedure.

Original Set in Ascending Numerical Order [1, 6, 7]

Constant Subtracted -1 -1 -1

Transposed Set In Prime Form [0, 5, 6]

Figure 4. 3-Set Transposed to Prime Form

' Thus, the prime form of set [1, 6, 7] Is [0, 5, 6], expressed In closest

possible ascending numerical order, with 0 as the first Integer (called

"level 0" by Forte13).

The second pitch class set In Example 18 is also reduced to its

prime form below, to Illustrate the reduction of a 4—set— a frequent

number encountered both melodlcally and simultaneously in the quartets.

The set [0, 5, 10, 4] Is first arranged in ascending order and then

successively reordered, as before:

12 Forte, The Structure of Atonal Music, p. 4.

13Ibid., p. 12. 68

[ 0, 4, 5, 10]

[ 4, 5, 10, 12]

[ 5, 10, 12, 16]

[10, 12, 16, 17]

Figure 5. Successive Reorderings of a 4-Set

Differences between first and last Integers are then calculated to de­ termine the closest order:

Difference Between First Ordering and Last Integer

0, 4, 5, 10 10

4, 5, 10, 12 8

5, 10, 12, 16 11

10, 12, 16, 17 7

Figure 6. Differences In Orderings for a 4-Set

As a last step, the ordering with the smallest difference [10, 12, 16,

17], is transposed to level 0:

Ordered Set [10, 12, 16, 17]

Constant Subtracted -10 -10 -10 -10

Transposed Set [0, 2 , 6, 7 ]

Figure 7, 4-Set Transposed to Prime Form

The prime form of set [0, 4, 5, 10], then, is set [0, 2, 6, 7], the fourth ordering transposed to level 0. 69

In his Appendix 1, Forte has listed 220 prime forms for pitch 14 class sets of three to nine pitch classes. These prime forms are arranged into groups according to the number of pitch classes, and each group of prime forms is ordered according to interval class content

(see next section). Numerical names are assigned to the forms, follow­ ing this scheme:

It is convenient to have names for the prime forms so that a pc set can be referred to without recourse to a cumber­ some description of some kind. Accordingly, each prime form has been assigned a name consisting of numbers sep­ arated by a hyphen. The number to the left of the hyphen is the cardinal number of the set; the number to the right of the hyphen is the ordinal number of the set— that is, the position of the prime form on the list. For example, 5—31 . . . is the thirty-first set on the list of sets with cardinal number 5 . ^

The prime forms for 3—sets and 4-sets are included in

Appendix A (see pp. 184-89), because they are the ones most frequently encountered in the horizontal and vertical planes in the four quartets.

The principal exceptions to this rule are the horizontally used 5-sets found in the first and third movements of the Coulthard work. However, because 5—sets are not used consistently throughout any of the quartets, prime forms for sets larger than 4-sets are not included. Both integer notation and musical notation are provided for the prime forms and their inversions. For example, set 3—1 is expressed as [0, 1, 2] in integer notation and c-c#—d in musical notation. (Forte does not supply either the inversions or the musical notation.)

■^Forte, The Structure of Atonal Music, pp. 179-81.

15Ibid.. pp. 11-12. There are twelve distinct prime forms for 3-sets and twenty-

nine prime forms for 4-sets.^ The numbers for these and larger sets

are shown In Table 3.

TABLE 3

Numbers of Prime Forms for Pitch Class Sets

Number of Pitch Classes Prime Forms

3 12 4 29 5 38 6 50 7 38 8 29 9 12

As one can see, the number of prime forms Increases with the number of pitch classes through number 6, and then declines symmetri­

cally from that point. Thus, pc sets with five or seven pitch classes both have 38 possible prime forms. The reason for this symmetrical alignment is explained below, using 3-sets and 9-sets for Illustration

For every 3-set designated as a prime form, there remains a

9-set that, when transposed properly, constitutes a prime form also.

16With Inversions the numbers Increase to twenty 3—sets and forty-six 4-sets (see Appendix A)• Some forms are equivalent to their prime form when Inverted and are not considered. For example, the set [0, 1, 2], when Inverted becomes [0, 11, 10]. Rearranged, the Inver­ sion is [10, 11, 12 (0)] and transposed up two Integers Is [0, 1, 2], the same set. The majority of prime forms, though, do have Inversions that are not equivalent. 71

These 9—sets are the complements of the original 3—sets. Since there are twelve distinct prime forms (and only twelve) for 3—sets, It follows that there are also only twelve complements containing nine pitches that will represent prime forms on that level. Perle clarifies this point as follows:

Since only the content of each chord is to be specified and not the particular order of arrangement of the notes consti­ tuting the chord, the elements which remain after each of the different three-note chords is In turn subtracted from the chromatic scale will constitute all the different nine- note chords, and vice-versa. Similarly, the number of eight—note combinations will be identical with the number of four-note combinations, the number of seven-note combin­ ations with the number of five-note combinations, etc., the only unique figure being the number of six-note chords.U

Inversion of a prime form (or any other set) results when each pitch class number Is subtracted from 12. This relationship may be conceptualized in notation by considering _£ the axis point, the single pitch classes being Inverted on the basis of distance from £. To Il­ lustrate this point, the set [0, 5, 6 ] from Figure A is Inverted as follows: ------1 > L f Vi, /■ ■ o i/ u h n 1 . -o- -e- [0, 5, 6] [0, 7, 6] 12-0 - 12(0) 12-5 - 7 12-6 - 6

Example 19. Prime Form and Inversion Notated Con­ ventionally and In Integer Notation

■^Perle, Serial Composition and Atonality. p. 106. 72

Interval Vector

A prime form, as well as any of Its variants assumed to be equivalent by Inversion or transposition, has a characteristic Inter­ val class content. The Interval class content is determined by comput­ ing the differences between pairs of Integers In the given set, reduc­ ing the resultant Integer Intervals, and assigning numerals to Interval classes. The expression of the total Interval class content for a set

1 ft 1 A In an "ordered array" is termed by Forte an "Interval vector."19

This label Is defined more precisely In his Glossary of Technical

Terms:

Interval vectors An ordered array of numerals enclosed in square brackets that represents the Interval content of a pc set. The first numeral gives the number of in­ tervals of Interval class 1 , the second gives the number of Intervals of Interval class 2 , and so on.20

Because there are six interval classes (see page 63), an Interval vector has six digits.

Using again the two pitch class sets in Example 18 (p. 64), let us determine the Interval vectors for sets [7, 1, 6] and [0, 5, 10, 21 4]. The procedure is shown in Figure 8 : the Integer Intervals are

^Forte, The Structure of Atonal Music, p. 15.

•*~9Ibid. Forte gives credit for ordered array to Donald Martino In a footnote. 20 Ibid.. p. 210. Use of the phrase "represents the Interval content" is perhaps misleading. "Specifies the Interval class content" is more accurate, because actual Intervals are subsumed first Into Integer Intervals and then into Interval classes (see pages 58-63). 21 Because each of these sets is considered the equivalent var­ iants of its prime form, the Interval vectors will be the same for each prime form. 73

3-Set; [7, 1, 6]

Step 1 Order into pairs: 7-1, 7-6, 1-6

Step 2 Obtain Integer Intervals by subtraction: 7—1 — 6 7-6 - 1 1-6 - 523

Step 3 Convert integer intervals into interval classes: (no conversion necessary here)2^

Step 4 Calculate Instances of 1 0 0 0 1 1 each interval class: (id)(ic2)(ic3)(ic4)(ic5)(ic6)

Step 5 Express as Interval vector: 100011

4-Set; [0, 5, 10, 4]

Step 1 Order into pairs: 0-5, 0-10, 0-4, 5-10, 5-4, 10-4

Step 2 Obtain Integer Intervals by subtraction: 0-5 - 5 0-10 - 10 0-4 - 4 5-10 - 5 5-4 - 1 10-4 « 6

Step 3 Convert integer intervals into Interval classes: — 5 (no conversion necessary) 10-2 (12-10 - 2)25 4 5 1 6

Step 4 Calculate instances of 1 1 0 1 2 1 each interval class: (lcl)(lc2)(ic3)(lc4)(ic5)(ic6)

Step 5 Express as interval vector: 110121

Figure 8 . Procedure for Determining Interval Vector

2^See page 59 for an explanation of negative difference

2^See page 63. 25 ‘‘■“'See page 63. 74 obtained by calculating the differences for each pair of integers In 22 the set (steps 1 and 2); the Integer Intervals are converted to In­ terval classes (step 3); and the results are expressed as an Interval vector (steps 4 and 5).

A point to consider in calculating total Interval class content

Is that, regardless of the specific configuration of the Interval vector, the sum of the instances remains constant within a given cate­ gory of pitch classes* For example, the sums of the numbers In the

Interval vectors In the preceding figure are 3 (100011) and 6 (110121), based on the number of pairs in each case. The larger the number of pitch classes In a set, the larger the number of comparisons in the

Interval vector. This progression Is illustrated In Table 4.

TABLE 4

Numerical Relationships Between Fitch Class Sets and Interval Vectors

Number of Number of Interval Fitch Classes Class Instances

3 3 4 6 5 10 6 15 7 21 8 28 9 36

22I have followed Forte's order of pair comparison for conven­ ience (see p. 15 In his text). 75

Any pitch class set may be described by an interval vector* The

interval vectors for all 3-sets and 4-sets are given in Appendix B on

pages 190-91. The tables In this appendix amplify Appendix A. To ex­

plain the format, the first three entries are given In Table 5.

TABLE 5

The Format for Entries in Appendix B

Numerical Prime Interval Name Form Inversion Vector

3-1 0 ,1,2 (12) 0,11,10 210000 3-2 0,1,3 0,11, 9 111000 3-3 0,1,4 0,11, 8 101100

The flr3t column, "Numerical Name," contains a simple numeri­

cal designation for each pitch class set. In the entry 3-1, "3" indi­

cates the number of pitch classes In the set, and "1" identifies this

entry as the first of the 3-sets. The second column gives the prime

forms in an ordered succession, and the third column gives the inver­

sions of the prime forms. The fourth column, "Interval Vector," pro­

vides the six-digit description of the Interval class content. For

example, the first entry means that 210000 is the interval vector for

the set [0, 1, 2], designated 3-1.

The systematic arrangement of Interval vectors provides the basis for the ordering of Appendix B. Specifically, those vectors In a given category, such as 3-sets, are ordered according to those vectors

that contain the greatest number of Interval class 1. As the example 76 above shows, the vector 210000 precedes both 111000 and 101100. In cases where the first digit (lei) Is Identical, the second digit, and then the third, and so forth, provide the basis for ordering. The re­ sulting arrangement Is In simple descending numerical order with the exception of Z—related pairs, explained below.

Although every pitch class set has a unique prime form, not every one has a unique Interval vector. There are cases In which two pitch class sets with different prime forms, nonequivalent regardless of transposition or Inversion, have identical Interval vectors. When such a relationship exists, Porte labels them Z-related pairs. ^

Chosen arbitrarily, the letter "Z” is simply convenient for designating this exceptional relationship. Z-related pairs are illustrated In

Example 20.

- 6-

Name PCS Vector

4-Z15 0, 1, 4, 6 111111 4-Z29 0, 1, 3, 7 111111

Example 20. A Comparison of Z-Related Prime Forms

As one can see, the letter Z is included In the designation for each set and the vector they share Is 111111.

2fi Forte, The Structure of Atonal Music, p. 21. 77

The particular sets In Example 20 are described as "all-Interval tetrachords," a label that is appropriate if "interval class" is under­ stood to be Implied by "interval" in this context. In Forte*s words,

• • • the two forms of the all-interval tetrachord, 4—Z15 and 4-Z29, are so constructed that they are not reducible by transposition or by inversion followed by transposition to a single prime form, yet they have identical vectors [111111].27

These two sets are unique among the 4—sets, being the only sets .in that category that either contain an instance of every Interval class or stand In the Z—relationship. Forte claims that the all-Interval tetra- 28 chord "has a very special place in atonal music," although his only specific reference is to Schoenberg's George Lleder. Opus 15/1. The quartets studied in this dissertation give no particular emphasis to the all-interval tetrachord.

Similarity Relations

Forte has developed measures of similarity for comparing pitch class sets that contain the same number of pitch classes. 29 In his

Glossary of Technical Terms he states that "similarity relations" con­ cern "ways in which two non-equivalent sets of the same cardinal number may be compared for structural similarity and difference."30

There are four basic types of similarity relations that may exist in­ dividually or In combination. The first of these, designated the "Bp

27 Forte, The Structure of Atonal Music, p. 21.

28Ibid.. p. 1.

2^Ibld.. p. 46.

30Ibid.. p. 211. 78 relation," concerns pitch class sets; the remaining three, termed "R^," 31 "Rj/* and "R0," concern interval vectors. Table 6 below presents a concise description of these four similarity relations. Following the table is an explanation of each relation.

TABLE 6

Basic Descriptions of Similarity Relations

Relation Meaning

*P Maximum similarity of pitch classes R1 Maximum similarity of interval class, with interchange feature

*2 Maximum similarity of interval class, with­ out Interchange feature

Minimum similarity of interval class

The first condition, the R. relation, involves maximum slmllar- P lty of pitch class content in a comparison of two non-equivalent pitch class sets. "Maximum similarity" means that at least one form, gener­ ally the prime form, of a pitch class set contains all but one of the pitch classes that comprise at least one form of another pitch class set (also generally the prime form). The critical point of Rp rela­ tions, regardless of the forms, is that all except one pair of pitch classes in two pitch class sets correspond-—if there were complete cor­ respondence, the pitch class sets would be identical, of course. An

3*ibid.. pp. 47-49. 79 example of two pitch class sets In the Rp relation with respect to their prime forms is presented In the following table:

TABLE 7

— Maximum Similarity of Two Pitch Class Sets

Name PCS Vector

4-1 0, 1, 2, 3 321000 4-3 0, 1, 3, 4 212100

Careful examination of these pitch class sets shows that they share three of the four pitch classes (0, 1, 3) and that the remaining ones

(pitch classes 2 and 4) are not shared. The Rp relation, then, signi­ fies a close similarity of pitch class content between two sets.

As indicated above, the prime forms do not always serve to indicate the presence of the Rp relation. The case given below is one

In which the prime forms do not show that the sets 4-3 and 4-4 are in­ deed In the R*. relation with each other.

TABLE 8

The R Relation Not Evident in Prime Forms P

Name PCS Vector

4-3 0, 1, 3, 4 212100 4-4 0, 1, 2, 5 211110 80

A comparison of pitch classes shows that the third and fourth pairs

(3,2) and (4,5) do not correspond. Nevertheless, there is a means for demonstrating the presence of the R^ relation for these two sets.

Successive reorderings of each set, followed by transposition to level

0, shown in Table 9, illustrate the relation. The Rp relation is demon­ strated by comparison of the transposed versions of the fourth order for the two sets: [0, 8, 9, 11] and [0, 7, 8 , 9]. Fitch classes 0,

8 , and 9 correspond, while pitch classes 7 and 11 do not.

TABLE 9

Rp Relation Revealed Through Transposition

Form Successively Reordered Set 4-3 Set 4-4

Prime Form (First Order) o, 1 , 3, 4 o, 1, 2 , 5

Second Order 1 . 3, A, 12 1. 2 , 5. 12

Second Order at Level 0 o, 2 , 3, 11 o, 1. 4, 11

Third Order 3, 4, 12, 13 2 , 5, 12, 13

Third Order at Level 0 o, 1, 9, 10 o, 3, 10, 11

Fourth Order 12, 13, 15 5, 12, 13, 14

Fourth Order at Level 0— R^ 0 , 8 , 9, 11 o, 7, 8 , 9

The second relation in Table 6, the Rj^ relation, involves max­ imum similarity of interval class content with the Interchange feature.

"Maximum similarity'* means that four pairs of the six pairs of digits in two interval vectors coincide (and two pairs do not coincide). The 81

"Interchange feature" means that the remaining two pairs of digits share

the same numbers, but that these same numbers occur In different digits*

The verbal description of this relation may be difficult to follow, but

the following table clarifies the meaning of the Rj. relation*

TABLE 10

R^— Maximum Similarity of Interval Class Content with Interchange Feature

Name PCS Vector

4-1 0 , 1, 2 , 3 321000 4-23 0, 2, 5, 7 021030 t 1 1 1 1 1 3 0 0 3

Comparison of the above two Interval vectors reveals that four of the six Interval class digits correspond: lc2 (2 Instances), lc3 (1 in­ stance), ic4 (0 Instances), and ic6 (0 instances)* The remaining digits, icl and lc5 do not correspond. However, the same pair of numbers, 0 and

3, are found In both digits* Because the same pair occurs, although in different digits, the pair is designated as Interchangeable. The R^ relation is a significant one in terms of Interval class content, be­ cause the Interval vectors correspond as closely as possible without being identical*

It follows, then, that the relation is like the R^, except that the Interchange relationship is lacking* To be explicit, the R2 82 relation, the third in Table 6 , is defined as a relation with a maximum

similarity of Interval class content, but without the interchange feature. An example of the ^ relation follows:

TABLE 11

1*2— Maximum Similarity of Interval Class Content Without Interchange Feature

Name PCS Vector

4-14 0, 2, 3, 7 111120 4-15 0, 1, 4, 6 111111 ( t 1 1 i 1 I 1 20 II

The digits in the two vectors above match in the first four Interval classes, but not in the last two. In this case, no matching pairs in the dissimilar digits can be found— hence, the interchange feature is not present. The K2 relation Is considered to be a close one, not, however, as close as the R-^ relation.

The fourth relation in Table 6, the RQ relation, involves a minimum similarity in the interval class content of two vectors.

"Minimum similarity1* denotes a complete lack of correspondence in the numbers for like Interval classes. An example of two Interval vectors in the RQ relation appears in Table 12. 83

TABLE 12

Rq —Minimum Similarity of Interval Class Content

Name PCS Vector

4-28 0, 3, 6, 9 004002 4-29 0, 1, 3, 7 111111

A comparison of the numbers for each Interval class reveals dissimilar­

ity throughout. An RQ relation, then, means that two interval vectors

are unrelated.

The combination of these relations provides a more sophisticated means of comparison between two sets than the relations used In isola­

tion. Forte treats these three combinations: 1. RX, Rp

2* ®2» Kp

3* ^o» Kp

In each of the three cases, a dual comparison of pitch class and Inter­

val class demonstrates the degree of similarity. Descriptions and ex­

amples of these combined conditions now follow.

Two pitch class sets stand In an R p Rp relation to each other

if the Interval vectors share maximum similarity with the Interchange

feature and if some order of the prime forms shares the same pitch

classes with only one exception. This maximum degree of similarity

with respect to both pitch class set and Interval vector content denotes 84 an even closer relationship then either of the two R-relatlons existing

in Isolation. An example of this combination appears below.

TABLE 13

The R^, Rp Relation

Name PCS Vector

4-2 0* 2, 4 221100 4^3 0, 1, 3, 4 212100 21 12

The second combination of similarity relations, R2, Rp, results when two Interval vectors share maximum similarity of content without the Interchange feature and some order of the prime forms shares the same pitch classes with only one exception. The degree of similarity is not quite as strong as it is in the R^, R_ combination:

TABLE 14

The R2 , Rp Relation

Name PCS Vector

4-17 0, 3, 4, 7 102210 4-18 0, 1, 4, 7 102111 1 1 2 0 1 1 85

The third dual relation, Rq , Rp, exists when two vectors share

no identical pairs of interval class content, yet do share common pitch

classes with only one exception. This somewhat paradoxical relation Is

shown below.

TABLE 15

The Rq , Rp Relation

Name PCS Vector

4-1 0, 1, 2, 3 321000 4-5 0 , 1 , 2 , 6 210111

Forte has graphically codified the four basic similarity rela­

tions and the three dual relations for all 4-sets, 5-sets, and 6-sets.

These codifications are Included In an appendix to The Structure of

Atonal Mu3ic but are not Included in this dissertation. Although simi­

larity relations are Identified several times In the course of this

study, their frequency is not sufficient to justify the Inclusion of

Forte's extensive tables.

Supplemental Procedures Employed In This Study

Hypothesis Testing

The principal objective of this study is to apply Forte's con­

cepts in an hypothesis-testing format. An orderly sequence of 1) Hy­

pothesis formulation, 2) Hypothesis testing, 3) Tabulation of results,

and 4) Interpretation of results provides the formal structure of

Chapter III (Horizontal Sets) and Chapter IV (Vertical Sets). Forte's 86

purpose In Structure is to set forth the procedures and give passing

illustrations^^ Qf how these procedures are relevant. The objective

here is to test those procedures in a comprehensive analysis of unord-

ered sets in the selected works.

One hypothesis to be Investigated is that there will be some

prime forms that, by their degree of frequency (usage), do not appear

to be randomly employed. The hypothesis must be treated In both the

horizontal plane (linear writing) and in the vertical plane (simultane­

ities) . Another hypothesis is that there will be musically significant differences in prime form frequencies among the composers. Statistical procedures are essential for testing the latter hypothesis (see below, pages 92-94.

The tabulation of the results provides (for each quartet) nu­ merical counts and percentage frequencies of prime forms for 3—sets and

4—sets. For example, the 3-sets in the horizontal plane of the

Coulthard quartet are considered to be a "population," so that the percentages for the twelve prime forms give a total of 100 percent.

The 4—sets in the horizontal plane 33 are considered to be another popu­ lation, as are vertical 3—sets and vertical 4—sets. In all, there are sixteen populations of prime forms— four for each quartet.

The percentage results in each population are interpreted ac­ cording to the degree of conformity to (or divergence from) a

32Forte does apply his procedures to his own condensed versions of passages in Stravinsky's "Sacrificial Dance" from The Rite of Spring and the third piece in Schoenberg's Five Pieces for Orchestra, Op. 16, 33 For simplicity the expressions "horizontal 3-sets" and "horizontal 4-sets" are used. 87

theroretical random distribution. For Instance, given twelve prime

forms for 3-sets, one might expect each form to occur randomly with an

8.33 percent frequency. A greater or lesser frequency than the theo­

retical one Is Interpreted as signifying that the frequency is probably 34 not random.

In relation to the hypothesis of non—random usage within the

quartets, a frequency cutoff approach is appropriate. The assumption

is that if the number of forms required to comprise at least 50 percent

of the frequencies is considerably less (or more) than half of the

total forms, the hypothesis appears to be confirmed. The hypothesis of

significant differences among composers is Interpreted by means of the

Y-Square statistic, described below on pages 92-94.

Distributions

In order to test the hypotheses described above, It Is necessary

to develop a system for calculating relative frequency distributions.

The system I have followed Involves five sequential steps that yield the required numerical information: 1) demarcation of sets, 2) encoding of

the sets Into integer notation, 3) Identification of the prime forms of encoded sets, 4) orderly tabulation of these prime forms, and 5) con­ version of tabulated counts to percentages for frequency distribution.

The demarcation of the sets In the music requires careful con­

sideration of the definition of pitch class set: "a group of differ­ ent pitch classes considered as a collection. *' The criteria used to

34 Given the exploratory nature of this Investigation, no rig­ orous statistical proof is attempted. 88 delimit collections were determined for this Investigation after a thorough examination of the music. Horizontally, five criteria were used to demarcate 3-sets and 4—sets: 1) phrasing, 2) articulation,

3) rhythm, 4) rests, and 5) melodic direction. Vertically, only one criterion was employed: pitch classes sounding at the start of each beat. The specific means for applying these criteria are discussed In

Chapters III (pages 101—06) and IV (pages 133-37)•

Encoding the delimited pitch class sets Into integer notation, the second step, Is most conveniently organized in chart form. The chart should Include the pitch class sets expressed In Integers, musi­ cal locations for the sets, and their prime form identifications (3-1,

3-2, etc.).

Identification of the sets according to prime form, the third step, is most rapidly accomplished by following the "improved proced­ ure for prime form Identification" described below on pages 89—92.

Special tables for Implementing this technique are Included In Appendix

C (pages 192-202). They are helpful for dealing with large numbers of sets, because they permit the prime forms to be identified quickly.

Tabulation and computation of percentages, the final step, in­ volves conversion of the tabulations to percentages to obtain a rela­ tive frequency distribution. After completing the simple arithmetic process necessary, a direct comparison of data for different pieces Is possible. Given a sufficient number of counts, It may be possible to demonstrate, for example, that the difference In frequency of set 3-1

In hypothetical work A, with many Instances, and work B, with few In­ stances, Is musically significant. Processing the tabulated counts by means of a computerized statistical program was most helpful for this step.

Improved Procedure for Prime Form Identification

I have developed a simple and efficient procedure for identify— ing pitch class sets. It evolved during the course of the investiga­ tion when hundreds of pitch class sets had to be examined and identi­ fied. Forte's procedure involves writing out successive reorderings of the set, obtaining differences between first and last integers of those reorderings, and transposing the version with the smallest dif— ference to level 0.

My procedure relies entirely upon the differences between the

Integers In the set when taken successively in numerical order. For example, the pitch class set [1, 2,5] contains the following differ­ ences between the successive numbers: 1^ (2 minus 1) and 3_ (5 minus 2).

The order of differences, then, is 1, 3. There is only one prime form in the 3-sets with this order of differences, and that is prime form

3-3. So, the order of differences is sufficient to identify the form, and for any given order of differences there is only one corresponding prime form, regardless of the number of Integers.

For quick reference, I have tabulated the orders of differences for all of the prime forms with three, four, six, seven, eight, and nine pitch classes. These tables are found in Appendix C (pages 192-

202). In the interest of simplicity only those orders derived from the normal order of the prime forms are given. (The technique for dealing

^This procedure is explained in detail on pages 64-68. 90 with sets la which the normal order Is not immediately apparent due to transposition or Inversion is explained below.) Thus, there are twelve orders of differences given for the 3-sets, twenty-nine orders given for the 4-sets, and so forth.

Actual examination of pitch class sets in the quartets revealed that many sets occur in such a form that a simple numerical ordering does not provide the pattern of differences necessary to Identify the prime form. In such cases it is necessary to reorder the set succes­ sively (see pages 65-66) to obtain an order of difference corresponding to the prime form. No transposition is necessary. For example, the

4—set used earlier, pc set [0, 4, 5, 10], contains the order of differ­ ences 4, 1, 5. Since this order does not correspond to any of the orders under 4-sets (Appendix C, page 193), it is necessary to succes­ sively reorder the set, so that an order of differences is found that will match. Table 16 shows the set with its reorderings and the dif­ ferences:

TABLE 16

Successive Reorderings and Differences

Set Differences

10, 12, 16, 17 91

Because prime forms are arranged In the closest possible ascend­ ing numerical order, It follows that the differences between the Inte­ gers will be small (as compared to larger differences for less closely ordered forms, or reorderings). Thus, the fourth order of differences in Table 16, which is 2, 4, 1, Is the likely choice, because its largest difference 13 only 4. This order does match prime form 4-16.

The original set [0, 4,5, 10] is, therefore, a variant of 4-16.

Although this technique of reordering a set and finding the orders of difference may appear to be time-consuming, in actual prac­ tice a quick visual examination generally suffices for obtaining the smallest order of differences. For example, in the original set [0, 4,

5, 10] it is apparent that any ordering with 10 following 5 will pro­ duce a difference of 5, which is not the small difference that one might expect in a closely ordered form (prime form). Placing 10 as the first digit (mentally) solves the problem and gives successive differ­ ences of 2_ (10 minus 0, or 12), 4^ (0 minus 4), and 1 (4 minus 5). Also, the addition of 12 to Integers (for reordering) is not necessary in the visual procedure, when 0 may be considered to mean 12 if it is appro­ priate (as in. 10 minus 12, the first difference). After some practice with the procedure, any 3-set or 4—set at any level of transposition or inversion may be quickly identified through a visual examination of its Integer notation expressed as it occurs in the music.

To facilitate this procedure for the use of others, I have arranged the tables of differences (technically, "order of differences," or "Integer interval sequence") by the orders of differences themselves, so that a difference of 1, 2, 3 immediately precedes a difference of 92

1, 2, 4, for example. Although most forms have two difference orders

(counting the inversion), those orders that are palindromes (e.g., 2,

1, 2) are of course listed only once to account for both the prime form and Its equivalent Inversion.

Statistical Comparisons

The Statistical procedure employed in this study is Goodman's 36 Y-Square. Previously employed In a musicological study by Richard

Sentierl, 37 this program has proven itself effective when applied to numerically expressed musical data. Here, the Y-Square program has effectively provided a measure of statistical confidence In comparisons between composers based on the Individual prime forms.

An understanding of the Y-Square Statistic may be achieved by referring to an explanatory model. This model gives the rationale for

Y-Square and explains the type of statement that can be made with the use of the statistic. The model Is purposefully general in order to separate It from the results obtained, which may then be interpreted with reference to the model. For this reason concrete numbers are not

Introduced In the following explanation.

36 Leo A. Goodman, "Simultaneous Confidence Intervals for Con­ trast Among Multinomial Populations," In The Annals of Mathematical Statistics. XXXV, No. 2 (June 1964), 716—25.

■^Alfred Richard Sentierl, "A Method for the Specification of Style Change in Music: A Computer-Aided Study of Selected Venetian Sacred Compositions from the Time of Gabrieli to the Time of Vivaldi" (unpublished Ph.D. dissertation, The Ohio State University, 1978). 93

Composer A P11 P12

X11 X12

Composer B P P21 22

X21 5[22

Figure 9. Y—Square Model

In the Y-Square model, the "P" figures represent theoretical propensities for Composer A and Composer B to use given prime forms.

These theoretical propensities are postulated as existing In the minds of the composers, whether consciously or Intuitively. To actually measure the propensity for a composer to employ given forms (Indicated by "F") Is not possible, but we may make an educated guess based on the "X" figures— the results obtained by counting prime forms and dividing the counts Into the total, giving a proportion of usage In some music of that composer. "X" is an approximation of "P", then. It is our best possible guess at the true value of "P", which we never confidently know. "P" is assumed to be Imbedded in the thought pro­ cesses of the composer, who has an "urge" to use given sounds repre­ sented by the prime forms.

To Interpret the model in terms of prime form usage, let P ^ equal the propensity for Composer A to use prime form 3-1, and let

P22 equal Composer B's propensity to use that same form. Then, X ^ and

X21 represent the actual proportions (percentages) obtained for usage 94

of prime form 3-1 by the two composers. Pritne forms 3-2 through 3-12,

as well as all the 4-set forms, are treated In like manner.

The purpose of the model is to locate musically significant differences In frequency of usage in two composers* use of the same

form. Although the quartets selected are not random samples of the composers* works, the statistical procedures are still employed to high­

light musically significant results. For the sake of prime form com­ parison, the assumption is temporarily adopted that the samples are random, and a 95 percent level of confidence is constructed to show where differences in prime form usage might lie. For an exploratory data analysis this procedure is Justifiable and has the power to

strongly indicate that there are musically significant differences in prime form usage. For this reason the statistical procedures for con­

structing a 95 percent level of confidence were employed. 7-Square is chosen because the statistic allows the construction of simultaneous

"confidence intervals" for comparison of every prime form between each pair of quartets.

Prime Form Comparison in the Quartets

A comparison between the total spectrum of possibilities of set usage (see page 70) and the actual sets that exist in a given work provides pertinent information. Although the specification of what is not used in a work may appear to be purely negative, there are appro­ priate reasons for the consideration of such information. First, the number of chosen sets in a work gives an idea of the extent that a given composer is restricting himself in that work. Second, the types of sets that are not employed may be compared with those that are

employed, giving insight into a composer's style when compared to other

composers. A description of what is not used, then, is the necessary

complement to the statement of what jLs used.

Prime form counts permit both a uniform description based on

systematic numerical relationships (see pages 64-65) and a quantitative

approach based on the numbers of occurrences of prime forms throughout

the quartets. The descriptions of the selected pitch class sets (see

pages 87-89) are uniform In that every set is reduced to a theoretical

(not necessarily musical) form of expression, of which the musically notated sets are regarded as variants. If the demarcation of pitch

class sets In both the horizontal and the vertical planes is methodic­

ally followed according to specified criteria, this methodical approach allows comparisons at two levels: 1) within composers individually and

2) between composers.

For the Individual composer, the prime form counts (numbers of

occurrences) may be compared, first, for randomness of usage, horizon­

tally and vertically. After compiling the counts and percentages de­

scribed under Testing and Results in Chapters III and IV, the highest

(and the lowest) percentages may be examined for their degree of cor­ respondence to theoretically random percentages. Second, the most fre­

quently used forms may be compared by way of examples in the music for

intervallic characteristics. Third, the results of prime form usage in

the horizontal plane and in the vertical plane for each composer may be compared to determine whether or not there exist significant relation­

ships. This third form of comparison is located at the end of Chapter IV. 96

Between composers, the prime form counts may be compared for each particular prime form in pairs of composers, in both the horizon­ tal and vertical planes. For exampleV^'Coulthard * s use of prime form

3—1 is compared with Champagne's use of 3-1, horizontally and vertic­ ally. The rationale for these sets of comparisons between each pair of composers is to determine if the composers exhibit stylistic differences that may be accounted for by prime form usage.

It should be noted that, although prime forms are by definition

(see page 64) numerically "ordered" sets, the sets are not considered

"ordered" In the sense that either the horizontal sequence of the ver­ tical arrangement would cause them to be regarded as nonequivalent (in reference to a given prime form). On the contrary, any arrangement of pitch classes reducible to a given prime form is equivalent to any other arrangement of those same pitch classes. Forte makes this point in a 1964 article:

If the order of succession of the members of a pitch-set is considered significant we are dealing with an ordered set. For example, if we regard [0, 1, 2] as distinct from [0, 2, 1] we are concerned with ordered sets. If the order of succession of the elements of a set is not of interest we are concerned with an unordered set. In this case we do not distinguish the set [0, 1, 2] from the set [0, 2, 1]. For example, when we consider simultaneous musical state­ ments of pitch-sets we are usually concerned only with un­ ordered sets.

The technique of reducing unordered sets to prime forms in the four quartets is intentionally limited to those sets with either three of four different pitch classes. After experimentation with

38 Allen Forte, "A Theory of Set-Complexes for Music," Journal of Music Theory (VIII, Mo. 2, Winter, 1964), p. 139. This article forms the basis for The Structure of Atonal Music. measurement of sets with up to eight pitch classes, it was found that

3—sets and 4—sets were most suited to consistent measurement throughout the string quartets. In the linear plane the great majority Can esti­ mated 98 percent) of pitch classes could be delimited by either 3-sets or 4-sets using criteria explained In Chapters III and IV. Vertically, most beats were found to contain either three or four pitch classes, although the sparseness of the Weinzweig quartet resulted In many sets of only two pitch classes. Nevertheless, sufficient numbers of 3-sets and 4-sets In Weinzweig *s quartet were found for comparison with the other composers.

Prime form comparisons, then, are systematic and based on quantification of unordered sets. Prime form usage in each composer's quartet is examined, and comparisons between composers are made.

The delineation of prime form patterns is of particular con­ cern. If there are underlying patterns in the Instrumental lines that are obscured by variant orderings, transpositions, and inversions, the reduction of these sets to their basic patterns (prime forms) may show recurring combinations. The assumption that patterns do exist In the horizontal plane is treated In the next two chapters. CHAPTER I I I

COMPARISON OP HORIZONTAL PRIME FORM SETS

This chapter presents a study of prime forms as they occur in

the horizontal, or linear, plane of the four selected string quartets.

The Investigation follows an experimental procedure, methodically con­

sidering the formulation of hypotheses, the collection of data to test

those hypotheses, and the Interpretation of the results.

Four Hypotheses

Four hypotheses are formulated for verification. The first

three refer to the quartets considered separately, and the last one refers to a comparison of the quartets.

Hypothesis 1— When the four Instrumental parts of a given quartet are segmented according to musically defined criteria, analysis of the resulting sets will yield meaningful results.

"Musically defined criteria" presupposes that the linear writ­

ing Is In a style that provides clear Indications of phrasing and articulation. This is the case in each of the selected quartets. The procedure followed, that of relying upon conventional criteria for grouping pitches, is consistent with Forte1s recommendations for "seg­ mentation," his approach to selecting sets for analysis.^ The approach

^See page 179 below.

98 99

here, however, differs from Forte in that all available, mutually ex­

clusive sets (i.e., no notes in one set are also In another set) are accounted for. Specific sets are not Isolated for study (as Forte does) but are Incorporated as an Instance of a given prime form. In

this Investigation the numbers of instances of prime forms are more

important than intricate relationships between specific sets. If the

tabulations of set occurrences exhibit musically important differ­

ences, one may then conclude that the application of conventional cri­

teria for segmentation has produced relevant sets for study.

Hypothesis 2— In a given quartet there will be manifest­ ly different frequencies in the usage of prime forms-—some will be used more often than others.

The second hypothesis implies that the composer has either con­

sciously or unconsciously selected those types of sets that may be reduced to certain prime forms. Whether the composers had any knowl­

edge or intention of set theory application is of no concern in this context. Rather, the conspicuous use of some prime forms over other prime forms is the condition to be affirmed or negated. The usefulness of prime form tabulations is not dependent upon the composerTs aware­ ness of the concept. Pitch class sets are merely groups of pitches that may be reduced to prime forms for the sake of analysis. Frequency and similarity comparisons are meaningful because the selected sets, by virtue of their existence within the equal-tempered scale system, may be considered as expressions of prime forms. In reference to the tabular comparisons of prime form usage between each composer, it is assumed that for any given form each composer has a certain degree of 100 propensity (which we can never confidently know) to use that form. The sets we can measure In his music give us an approximation of his pro­ pensity for given forms (whether conscious or intuitive).

Hypothesis 3--In a given quartet the most frequently used set forms will have close similarity rela­ tions. ^

The third hypothesis asserts that certain frequently occurring set forms will reveal patterns, such as a similarity of Interval class content, as measured by similarity relations.^ This similarity may take two forms. First, frequently used prime forms will contain con­ sistently high counts of particular interval classes (as shown in the interval vectors), and necessarily lower counts of other interval classes. For example, prime forms with high counts of icl and ic2 will have lower counts of ic3, ic4, lc5, or lc6. In a 3-set that has three intervallic comparisons^ 2 counts of any single interval class must be judged "high." In a 4—set (six intervallic comparisons), a 3-count Is

"high," although 2 counts of an interval class may still be taken as

"reasonably high."

As a second form of comparison, prime forms may be contrasted using Forte's similarity relations. If, for example, two given prime forms are the most frequently used in a particular composer's quartet, it is hypothesized that these forms will share similarities of pitch

9 "Set forms" is taken to mean "prime forms" with an emphasis on their specific representation in the music, appropriate to this hypothesis• 3 See pages 77-84.

^See page 74. 101 or Interval class. As with the other hypotheses, this assertion will be either confirmed or negated to some degree by the tabulations.

Hypothesis 4-— A comparison of the quartets will reveal significantly different frequencies in the usage of prime forms.

In testing the fourth hypothesis, the Y-Square statistic^ en­ able us to state with some degree of confidence whether or not differ­ ences in prime form frequency between quartets 13 musically significant.

It is hypothesized that reliable differences will be found between the quartets, because they represent different styles of composition. Ob­ jective measurement provides a tool for both stylistic description and stylistic contrast in this context.

Procedures for Linear Segmentation

In order to either confirm or negate these four hypotheses, it was necessary to select appropriate sets in the horizontal plane for reduction to prime forms. Groupings (or Forte's "segmentation”) in the instrumental lines** were delineated on the bases of musical cri­ teria— articulation, melodic direction, phrase marks, rests, and rhythm. For the many passages in which these factors occur in combin­ ation with each other, it was necessary to construct a hierarchy, or order of preference, for segmenting the passages consistently. Phrase marks were considered the clearest indication of the composer's intent that certain pitches are to be thought of as a unit. Articulation and rhythm were considered next, with rests and melodic direction employed

^See pages 92-93 above. ^"Horizontal plane," "instrumental lines," and "linear writing" are considered as approximate equivalents in the context of this study. 102

Co a leaser extent as Indicators of set groupings. These five criteria

were sufficient to account for the great majority of the pitch classes

in the instrumental lines. In some problem cases two or more criteria might apply, resulting in conflicting indicators. In such cases the

single criterion judged to he the most applicable was chosen. The

hierarchy of criteria is shown below, followed by examples from the

quartets to Illustrate their application in set selection.

1. Phrase marks

2. Articulation

3. Rhythm

4. Rests

5• Melodic direction

Given the linear nature of the string quartets, phrase marks were the appropriate criterion for the segmentation of a great many passages. An example from Coulthard's quartet shows a typical applica­

tion of this criterion:

motto crwe.

Example 21. Phrase Marks as Segmenting Criterion, Coulthard, String Quartet Mo. 2. First Movement, m. 143, first violin

In the above example the first phrase Is designated numerically as [7,

10, 9, 6] and the second as [7, 10, 1, 3]. Note that In the first phrase the repetition of £ is disregarded (as are octave repetitions 103 also)• By the systematic reduction of sets to prime forms according to intervallic differences,7 the fir3t set [7, 10, 9, 6] is identified as prime form 4—3, and the second set [7, 10, 1, 3] is identified as prime form 4-27.

Articulation as a segmenting criterion was useful especially when phrase marks were absent. The excerpt below shows three staccato notes followed by four notes not so sharply articulated:

Example 22. Articulation as Segmenting Criterion, Champagne, Quatuor 5 Cordes. Second Movement, mm. 200-01, viola

The staccato notes are combined as set [2, 3, 4] (prime form 3-1), separate from the following set [5, 6, 1, 4] (prime form i—4).

An example from Somers* quartet shows the use of rhythm to segment notes Into two 4-sets:

Example 23. Rhythm as Segmenting Criterion, Somers, String Quartet No. 3. m. 17, first violin

7See pages 89-91. 104

The two rhythmically derived 4-sets [4, 7, 6, 8] and [5, 2, 3, 1] are

both instances of prime form 4-2. This passage typifies the many in­

stances throughout the quartets of adjacent expressions of the same .

set.

Using rests as a means of grouping is particularly useful in

the Weinzweig quartet, because short silences are consistently used to ventilate the linear writing. The opening measures provide an example

in which this criterion is the most logical justification for consider­

ing the notes as a collection:

Example 24. Rest as Segmenting Criterion, Weinzweig, String Quartet Ho. 3. First Movement, mm. 1-2 , cello

The two dotted-quarter rests in measure 2 suffice to set apart the set

[2, 1, 11], prime form 3-2. The sixteenth rests are not considered as

Important In this cent ext.

Melodic direction as a criterion is Infrequently required. An example from Somers' quartet that Illustrates this criterion is shown below:

fctjOOf f * U = ^ k JV j tlfV H * JB3ET- .------— — r

Example 25. Melodic Direction as Segmenting Criterion, Somers, String Quartet No. 3. mm. 449-50, second violin 105 The set [6, 4, 2, 1] is grouped according to descending melodic

motion. Sets [3, 2, 1, 0] and [11, 10, 8 , 9] are segmented by melodic motion and rhythm, respectively.

Not reckoned in the tabulations are the occasional pitch

classes that could not reasonably be grouped with other pitch classes

to form a set. An example of two clearly Isolated pitches Is found in

Weinzweig1s quartet:

Iff .. ■ -rt 1\ J . |-2 —ln-a— ----r~.------—f tp— A ...... d :'y.7. T. ---*- --- # 7-'------

Example 26. Isolated Pitches, Weinzweig, String Quartet No. 3. Third Movement, mm. 8-9, first violin

The "dyad" f e is Isolated by phrasing and rests from the group in

the next measure. The pair f#-e is ignored and only the larger group, set [4, 0, 1, 2] is tabulated in this instance.

In some passages a choice between criteria is necessary, and the standard approach was to select a criterion (and consequent group­ ing) that would be consistent with previous choices and allow formation of either a 3-set or 4-set. The following passage may be grouped ac­ cording to either the rhythm or the rest: Example 27. Conflicting Segmentation Criteria, Champagne, Quatuor & Cordes. Second Movement, mm* 344-45, second violin

If the criterion of rests is applied in this Instance, a 5—set results:

[3, 9, 4, 5, 1] (the lower notes of double stops are not considered, because they are not consistently present in any of the quartets)•

However, the criterion of rhythm may also apply, giving [3, 9, 4] and

[4, 5, 1] (the first pitch in the second measure begins the second set). The latter procedure appears more suitable, because both rhythmic values and articulation marks indicate the grouping.

The five criteria discussed above proved to be sufficient for grouping the overwhelming majority of notes. The large majority of

the sets were 3—sets and 4-sets, with both types being sufficient in

Q numbers to permit tabular comparison. Thus given the musical criteria employed for the selection of sets, the sets reported in the results represent a thoroughly and uniformly measured data base derived from the linear writing of the four quartets.

o See pages 92—93.

^Each of the four Instrumental parts was segmented throughout each of the four string quartets. 107

Tabulation of Results

The process of linear segmentation yielded large numbers of

3-sets and 4—sets In each quartet* However, the counts had to be com­

posites of all four Instrumental lines because the numbers yielded by

Individual lines were insufficient to consider statistically. The

totals were divided according to occurrences of the twelve prime forms

for 3-sets and the twenty-nine prime forms for 4-sets.

These counts are presented In the two tables that follow.

Table 17 gives 3-set prime form occurrences for each of the four quar­

tets.^® The 4-set prime form counts appear In Table 18.

TABLE 17

Horizontal Plane - Prime Form Counts, 3-Sets

Count by Composer Form Co Ch So Wz '

3-1 73 37 183 61 3-2 128 81 195 42 3-3 ' 136 30 24 31 3-4 67 17 12 16 3-5 32 16 26 17 3-6 7 40 10 10

3-7 17 28 20 7 3-8 6 32 7 8 3-9 5 15 3 4 3-10 12 24 11 1 3-11 35 23 3 6 3-12 5 7 0 1

^For convenience, the following abbrebiatlons are used: Co (Coulthard), Ch (Champagne), So (Somers), Wz (Weinzweig). TABLE 18

Horizontal Plane - Prime Form Counts, 4-Sets

Count by Composer

Wz

31 57 5 10 14 10

4 5 2 2 9 3

2 2 5 29 0 2

4 1 1 4 0 1 0 1 1 0 2 109

The counts In these tables represent the 3-sets and 4-sets

reduced to their prime forms. Inversions are considered equivalent

to the "normal order" prime forms, so that any given count may contain

both the normal order and the inversion. Since the total intervallic

class content (as represented by the Interval vector) is equivalent

for any expression of a prime form, Inversions are not counted separ­

ately.

Because the total counts were different for each composer, it was necessary to convert the item counts for specific prime forms Into percentages. These percentages make uniform comparisons possible.

For uniformity the percentages found in the following tables are ex­

tended to four digits.

TABLE 19

Horizontal Plane - Prime Form Percentages, 3-Sets

Percentages by Composer Form Co Ch So Wz

3-1 13.95 10.57 37.04 29.90 3-2 24.47 23.14 39.47 20.58 3-3 26.00 08.57 04.85 15.19 3-4 12.81 04.85 02.42 07.84 3-5 06.11 04.57 05.26 08.33 3-6 01.33 11.42 02.02 04.90

3-7 03.25 08.00 04.04 03.43 3-8 01.14 09.14 01.41 03.92 3-9 00.95 04.28 00.60 01.96 3-10 02.29 06.85 02.22 00.49 3-11 06.69 06.57 00.60 02.94 3-12 00.95 02.00 00.00 00.49 110 TABLE 20

Horizontal Plane - Prime Form Percentages, 4-Sets

Percentages by Composer Form Co Ch So Wz

4-1 15.32 09.79 41.13 14.97 4-2 06.13 05.67 19.34 27.53 4-3 15.32 07.21 08.29 02.41 4-4 04.59 04.12 01.95 04.83 4-5 02.29 01.03 00.97 06.76 4-6 00.00 02.06 00.97 04.83

4-7 14.55 01.54 00.16 01.93 4-8 10.34 01.54 00.32 02.41 4-9 01.14 00.51 00.16 00.96 4-10 02.68 07.21 08.29 00.96 4-11 01.53 16.49 02.43 04.34 4-12 01.14 03.60 02.43 01.44

4-13 01.14 02.06 08.13 00.96 4-14 00.76 05.15 00.32 00.96 4-15 03.06 01.54 00.00 02.41 4-16 02.29 03.09 00.00 14.00 4-17 01.14 00.51 00.00 00.00 4-18 01.91 02.57 00.32 00.96

4-19 08.42 01.03 00.16 01.93 4-20 00.76 00.51 00.00 00.48 4-21 00.00 07.21 00.65 00.48 4-22 00.00 04.12 00.48 01.93 4-23 00.00 03.09 00.00 00.00 4-24 00.38 01.54 00.16 00.48

4-25 00.00 01.54 00.00 00.00 4-26 00.00 02.06 00.32 00.48 4-27 01.53 02.06. 00.81 00.48 4-28 00.00 00.51 00.00 00.00 4-29 03.44 00.51 02.11 00.96 Ill

Interpretation of the above results Is given In the next section.

Each of the four hypotheses (pages 98—101) Is tested against the data to determine the extent of either confirmation or negation. Selected musical examples are presented to aid In the Interpretation of the results.

Due to their much lower frequency of occurrence, 5-sets and

6-sets are not Included In the results. However, the following table is provided to show the total counts for 5- and 6—sets compared to those for 3- and 4-sets. In each of the sixteen totals for 3-sets and

4-sets (eight in the horizontal plane and eight in the vertical plane), the number of instances Is greater than 100. However, In the sixteen totals for 5-sets and 6-sets, only four of the totals are greater than

100. The remaining twelve totals for 5- and 6-sets are each less than

100, with as few Instances as 1 and 0 found in the vertical plane of the Somers quartet. Consequently, the data for 3-sets and 4-sets may be judged to be adequate for comparisons between composers' quartets, whereas the data for 5-sets and 6-sets is insufficient.

As a visual aid to interpretation of the percentages given for

3-sets and 4-sets in this section, the following graphs were derived from the percentages. Graphs for the percentages in the 3-sets are given first— one for each quartet. These are followed by graphs repre­ senting the 4-sets. Similar graphs for the vertical sets will be found in Chapter IV. The vertical axis represents the percentages of usage for the prime forms. The horizontal axis gives the individual prime form designations. For example, column 1 in the first graph (Coulthard, TABLE 21

Prime Form Totals: 3-, 4-, 5-, 6-Sets

Horizontal Plane

ChCo Wz

3-Sets 523 350 494 204

4-Sets 261 194 615 207

5-Sets 113

6-Sets 11 42 100 15

Vertical Plane

3-Sets 348 330 385 160

4-Sets 312 519 511 137

5-Sets 61 137 57

6-Sets 12 20 50

Horizontal 3-Sets) reads as approximately 14 percent usage of prime form 3-1 In Coulthard*s quartet. Figure 10. Graph for Coulthard, Horizontal 3-Set3

45

30

uo « put

15

Figure 11. Graph for Champagne, Horizontal 3-Sets 114

4 St-

30' u n oQ> u

10

Figure 12. Graph for Somers, Horizontal 3-Sets

30 4J CJS a) cu 15

5 10

Figure 13. Graph for Weinzweig, Horizontal 3-Sets 115

45,—

30- ■u C 4)O «U P-

15:

0L ■n=t 1 I \ 10 15 20 25

Figure 14. Graph for Coulthard, Horizontal 4-Sets

45

30 ■M S CJ fuS 15

5 10 15 20 25

Figure 15. Graph for Champagne, Horizontal 4-Sets 116

45

30

aJ 0(J o! cu 15

L tJ; 10 15 20 25

Figure 16. Graph for Somers, Horizontal 4-Sets

45t--

10 15 20 255

Figure 17. Graph for Welnzweig, Horizontal 4-Sets Interpretation of Results

The first hypothesis^ appears to be validated. The 3—sets,

containing fewer forms than the A—sets, show more readily that music­

ally 'meaningful results have been obtained. Differences of prime form

usage within the separate quartets relate to Hypothesis 2 and are dis­

cussed below. The validity of Hypothesis 1 is seen best by examining

the prime form percentages for each composer in turn. An arbitrary

figure of 50 percent is used for each composer's percentages with ref­

erence to the following question, "How many forms are required to reach

50 percent?" The implication of the question is that if the number of

forms is manifestly less than half of the available forms, there is a

suggestion that the distribution is not random.

The second hypothesis 12 is confirmed in varying degrees for

each quartet. In Coulthard's quartet only two (of twelve) 3-sets ac­

count for 50 percent of the total 3-sets: 3-2 (24.47%) and

3—3 (26.00?). Further, two more forms, with the preaeding two, account

for 77.23 percent of all sets: 3-1 (13.95%) and -3-4 (12.81%). These

four forms all contain i d and ic2 , reflecting an emphasis upon step­ wise motion In Coulthard's quartet. Table 22 shows the structure of

these forms.

^ftien the four Instrumental parts of a given quartet are seg­ mented according to musically defined criteria, analysis of the result­ ing sets will yield meaningful results.

12In a given quartet there will be manifestly different frequencies in the usage of prime forms— some will be used more often than others. 118

TABLE 22

Frequent Horizontal 3—Set Forms, Coulthard

Form PCS Vector Percentage

3-3 0, 1, 4 101100 26.00 3-2 0, 1, 3 111000 24.47 3-1 0, 1, 2 210000 13.95 3-4 0, 1, 5 100110 12.81

The following example based on the first two forms shows this stepwise motion:

ritord £OCO

' 1

Example 28. Horizontal Forms 3-2, 3-3, Coulthard, String Quartet No. 2. First Movement, mm. 6—7, first violin

The pattern of emphasis on icl and lc2 Is also found In the most frequent 4-sets. As Table 23 shows, four of the twenty-nine forms comprise over half the sets: 4-1 (15.23%), 4-3 (15.32%), 4-7 (14.55%), and 4-8 (10.34%). The following musical example Illustrates two of these forms.

erase, e. nno'tp occp.l. Allegro con Fuoco > _ >

Example 29. Horizontal Forms 4-3, 4-1, Coulthard, String Quartet No. 2. First Movement, mm. 73-75, first violin 119

TABLE 23

Frequent Horizontal 4-Set Forms, Coulthard

Form PCS Vector Percentage

4-1 0, 1, 2, 3 321000 15.32 4-3 0, 1, 3, 4 212100 15.32 4-7 0, 1, 4, 5 201210 14.55 4-8 0, 1, 5, 6 200121 10.34

High percentages for particular forms Is not as evident In

Champagne's quartet. Four of the 3—set forms are required to comprise half the sets: 3-2 (23.14Z), 3-6 (11.42%), 3-1 (10.57%), and 3-8

(09.14%). Form 3-2 Is more prominent than the other three. The hypoth­ esis of frequently used forms Is confirmed to a lesser degree than In the Coulthard work, where only two forms made up half of the sets. An example showing Champagne's use of 3-2 follows the Information on the forms:

TABLE 24

Frequent Horizontal 3-Set Forms, Champagne

Form PCS Vector Percentage

3-2 0, 1, 3 111000 23.14 3-6 0, 2, 4 020100 11.42 3-1 0 , 1 , 2 210000 10.57 3-8 0, 2 , 6 010101 09.14 120

.iiyv p -I — = 3 ! = =±— ■■ Y 3-2 3-2.

Example 30, Horizontal Form 3-2, Champagne, Quatnor & Cordes. Second Movement, mm. 255—56, second violin i Champagne*s usage of 4—sets is also less clear-cut than

Coulthard*s. Six 4-set forms are required to Include 50 percent of the sets, with 4-11 as the most frequent form: 4-11 (16.49%), 4-1

(09.79%), 4-3 (07.21%), 4-10 (07.21%), 4-21 (07.21%), and 4-2

(05.67%). Again, the hypothesis Is validated to a lesser degree. A table of these forms and a musical example Involving two of them Is shown below.

TABLE 25

Frequent Horizontal 4-Set Forms, Champagne

Form PCS Vector Percentage

4-11 0, 1, 3, 5 121110 16.49 4-1 0, 1, 2 , 3 321000 • 09.79 4-3 0, 1, 3, 4 212100 07.21 4-10 0, 2, 3, 5 122010 07.21 4-21 0 , 2 , 4, 6 030201 07.21 4-2 0, 1, 2 , 4 221100 05.67 121

,-- 3 ljp| H j A ~ * r

Example 31. Horizontal Forms 4-11, 4-3, Champagne, Quatuor & Cordes. First Movement, mm. 103—05, cello

Very strong confirmation of Hypothesis 2 is found In the

Somers quartet for both 3-sets and 4-sets. Only two of the 3—set forms yield 76.51 percent of all sets, an even clearer delineation of non- randomness than shown in the Coulthard quartet (where four forms are required to reach a comparable percentage): 3—1 (37.04%) and 3—2

(39.47%)•

TABLE 26

Frequent Horizontal 3—Set Forms, Somers

Form PCS Vector Percentage

3-2 0, 1, 3 111000 39.47 3-1 0 , 1 , 2 210000 37.04 Example 32. Horizontal Forms 3—2, 3—1, Somers, String Quartet No. 3. mm. Ill, viola

In the 4-sets only two forms are required to account for 60 per cent of the sets, and three forms account for 68.8 percent: 4-1

(41.13%), 4-2 (19.34%), 4-3 (08.29%). A strong emphasis upon i d and ic2 Is to be found, as shown In the example following Table 27.

TABLE 27

Frequent Horizontal 4-Set Forms, Somers

Form PCS Vector Percentage

4-1 0, 1, 2, 3 321000 41.13 4-2 0, 1, 2, 4 221100 19.34 4-3 0, 1, 3, 4 212100 08.29 Example 33. Horizontal Forms 4-1, 4-2, 4-3, Somers, String Quartet No. 3. mm. 407-09, second violin

In both 3—sets and 4—sets Welnzwelg's usage also confirms

Hypothesis 2, although not quite so strongly as Somer's usage. Two

3-set forms comprise 50 percent of the sets, and a third gives

65 percent: 3—1 (29.90Z), 3—2 (20.58Z), and 3—3 (15.19Z). The emphasis on lcl and lc2 found In Somers and Coulthard Is shared by Welnzwelg, as shown In the example that follows Table 28.

TABUS 28

Frequent Horizontal 3-Set Forms, Welnzwelg

Form PCS Vector Percentage

3-1 o, 1, 2 210000 29.90 3-2 0, 1, 3 111000 20.58 3-3 0, 1, 4 101100 15.19 124

3-t 3-/ 3 -a

Example 34. Horizontal Forms 3—lt 3—2, Welnzwelg, String Quartet No. 3. Second Movement, mm. 114—15, first violin

The pattern of emphasis on lcl and ic2 Is apparent to a degree in Welnzwelgfs 4-set usage also. Here, three of the forms account for over 50 percent of the sets: 4-2 (27.53%), 4-1 (14.97%), and 4-16

(14.00%). A musical example Illustrating the more frequently occurring form 4-2 follows Table 29.

TABLE 29

Frequent Horizontal 4-Set Forms, Welnzwelg

Form PCS Vector Percentage

4-2 0 , 1, 2, 4 221100 27.53 4-1 0, 1, 2, 3 321000 14.97 4-16 0, 1, 5, 7 110121 14.00 125

V -----j-.— -- f t f r r ~ b 411 ? U ' — ■fifi sub.

f -3. f - /

Example 35, Horizontal Forms 4-2, 4-1, Welnzwelg, String Quartet No. 3. First Movement, mm. 73-74, cello

The third hypothesis asserts that "in a given quartet the most

frequently used set forms will have close similarity relations'* (page

100)• In this context "most frequently used" will be taken to mean

those 3-set and 4—set forms discussed above that comprise 50 percent of

the sets for each quartet. A further limitation is also necessary:

the consideration of only the R^, Rp and the R2 , Rp relations (see pages 83-84)• These are the two relation combinations that 3how where

two prime forms are very similar both in pitch class and interval class content.

In Coulthard's work the R^, Rp relation (termed Rp" for convenience) holds for 3-3 and 3-2 (see Table 22), thus showing the two

forms to be closely related and confirming Hypothesis 3. Including the

"secondary" forms (the remaining two in the table), icl is present in every form, and lc5 and lc6 are infrequent. The hypothesis is not sup­ ported in the 4-set forms, where no close relations are found. Interval 126 class 1 is still dominant in the most frequent 4-set forms, however, as it is In the 3-sets. Stylistically, this dominance is expressed as frequent linear motion by half steps (both diatonic and chromatic).

The 3-set similarity relations In Champagne's quartet reveal an interesting dichotomy in half-step and whole-tone content. First,

3-2 and 3-1 (see Table 24) are related by R2 , Rp, and both forms have 1 3 icl content. Second, 3-6 and 3-8 are also related by R2 , Rp, both of which contain ic2 but no icl. Expressed conventionally, they are both segments of a whole tone scale: [0, 2, 4] and [0, 2, 6]. Stylistic­ ally it may be said that motion by half steps is preferred somewhat over motion by whole steps (34 percent versus 21 percent frequencies for the two pairs) in 3-sets.

The 4-sets reverse this pattern: the six most frequent forms contain more of lc2 than Icl, Including the highest frequency form

4—11 (vector: 121110). From the data gathered it appears that both whole tones and half steps are important in the linear writing.

Of the six forms, R^, Rp holds for 4-2 and 4-3 and R2 , Rp holds for 4-1, 4-2 and 4-10, 4-11. In terms of similarity relations

Hypothesis 3 is not clearly substantiated. With regard to Interval class content there appears to be confirmation with regard to icl and lc2 .

Interval classes 1 and 2 are also dominant In Somers' most fre­ quent forms (both 3—sets and 4—sets)• R2 , Rp relates the two 3—set forms (see Table 26), and the 4-set forms contain these similarity

13 "Icl content" in the linear writing of each quartet is gen­ erally expressed as a tendency toward motion by half steps, although occasional leaps of a major seventh also fall into the category "icl content.” 127 relations: R^, R^ (4-2, 4-3) and R2 , Rp (4-1, 4-2), Note that 4-2 is in close relation to both 4-1 and 4-3. As with Hypothesis 2, Somers* quartet quite strongly confirms Hypothesis 3, both with regard to In­ terval class content and the similarity relations.

Strong confirmation of Hypothesis 3 is also found in

Welnzwelg*s work. The most frequent 3-set forms each contain icl, as well as the 4-set forms. The R^, Bp relation is found for 3-2 and 3-3

(see Table 28), and R2 , Rp holds for 3-1, 3-2 and 4-1, 4-2. As with the Somers quartet, form 3-2 relates to both 3-1 and 3-3 (form 4-2 In

Welnzwelg * s case) .

Statistically, there are two Instances (see Table 30, follow­ ing) of musically significant differences In usage of a given prime form, according to the Y-Square statistic. The first is prime form

3-3 In the Coulthard and Champagne quartets (26.00% and 08.57%, respec­ tively). Coulthard employs this form 17.43 percent more than does

Champagne, judged to be significant according to the Y—Square statistic

(see pages 92-93). In terms of the general model used to explain

Y-Square on page 93, 26.00 percent and 08.57 percent are the "X" fig­ ures. The actual difference between the two percentages, 17.43 percent must be Interpreted not to be the "real" difference, but to represent the midpoint of a probable range In which a real difference is thought to exist: somewhere between 00.68 percent and 34.18 percent.

The second Instance is prime form 3-3 In the Coulthard and

Somers quartets (26.00 % and 04.85%). The difference In frequencies,

21.14 percent again represents the midpoint of the probable range of the difference: 06.35 percent to 35.93 percent. 128

Coulthard's relatively high use of form 3-3 accounts for the two differences, although these are the only ones found In all the comparisons between pairs of composers for each item in the 3-set forms and the 4-set forms. Given larger numbers of counts, such as might be obtained by examination of several works for each composer, one might hypothesize that more differences would be found. Given only the two

Instances above, there are not enough differences revealed to be able to make any general statements regarding stylistic comparisons based on these statistical procedures. To a very limited degree, the hypoth­ esis regarding musically significant differences in prime form usage is confirmed in these two Instances, but there are not enough differ­ ences to enable us to say that the hypothesis is generally substan­ tiated.

The statistical data for the above conclusions is given in the following table. The first two columns give the appropriate prime form and the pair of quartets compared in each instance where a musically significant difference was found. The third and fourth columns give the observed frequencies of prime form usage from "Results" and the differences between the frequencies. The fifth column supplies the probable range of difference as interpreted by the Y-Square sta­ tistic. The fourth column is the midpoint of the frequency range given in the fifth column. Numbers in the first four columns, which may be derived directly from the text, are given in whole integers, such as 26.00 percent In the first row. Numbers in the fifth column, however, are given as decimal percentages ranges, such as 0.006836, 129

Interpreted as 0.68 percent. These latter numbers are derived from the

Y-Square statistic and are presented unaltered.

TABLE 30

Horizontal Y-Square Data for Musically Significant Differences

Y-Square Quartets Observed Confidence Form Compared Frequencies Difference Intervals

3-3 Co, Ch Co, 26.00% 0.006836, Ch, 08.57% 17.43% 0.341812

3-3 Co, So Co, 26.00% 0.063556, So, 04.85% 21.14% 0.359354

In order to summarize the evaluations concerning the degree of confirmation for the hypotheses In this chapter, the following table presents the conclusions for each hypothesis relative to the 3-sets and

4-sets In each quartet. In descending order of strength of confirma­ tion, the teems employed are: very strong, strong, fairly strong, fairly weak, weak, very weak. For example, the degree of confirmation for Hypothesis 1 In Coulthard's 3-sets is "fairly strong" and "very strong" In the 4-sets contained In that quartet. 130

TABLE 31

Degree of Confirmation for Hypotheses Relative to Horizontal Plane

Hypotheses Quartets 1 2 3 4

Co 3-Sets fairly fairly strong fairly strong strong strong

4-Sets very very very very strong strong weak weak

Ch 3-Sets fairly fairly fairly weak strong strong strong

4-Sets strong strong fairly very weak weak

So 3—Sets very very very weak strong strong strong

4-Sets very very very very strong strong strong weak

Wz 3-Sets strong strong strong very weak

4-Sets very very strong very strong strong weak 131

Examination of the table shows that Hypotheses 1 and 2 are well substantiated in the four quartets. The third hypothesis is fairly well supported, and the fourth hypothesis is weakly supported. One may then say that the selection of sets defined according to musical criteria has revealed patterns of usage In each quartet that do not appear to be random and that the selection process Is therefore mean-

Ingful (Hypotheses 1 and 2). The most frequently employed forms tend to be related by pitch class content and interval class content

(Hypothesis 3). However, the differences between prime form usage as compared between pairs of quartets do not appear to be musically sig­ nificant according to the statistical procedure employed (Hypothesis

4). CHAPTER XV

COMPARISON OF VERTICAL PRIME FORM SETS AND SUMMARY

OF SIMILARITY RELATIONS

As a parallel to the focus on horizontal (linear) structures

In the preceding chapter, this chapter begins with an Investigation of the vertical (simultaneous) prime form usage In the quartets. The same sequence of discussion is followed here as before. Because aspects of the previous discussion apply to both chapters, unnecessary dupli­ cation is avoided whenever possible. The objectives here are to syste­ matically quantify the prime form usage with regard to simultaneities, to set forth relevant hypotheses, and to test and interpret those hypoth­ eses in light of the data. At the close of the chapter, comparisons are made between the prime form usage in the horizontal and vertical planes of the quartets to show the strength of similarity relations in these two dimensions of pitch organization.

Four Hypotheses

The hypotheses set forth In this chapter are similar to those in the preceding chapter. They concern the frequencies and relation­ ships of the prime forms in each quartet and make comparisons among the quartets. The four hypotheses stated below refer to set forms in the vertical plane:

132 133

Hypothesis 1——'When the simultaneities occurring at the start of beats are treated as sets, anal­ ysis of the sets will yield meaningful results.

Hypothesis 2-—In a given quartet there will be mani­ festly different frequencies in the usage of prime forms— some will be used more often than others.

Hypothesis 3— In a given quartet the most frequently used set forms will have close similarity relations.

Hypothesis 4— A comparison of the quartets will reveal significantly different frequencies in the usage of prime forms.

Procedures for Vertical Segmentation

Only one criterion proved to be necessary for selecting verti­ cal sets, whereas five were required for choosing melodically gener­ ated sets. Viewing simultaneities as collections provided a tradition­ al means for grouping notes. Further, it was found that the instru­ mental lines of the selected quartets comprised sets of primarily three or four pitch classes at the start of most beats. By limiting the tab­ ulations to only those pitches that sound simultaneously at the begin­ ning of beats a representative sample of vertical sets was provided— a sample that was manageable and yet large enough to investigate by the methods used in this dissertation.

Although the selection of pitch classes occurring on beats would appear to be a straightforward process, some passages required selective judgment. The following passage from Coulthard's quartet il­ lustrates the sort of interpretation required to apply the criterion: Example 36. Beat Defined for Set Selection, Coulthard, String Quartet No. 2. First Movement, mm. 9-10, all parts

In this passage the "beat" occurs on the first and the fourth eighth notes. Although a very slow 6/8 passage might require every eighth note to be treated as a beat, this passage was deemed suffici­ ently typical to define the bar as having only two beats.^ The pitch classes selected as sets, then, were a#-g(#)-e-ctf-d (10-8-4-1-2, form

5-28), b-gtf—e-b-d (11-8-4-2, form 4-27), fff-bb-ai?-d (6-10-2, form 3-12), and g-bb-e-g-g (7-10-4, form 3-10). In the third set (3-12) the bb and a# are both represented as "10." Sets of five or more pitch classes occurred occasionally in the quartets, but three or four pitch classes were much more common.

In order to represent as many of the beats as possible In the tabulations, any notes sounding on a beat were considered, even where

^There were no slow passages In compound meter In any of the four quartets. 135 they have been articulated before the beat. An excerpt from Somers* quartet Is relevant here to Illustrate this distinction:

Example 37. Notes Articulated Prior to Beat as Set, Somers, String Quartet No. 3. ra. 23, all parts

The same set "occurs" (for purposes of quantifying sets on beats) on the second beat as on the first beat: [2, 8 , 11, 10] (form 4-12). Al­ though every note is articulated prior to the second beat, for the sake of tabulation it receives Its own count. A number of sets are "carried over" into beats, but more frequently there are different sets on ad­ jacent beats. Beat 3 above [3, 5, 2, 9], is form 4-29.

Occasionally the sparsity of texture limited the numbers of

3-sets and 4-sets, particularly in the Welnzwelg quartet. This limita­ tion did not, however, reach the point where the numbers were insuffic­ ient for drawing valid conclusions. An example from Welnzwelg illus­ trates the problem: Example 38. Set Selection In Sparse Texture, Welnzwelg, String Quartet No. 3 . First Movement, mm. 20—21, all parts

The only beats (dotted quarter) containing 4—sets are beat 1 of measure

20: [7, 2, 3, 9] (form 4—16) and beat 1 of measure 21: [4, 2, 3, 10]

(form 4-5)•

A final problem to be faced was that of how to interpret tremolos. An example from Coulthard shows two-note tremolos In the cello and the first violin: 137

Example 39. Tremolos In Set Selection, Coulthard, String Quartet No. 2. First Movement, mm. 86—87, all parts

For practical purposes It was decided that both notes of a tremolo should be considered as belonging to the collection. Although technically speaking, the notes are not sounded simultaneously, the aural effect is that the two notes are sounding as one. Moreover, It would not be practical in tremolos to assert which note would sound on beats 2, 3, and 4, as this would vary according to the performer.

Thus, and b^ in the cello and _b and cl in the first violin were con­ sidered as set members. The sets in the above excerpt were found to be [5, 10, 11, 2] (4-15), [5, 10, 8 , 11, 2] (5-31), [5, 10, 4, 11, 2]

(5-19), [5, 10, 11, 2] (4-15) in measure 86 and [6, 11, 10, 2] (4-19),

[6, 11, 8 , 2] (4-27), [6, 11, 4, 2] (4-22), [6, 11, 5, 2] (4-18) in measure 87. It Is often the case that a single note change from one set to the next results in different prime form Identifications. 138 Tabulation of Results

As with the horizontal sets, counts were tabulated for each

3-set prime form and each 4-set prime form. Measurement of every beat throughout each of the four quartets yielded the counts in Tables 32 2 and 33.

TABLE 32

Vertical Prime Form Counts, 3-Set Forms

Count by Comiposer Form Co Ch So Wz

3-1 0 21 36 9 3-2 12 16 47 19 3-3 24 61 34 33 3-4 14 60 60 19 3-5 18 42 44 17 3-6 4 13 23 17

3-7 30 31 56 16 3-8 37 21 25 14 3-9 16 9 30 6 3-10 46 14 6 0 3-11 114 31 19 10 3-12 33 11 5 0

5 Abbreviations are again employed to represent the quartets: Co (Coulthard), Ch (Champagne), So (Somers), Wz (Welnzwelg)• 139

TABLE 33

Vertical Prime Form Counts, 4-Set Forms

Count by Composer Form Co Ch So W*

4-1 0 12 19 9 4-2 1 27 19 18 4-3 2 11 20 3 4-4 2 14 16 7 4-5 1 19 11 4 4-6 0 6 24 15

4-7 3 10 • 24 5 4-8 2 15 10 2 4-9 4 13 3 2 4-10 1 2 19 1 4-11 9 12 26 9 4-12 17 22 16 1

4-13 9 36 21 6 4-14 9 16 29 10 4-15 13 26 50 7 4-16 4 30 29 8 4-17 16 22 6 2 4-18 28 32 19 9

4-19 52 58 37 4 4-20 7 21 39 1 4-21 5 9 5 0 4-22 17 13 16 2 4-23 7 24 22 1 4-24 4 26 3 0

4-25 9 5 1 3 4-26 17 11 6 9 4-27 56 17 5 1 4-28 2 1 0 0 4-29 15 9 16 7 140

As before, conversion of the counts Into percentages was neces­ sary In order to make comparisons.^ The appropriate percentages are shown In Tables 34 and 35. These are the tables used for the Interpre­ tative conclusions in the next section.

TABLE 34

Vertical Prime Form Percentages, 3-Set Forms

Percentage by Composer Form Co Ch So Wr

3-1 00.00 06.36 09.35 05.62 3-2 03.44 04.84 12.20 11.87 3-3 06.89 18.48 08.83 20.62 3-4 04.02 18.18 15.58 11.87 3-5 05.17 12.72 11.42 10.62 3-6 01.14 03.93 05.97 10.62

3-7 08.62 09.39 14.54 10.00 3-8 10.63 06.36 06.49 08.75 3-9 04.59 02.72 07.79 03.75 3-10 13.21 04.24 91.55 00.00 3-11 32.75 09.39 04.93 06.25 3-12 09.48 03.33 01.29 00.00

Graphs depicting the percentage frequencies for 3-sets and

4-sets are supplied. The first four graphs show the simultaneous 3-set results, and the second four deal with the 4-sets.^

% e e page 109.

^Directions for interpreting the graphs are found on page 111. 141

TABLE 35

Vertical Prime Form Percentages, 4-Set Forms

Percentage by Composer Form Co Ch So Wz

4-1 00.00 02.31 03.71 06.56 4-2 00.32 05.20 03.71 13.13 4-3 00.64 02.11 03.91 02.18 4-4 00.64 02.69 03.13 05.10 4-5 00.32 03.66 02.15 02.91 4-6 00.00 01.15 04.69 10.94

4-7 00.96 01.92 04.69 03.64 4-3 00.64 02.89 01.95 01.45 4-9 01.28 02.50 00.58 01.45 4-10 00.32 00.38 03.71 00.72 4-11 02.88 08.31 05.08 06.56 4-12 05.44 04.23 03.13 00.72

4-13 02.88 06.93 04.10 04.37 4-14 02.88 03.08 05.67 07.29 4-15 04.16 05.00 09.78 05.10 4-16 01.28 05.78 05.67 05.83 4-17 05.12 04.23 01.17 01.45 4-18 08.97 06.16 03.71 06.56

4-19 16.66 11.17 07.24 02.91 4-20 02.24 04.04 07.63 00.72 4-21 01.60 01.73 00.97 00.99 4-22 05.44 02.50 03.13 01.45 4-23 02.24 04.62 04.30 00.72 4i»24 01.28 05.00 00.58 00.00

4-25 02.88 00.96 00.19 02.18 4-26 05.44 02.11 01.17 00.00 4-27 17.94 03.27 00.97 00.72 4-28 00.64 00.19 00.00 00.00 4-29 04.80 01.73 03.13 05.10 142

30 4 -1 *3 O0) 0u)

15

10

Figure 18. Graph for Coulthard, Vertical 3-Sets

45 —

30

10

Figure 19. Graph for Champagnet Vertical 3—Sets 143

45

30 4J c o01 M tu

5 10

Figure 20. Graph for Somers, Vertical 3—Sets

45 —

30

su h P4w 15

1 u 5 10

Figure 21. Graph for Welnzwelg, Vertical 3—Sets Percent * Percent 30 45 45 15 15 < —r i— o< Figure 23. Graph for Champagne, Vertical 4—Sets Champagne,forVertical Graph 23. Figure Figure 22. Graph for Coulthard, Vertical 4-Sets Coulthard, Vertical for Graph 22. Figure 10 10 52 25 20 15 15 20 25 144 145

45]—

15 n 10 15 20 25

Figure 24. Graph for Somers, Vertical 4—Sets

45

30

15

m 10 15 20 25

Figure 25. Graph for Welnzwelg, Vertical 4-Sets Interpretation of Results

The first hypothesis^ asserted that analysis of a larger sample of sets would lead to meaningful results. The results obtained from the vertical plane were not as rewarding as those of the horizontal plane,*’ where fairly high frequencies for certain prime forms occur In all but Champagne's quartet. Perhaps the linear organization of the tonal/non-tonal quartet writing accounts for the greater applicability of this hypothesis to the horizontal plane. As the discussion of the second hypothesis will show, three of the four quartets do not exhibit prime form frequencies as conspicuously high as those for the horizon­ tal writing. Coulthard's quartet is the exception In this case.

The second hypothesis^ may be examined for each composer by means of the 50 percent frequency cutoff technique explained earlier.®

If a small number of set forms accounts for half of the sample, then these are the ones that are structurally Important. In the Coulthard work only three of the twelve 3-set prime forms are required: 3-11

(32.75%), 3-10 (13.21%), and 3-12 (09.48%). Form 3-11, the minor triad

(and its mirror inversion, the major triad), accounts for one-third of all sets, a percentage that is very high. Two musical examples of

^"When the simultaneities occurring at the start of beats are treated as sets, analysis of the sets will yield meaningful results" (page 133) .

®See page 117.

^"In a given quartet there will be manifestly different fre­ quencies in the usage of prime forms— some will be used more often than others" (page 133) .

®See page 117. Coulthard1s usage of this form follow the structural information given in Table 36.

TABLE 36

Frequent Vertical 3-Set Forms, Coulthard

Form PCS Vector Percent

3-11 0, 3, 7 001110 32.75 3-10 0, 3, 6 002001 13.21 3-12 0, 4, 8 000300 09.48

Example 40. Vertical Form 3-11, Coulthard, String Quartet No. 2. First Movement, mm. 140-41, all parts

fee, x-

Example 41. Vertical Form 3-11, Coulthard, String Quartet No. 2. Second Movement, mm. 15-16, all parts 148

Five of the twenty-nine 4—set forms are required to account for

50 percent of all the 4—sets In Coulthard*s quartet. These are 4—27

(17.942), 4-19 (16.662), 4-18 (08.972), 4-12 (05.442), and 4-22

(05.442). Musical examples of the first two of these forms follow the table below.

TABLE 37

Frequent Vertical 4-Set Forms, Coulthard

Form PCS Vector Percentage

4-27 0, 2, 5, 8 012111 17.94 4-19 0, 1, 4, 8 101310 16.66 4-18 0, 1, 4, 7 102111 08.97 4-12 0, 2, 3, 6 112101 05.44 4-22 0, 2, 4, 7 021120 05.44

Example 42. Vertical Form 4-27, Coulthard, String Quartet No. 2. First Movement, mm. 38-39, all parts 149

Furioso

Funooa

Furioso

Furioso

f -J9 fy-W H'-tf

Example 43. Vertical Form 4-19, Coulthard, String Quartet No. 2. Third Movement, m. 95, all parts

Hypothesis 2 appears to be clearly substantiated in Coulthard*s

quartet. Only one—fourth of the 3-set forms comprise 50 percent of

the 3-sets, and a sixth of the 4-set forms account for 50 percent of

the 4-sets.

In the Champagne quartet four of the 3-set forms are needed to

comprise 50 percent of the 3-sets: 3-3 (18.48%), 3-4 (18.18%), 3-5

(12.72%), and 3—7 (09.39%). These forms and musical examples of them are shown below.

TABLE 38

Frequent Vertical 3-Set Forms, Champagne

Form PCS Vector Percentage

3-3 0, 1, 4 101100 18.48 3-4 0, 1, 5 100110 18.18 3-5 0 , 1, 6 100011 12.72 3-7 0, 2, 5 011010 09.39 150

Example 44. Vertical Forms 3-7, 3-3, 3-5, Champagne, Quatuor & Cordes. First Movement, mm. 20—21, all parts

i = ! i

N r& r * i

a

a r=p-f *> ftp ‘if *r

3-3 3 -9-

Example 45. Vertical Forms 3-3, 3-4, Champagne, Quatuor 2t Cordes. Second Movement, mm. 262-63, all parts

In the case of 4-sets, eight of the twenty-nine forms avail­ able are required to comprise 50 percent of the sets. None of the eight show particularly high percentages: 4—19 (11.17Z), 4—11 (08.31Z), 151

4-13 (06.93%), 4-18 (06.16%), 4-16 (05.78%), 4-2 (05.20%), 4-15

(05.00%), and 4—24 (05.00%). An example of 4—19 follows the forms below.

TABLE 39

Frequent Vertical 4—Set Forms, Champagne

Form PCS Vector Percentage

4-19 0, 1, 4, 8 101310 11.17 4-11 0, 1, 3, 5 121110 08.31 4-13 0 , 1, 3, 6 112011 06.93 4-18 0, 1, 4, 7 102111 06.16

4-16 0, 1, 5, 7 110121 05.78 4-2 0, 1, 2 , 4 221100 05.20 4-15 0, 1, 4, 6 111111 05.00 4-24 0, 2, 4, 8 020301 05.00

f-/7 H'-W

Example 46. Vertical Form 4-19, Champagne, Quatuor £ Cordes. Second Movement, mm. 216-17, all parts 152

Unlike Coulthard1s quartet, there is no strong confirmation of

Hypothesis 2 to be found in Champagne*s quartet. One-third of the

3—set forms and almost one-third of the 4-set forms are required to account for half the sets.

One-third of the 3-set forms are also required to comprise *

50 percent In the Somers work: 3-4 (15.58%), 3-7 (14.54%), 3-2

(12.20%), and 3-5 (11.42%). None of the percentages appear to be noteworthy. Tabulations and illustrations of these forms are given below. - TABLE 40

Frequent Vertical 3-Set Forms, Somers

Form PCS Vector Percentage

3-4 0, 1, 5 100110 15.58 3-7 0, 2 , 5 011010 14.54 3-2 0, 1, 3 111000 12.20 3-5 0, 1, 6 100011 11.42 m i r ~ E

i

§ = ^ W ^

j?.fy ■ k g frfc

3-7 3-t Example 47. Vertical Forms 3-7, 3-4, Somers, String Quartet No. 3. m. 134, all parts 153

la the Somers quartet, eight forms are needed to account for

50 percent of the 4-sets: 4-15 (09.78%), 4—20 (07.63%), 4-19

(07.24%), 4-14 (05.67%), 4-16 (05.67%), 4-11 (05.08%), 4-6 (04.69%), and 4-7 (04.69%). An example of 4-15 follows the forms in Table 41.

TABLE 41

Frequent Vertical 4-Set Forms, Somers

Form PCS Vector Percentage

4-15 0, 1, 4, 6 111111 09.78 4-20 0, 1, 5, 8 101220 07.63 4-19 0, 1, 4, 8 101310 07.24 4-14 0, 2, 3, 7 111120 05.67

4-16 0, 1, 5, 7 110121 05.67 4-11 0, 1, 3, 5 121110 05.08 4-6 0, 1, 2, 7 210021 04.69 4-7 0, 1, 4, 5 201210 04.69

.1 hail

Example 48. Vertical Form 4-15, Somers, String Quartet No. 3. m. 53, all parts 154

As in Champagne's quartet, Hypothesis 2 is only partially sup­ ported here. The numbers of forms required to supply half the sets are equivalent to those for Champagne: four 3-set forms and eight 4-set forms.

Four of the 3-set forms are likewise necessary ±n Welnzwelg*s quartet to comprise half of the 3-sets, the first of which is somewhat higher in frequency: 3-3 (20.62%), 3-2 (11.87%), 3-4 (11.87%), and

3-5 (10.62%). An example of 3-3 follows Table 42.

TABLE 42

Frequent Vertical 3-Set Forms, Weinzweig

Form PCS Vector Percentage

3-3 0, 1, 4 101100 20.62 3-2 0, 1, 3 111000 11.87 3-4 0, 1, 5 100110 11.87 3-5 9, 1, 6 100011 10.62

b ^ ------aureo ----- 1------1— rr-j------...... =J

' S k i ' } J cur co -ft------

A t J j y ? : £> — = = = . ------

. ,t j . , J _ J j J .

3-3 3-3 Example 49. Vertical Form 3—3, Weinzweig, String Quartet No. 3. Fourth Movement, mm. 31-32, all parts 155

Only six of the 4—set forms are needed to make up 50 percent of the forms. These forms are 4-2 (13.13%), 4-6 (10.94%), 4-14

(07.29%), 4—1 (06.56%), 4-11 (06.56%), and 4—18 (06.56%). Two examples follow the forms In Table 43.

TABLE 43

Frequent Vertical 4-Set Forms, Weinzweig

Form PCS Vector Percentage

4-2 0, 1, 2, 4 221100 13.13 4-6 0, 1, 2, 7 210021 10.94 4-14 0, 2, 3, 7 111120 07.29

4-1 0, 1, 2, 3 321000 06.56 4-11 0, 1, 3, 5 121110 06.56 4-18 0 , 1, 4, 7 102111 06.56

.vft- - -\PjF ------— m &-- r- m

i EEES

V a T " y " j — J . ■ ■ "2? -5- J l

if-IS.

Example 50. Vertical Form 4-2, Weinzweig, String Quartet No. 3. Fir3t Movement, mm. 14—15, all parts 156

poco CL

— flr- p°«l

$-6

Example 51. Vertical Form 4-6, Welnzwelg, String Quartet No. 3. Second Movement, mm. 152-53, all parts

Hypothesis 2 Is reasonably supported In Welnzwelg*s quartet, although not as clearly in 3—sets as In 4-sets. In this respect his work Is most similar to Coulthard*s work: both exhibit manifest differences In prime form usage in the horizontal and vertical planes.

The third hypothesis^ was again examined with respect to the two pairs of similarity relations, R^, Rp and R2 , Rp* Only the most frequent forms of those listed in each table were selected for similar­ ity relation content, whereas all forms were viewed with respect to intervalllc content. Because the higher frequency forms in the hori­ zontal plane were generally more conspicuous than those In the vertical

^”In a given quartet the most frequently used set forms will have close similarity relations” (page 133)• 157 plane, the selection of "most frequent forms" may seem somewhat arbi­ trary here. An effort has been made to find a reasonable "break point" in each case, with an attempt to limit the selected forms to four or less (to increase the chance that any relationships found are meaning­ ful ones) •

Neither pair of similarity relations are found in the 3—set forms for Coulthard,1® although It may be noted that none of the three most frequent forms contain either icl or lc2 , the interval classes which were employed in the horizontal plane. If icl, Ic2, and ic6 may be taken to represent dissonance, and lc3, ic4, and ic5 to represent consonance, it may be said that dissonant sets are employed horizon­ tally and consonant sets are employed vertically.

There are also no similarity relations in the first three 4-set forms. Unlike the 3-sets, icl or lc2 occur in every one of the five highest frequency 4—set forms, although ic3, ic4, and lc5 are still used more frequently. To an extent, then, the tendencytoward "conson­ ant" interval classes in the vertical plane is supported in the 4-sets.

As defined by similarity relations. Hypothesis 3 is not sup­ ported by the data for Coulthard in the vertical plane. Stylistically,

"consonant" pitch class sets are more frequent vertically than "dis­ sonant" ones.

In the 3-set forms for Champagne's quartet,11 the R^, Rp rela­ tion holds for 3-3, 3-4, and 3—4, 3—5. There is no predominant

■^See Table 36, page 147.

11See Table 38, page 149. 158

Interval class shown in the interval vectors. Of the four highest frequency 4—set forms R^, Rp holds for 4—13, 4—18, and R2 , Rp holds for

4-11, 4-19. Again, no interval classes predominate.

Strong support is evident for Hypothesis 3, in that the two highest frequency forms in both 3-sets and 4-sets are closely re­ lated. Stylistically, there is no strong preference shown for any given interval class. The results have suggested that some control of the total sonority aggregate is present as demonstrated by the similar­ ity relations— but that control of specific interval classes is not entailed here.

There are two pairs of R^t Rp relations in the 3-set forms12 13 for Somers: 3—4, 3-5 and 3—7, 3—2. No particular interval classes predominate. Xn the three highest frequency 4-set forms there is one

R2 , Rp relation: 4—20, 4-19. There is no single dominant interval class. For Somers Hypothesis 3 is supported to a degree, although not as well as for Champagne. Stylistically, there is only limited verti­ cal control of pitch class sets.

In the Weinzweig quartet there are three R^, Rp relations in the 3-set forms:^ 3-3, 3-2; 3-3, 3-4; and 3-4, 3-5. In the 4-set forms no similarity relations occur among the three most frequent ones, and Interval class 1 predominates.

■^See Table 40, page 152.

13Successive reorderings are necessary to demonstrate R^— see pages 79-80.

^ S e e Table 42, page 154. 159

For Weinzweig Hypothesis 3 is supported in the 3-set forms, but not in the 4-set forms. It is interesting to note that interval class 1 is the dominant one in both the horizontal and vertical planes. Stylis­ tically, according to this measure, the Weinzweig quartet is the most dissonant of the four.

With regard to Hypothesis 4, there are only four instances of musically significant different frequencies in the usage of prime forms when compared between composers.^"* The first three of these compare

Coulthard*s use of form 3—11 (32.75%) with each of the other three com­ posers use of the same form: Champagne, 09.39%; Somers, 04.93%; and

Weinzweig, 06.25%. The differences in frequencies are 23.36%, 27.82%, and 26.50%, respectively, with Coulthard*s frequency higher in each case. Thus, Coulthard employs 3-11 23.36% more than does Champagne, for Instance. The probable ranges^ of the differences are, in order,

02.81% to 43.91%, 08.90% to 46.73%, and 04.74% to 48.27%.

The fourth significant comparison involves prime form 3—10 in

Coulthard and Weinzweig. The frequencies are 13.21% and 0.0%, and the probable range of difference is 00.71% to 25.71%.

Again, there are not enough differences to make conclusive statements regarding the stylistic approaches of the composers based on statistical comparisons. It may be remarked that Coulthard employs the form for the major or minor triad (3—11) much more than either

Champagne, Somers, or Welnzwelg, perhaps justifying the conclusion that

^ ”A comparison of the quartets will reveal significantly dif­ ferent frequencies in the usage of prime forms" (page 133).

■^See page 128. 160

her quartet is more consonant than any of the others. Higher counts,

especially In the 4-set forms, would be required to locate more dif­

ferences In prime form usage between the composers. The hypothesis regarding significant differences in prime form usage is confirmed

only in the instance of form 3-11.

The statistical data for these conclusions is presented below.

Format is the same for Table 30, explained on page 129.

TABLE 44

Vertical Y—Square Data for Musically Significant Differences

Y-Square Quartets Observed Confidence Form Compared Frequencies Difference Intervals

3-11 Co, Ch Co, 32.75% 23.36% 0.028152, Ch, 09.39% 0.439142

3-11 Co, So Co, 32.75% 27.82% 0.089083, So, 04.93% 0.467388

3-11 Co, Wz Co, 32.75% 26.50% 0.047460, Wz, 06.25% 0.482712

3-10 Co, Wz Co, 13.21% 13.21% 0.007186, Wz, 0.0 % 0.257182 161

As In Chapter III a table summarizing the degree of support or negation of the hypotheses is presented as a visual aid in interpreta­ tion. Each of the four hypotheses is evaluated with reference to the

3—sets and 4-sets in each quartet. In descending order of strength of confirmation, the terms employed are very strong, strong, fairly strong, fairly weak, weak, very weak. For example, the degree of con­ firmation for Hypothesis 4 in the 3-sets in Coulthard's work is

"strong." Table 45 follows below: 162

TABLE 45

Degree of Confirmation for Hypotheses Relative to Vertical Plane

Quartets Hypotheses 1 2 3 4

Co 3-Sets strong strong very strong weak

4-Sets strong strong very very weak weak

Ch 3-Sets fairly fairly strong weak strong strong

4-Sets fairly fairly strong very strong strong weak

So 3-Sets fairly fairly fairly weak strong strong strong

4-Sets fairly fairly fairly very strong strong weak weak

Wz 3-Sets fairly fairly very fairly strong strong strong weak

4-Sets strong strong very very weak weak 163

To conclude the discussion of prime form usage in the vertical plane, these conclusions can be drawn. Hypotheses 1 and 2 have been well supported in the four quartets. Distinct differences in frequency of various prime forms in each quartet were observed. Hypothesis 3 is strongly supported in Champagne's quartet, moderately supported in the

Somers and Welnzweig works, and negated in Coulthard's quartet. Simi­ larity relations are effective in some instances for relating the most frequent prime forms. The fourth hypothesis is strongly supported only in the 3—sets in Coulthard's quartet. The Y-Square statistic used in testing the fourth hypothesis did not yield positive conclusions for the most part.

Similarity Relations Between the Two Planes

This section draws upon data for both the horizontal plane and the vertical plane in order to compare prime form usage in the two planes. These comparisons are made by testing for similarity relations in order to support (or negate) the following hypothesis: Hypothesis 5— In a given quartet the set forms used most fre­ quently in the horizontal plane will have an equivalent or a close similarity relation to those used most frequently in the vertical plane.

The purpose of such comparisons is to explore the possibility that a given composer may "prefer" the same sets or closely related ones in both linear and simultaneous writing. Although the question of his

"preference" cannot be answered directly, it may be approached oblique­ ly by means of determining relations in the selected string quartet.

The number and kinds of relations necessary to substantiate the hypoth­ esis require critical interpretation in each situation, as the follow­ ing discussion shows. 164 Of the 3-set forms In Coulthard, there are four frequent ones

In the horizontal plane (3-3, 3-2, 3-1, 3-4) and three in the vertical

plane (3-11, 3-10, 3-8). When the two groups are compared with each

other for equivalent forms and similarity relations, two pairs of R^,

Rp relations, 3-3, 3-11 and 3-4, 3-11, emerge. The former relation is

more significant in this context: 3—3 is the most frequent form hori­

zontally (26.007), and 3-11 is by far the most frequent form vertically

(32.757). In the most important instance (i.e., the most frequent

forms in either plane), then, the hypothesis is confirmed for 3-sets.

Of the horizontal 4^set forms, 4-1, 4-3, 4—7, 4-8, and the vertical

forms, 4-27, 4-19, 4-18, that must be considered for Coulthard, R2 , Rp

holds only for 4—7, 4—19 (14.557, 16.667). Although the one relation­

ship may be noteworthy, there is not enough support here to confirm

the hypothesis.

The most frequent horizontal 3-set forms in Champagne's quartet

are 3-2, 3-6, 3-1, and 3—8; the vertical forms are 3-3, 3-4, 3-5, and

3—7. R^, Rp relations exist between the two highest frequency forms,

3—2 (23.14%) and 3—3 (18.487), and for the less consequential pair

3—2, 3-7. Thus for Champagne, Hypothesis 5 is confirmed in the most

important instance. The most frequent 4-set forms are also related:

R2 , Rp holds for 4-11 (16.497) and 4-19 (11.177). In comparing the

horizontal 4-set forms, 4-11, 4-1, with the vertical 4-set forms, 4-19,

4-11, 4-13, 4-18, one form, 4-11, Is used in both planes (16.497 hor­

izontally, 08.31% vertically). Thus Hypothesis 5 is confirmed by one similarity relation and one "equivalence" relation (i.e., use of the same form). 165

The hypothesis Is confirmed to a lesser degree In the horizon­

tal 3—set forms, 3-2, 3—1, and the vertical 3-set forms, 3-4, 3-7,

3-2, 3-5, for Somers. 3—2 is used in both planes, 3-2 and 3-7 are in

the R^, relation, and 3—1, 3—2 are related by R£, Rp. Because the

two most frequent forms are unrelated, the hypothesis is not confirmed, although the relations mentioned do lend some support to it. No rela­

tions exist in the 4—set forms for Somers.

The strongest confirmation of Hypothesis 5 may be found in

Welnzweig's quartet. Between the four most frequent vertical horizon­ tal 3-set forms (3-1, 3-2, 3-3, 3-5) and the four most frequent verti­ cal 3-set forms (3—3, 3—2, 3-4, 3—5), there are three equivalence rela­ tions, and five similarity relations. Three of the four horizontal forms are also employed vertically: 3-2 (20.58%, 11.87/S), 3-3 (15.19%,

20.62%), and 3-5 (08.33%, 10.62%). Furthermore, four pairs of forms are related by R^, Rp (3-2, 3—2; 3—3, 3-2; 3-3, 3-4; 3—5, 3-4) and one pair by R2 « Rp (3—1, 3-2). Although the most frequent pair is not related, the number and type of relations among Welnzweig's 3-set forms give very strong support to Hypothesis 5.

A similarly firm substantiation of the hypothesis is found in his 4-set forms. Form 4-2 is employed most frequently in both planes:

27.53% and 13.13%, a notable Instance of the equivalence relation.

Also, 4-1, 4-2 and 4—16, 4-6 are related by R2 , Rp; and 4-16, 4-14, by

R^, Rp. Of the most frequent forms In the two planes, each is related to at least one form In the opposite plane by either an equivalence or a similarity relation. Thus, Weinzwelg shows the strongest and most consistent confirmation of Hypothesis 5 among the four composers. 166 Graphs comparing 3-sets and A-sets between planes for each

quartet are presented on the following pages. The first pair shows

Coulthard's use of horizontal 3-sets and vertical 3-sets, followed by horizontal A—sets and vertical A—sets. The third pair contrasts

Champagne's horizontal 3—sets and vertical 3—sets, and so forth. Also based (as in Chapter XII) on the percentages of prime form usage,

these graphs provide a visual picture of the pitch class and interval

class organization in both planes for each quartet. Because the forms

listed first (e.g., set 3-1, 3—2, etc. in the 3-set graphs) contain high icl counts and those listed last (e.g., set 3-11, 3-12) contain no

icl counts, the graphs may be considered to provide clues to the degree of "consonance" or "dissonance" as measured by icl content in the prime forms. Figure 26. Comparative Graph for Coulthard, Horizontal 3-Sets

45

30

15

0 10

Figure 27. Comparative Graph for Coulthard, Vertical 3—Sets 168

4 -1 01C Cl 0M) Pn 15

10 15 20 25

Figure 28. Comparative Graph for Coulthard, Horizontal 4-Sets

45

30

os - n « p-t 15

d 10 15 20 25

Figure 29. Comparative Graph for Coulthard, Vertical 4-Sets 169

45

30

3 o 0J Pn 15

10

Figure 30, Comparative Graph for Champagne, Horizontal 3-Sets

451—

30

o3 M 0) A< 15

10

Figure 31. Comparative Graph for Champagne, Vertical 3-Sets 170

3C a)a o M (ua) 15' n D = c rrizbd ] = a i ~ n ~ n - 10 15 20 25

Figure 32. Comparative Graph for Champagne, Horizontal 4-Sets

45

3 0

4 -1 S UCJ 0) p-> 15

10 15 20 25

Figure 33. Comparative Graph for Champagne, Vertical 4-Sets 171

Figure 34. Comparative Graph for Somers, Horizontal 3—Sets

45

30 e Cl u

15

Figure 35. Comparative Graph for Somers, Vertical 3-Sets 172

45

30

4J e) o0 a p- 15

10 15 20 25

Figure 36. Comparative Graph for Somers, Horizontal 4-Set3

451—

30 o§ a)H Pi 15

m ~i— i 10 15 20 25

Figure 37. Comparative Graph for Somers, Vertical 4-Sets 173

30 a 0a> u 01 &4

15

~ ) - T l — 10

Figure 38. Comparative Graph for Weinzweig, Horizontal 3-Seta

45 —

30 4J g uCJ 0> P-i 15

10

Figure 39. Comparative Graph for Weinzweig* Vertical 3—Sets 174

45f—

3C

AJ so VI

<— r u - n l— l or 10 15 20 25

Figure 40. Comparative Graph for Weinzweig, Horizontal 4-Sets

451—

30

S u

10 15 20 25

Figure 41. Comparative Graph for Weinzweig, Vertical 4-Sets 175

A summary of the support for Hypothesis 5 is provided in the following table. Each of the four quartets Is classified with regard to 3—sets and 4-sets in these terms: very strong, strong, fairly strong, fairly weak, weak, very weak. For example, Hypothesis 5 is strongly confirmed In Coulthard's 3-sets.

Examination of the table shows that forms are closely related in the 3—sets but not in the 4—sets in Coulthard's work. Considering the emphasis upon the minor and major triad (form 3-11) discussed earlier, we may assume that organization of 3-sets Is more evident than that for 4—sets in this work. Champagne's quartet shows fairly strong support of the hypothesis, while the reverse (lack of support) occurs in Somers' quartet. The most conclusive support Is found In both

3—sets and 4-sets In Welnzweig's work. As judged by relationships among prime forms compared between planes, this quartet shows the clearest Indicators of pitch class and interval class organization. 176

TABLE 46

Degree of Confirmation for Hypothesis Relative to Horizontal and Vertical Planes

Quartets Hypothesis 5

Co 3-Sets strong

4—Sets weak

Ch 3-Sets fairly strong

4—Sets strong

So 3-Sets fairly weak

4-Sets very weak

Wz 3-Sets very strong

4-Sets very strong CHAPTER V

EVALUATION AND CONCLUSIONS

The mechodology used In this dissertation has made it possible to analyze the pitch organization of four quartets, which though they are individually distinctive, fall within a spectrum of styles ranging from "tonal/non-tonal" to "atonal." Although the level of the analytic findings that have been obtained may be viewed as elemental, the strength of the method rests in the basis it provides for comparing the quartets in ways that are not possible with other systems of analysis.

A number of improvements in methodology were developed for this study.* Examination of a broad sample of prime forms and Interval vectors (versus pitches and intervals) throughout each composition proved to be a more rigorous procedure and provided a wider scope and was more productive than the typical practice of selecting only certain sonorities from sample passages for ancedotal discussion.

In this study, traditional means for defining pitch collections were respected to a high degree. Given the particular quartets, the selection of the beat as a norm for sampling vertical groups coincides for the most part with the perceived rhythmic structures. Likewise,

*See pages. 85-97.

177 178

the selection of horizontal sets followed a traditionally determined

set of criteria, detailed fully in Chapter III.^

Consistency was assured by means of absolutelyuniform measure­ ment. Sets were measured by numerical ordering of pitch classes when

transposed (theoretically) to "c-level"^— that is, to the prime forms.

The accompanying interval vectors uniformly classify the sets according

to their Interval class content. Accordingly, prior analytic interpre­

tations regarding constructs such as altered triads, clusters, and

basic cells are considered premature. The percentage distributions may

point the way toward such analytic Interpretations, but these become

explicit only after measurement of a broad sample.

Systematic methods provided control In interpreting the data.

A valuable means of evaluating the degree of confirmation (or negation)

of certain hypotheses is found in the Y-Square statistic. Given the

large numbers of sets measured, systematic Interpretation proved to be

a reliable means for specifying supportable conclusions.

An equal classification of both "positive" (manifest differ­

ence) and "negative" (no manifest difference) Information is another

strength of the method. Thus the emphasis is placed on the process of analysis, as opposed to the product of that analysis. Although the

question of whether the information obtained is positive or negative may be a matter of some concern, this concern need not be paramount.

Much of the information In this study may be interpreted as "negative"

2 See pages 101-06. 3 See pages 64—65. 179

in character— that is, a number of the conclusions are of the type "no

relationship exists.” There are two reasons for respecting negative

Information, however. First, all statements are based upon quantita­

tive measurement. Hence, both positive and negative findings are use­

ful. Second, the reporting of a negative conclusion with regard to a given question may provide guidance to future investigators.

The criteria employed for segmentation (summarized above) will

influence the degree of strength latent in the "positive" or "negative" findings. Those criteria employed in this study have yielded suffic­

ient differences in prime form usage to allow for the stylistic com­ parisons reported in Chapters III and IV. Insofar as it is stated,

Forte's criteria for grouping pitch classes into sets appears to be similar to the criteria employed successfully in this investigation:

. . . segmentation. By segmentation is meant the pro­ cedure of determining which musical units of a composi­ tion are to be regarded as analytical objects. Whereas this process is seldom problematic in tonal music, due to the presence of familiar morphological formations (harmonies, contrapuntal substructures, and the like), it often entails difficulties in the case of an atonal work . . . the term primary segment will be used to designate a configuration that is isolated as a unit by conventional means, such as a rhythmically distinct melodic figure. For the most part such segments are indicated by some notatlonal feature, for example, by a rest or beamed group, and offer no novel problems. Similarly, chords, in the sense of vertical groupings, and ostinato patterns are not difficult to identify as primary segments.

The influence of conventions derived from tonal practice, such as metric beats and melodic phrases that have traditional harmonic implications

^Forte, The Structure of Atonal Music, p. 83. 180

Intrudes upon, the selection of sets In "atonal" music. Forte's approach to segmentation Implies the retention of familiar units as groupings to be studied in a less familiar context— one where functional harmony is absent. In the act of selection, then, units recognized originally In tonal contexts are retained. The same reasoning was followed in the present study, and at this point the best that can be said regarding the apparent paradox Is that the technique successfully provides analy­ tic Information not available with other methods.

The difficulties related to segmentation in the linear plane suggest similar problems regarding the concept "interval vector." An interval vector can be literally present (sound) only when all the notes in a set are sounded simultaneously. Although a prime form may occur in linear fashion, its associated interval vector is by no means obvious in a horizontal statement. For this reason the term may be considered to be more harmonically than linearly oriented. In a review of Forte's text Eric Regener makes this point, although perhaps a bit too strongly:

A theory of harmony differs from a theory of chords, pre­ sumably, in that it deals with relationships based on succession in time ("progressions"). Since Forte's theory does not deal with temporal relationships in any such sys­ tematic way, It therefore seems appropriate to consider it basically a theory of chords.^

Regener exaggerates when he limits Forte's theory to chords. Although the concept "interval vector" is more applicable to simultaneities

^Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music. XIII (Fall-Winter, 1974), p. 193. 181

(a more neutral term than "chord*'), the concept of "prime form" is

applicable to linear writing, as evidenced by the large frequencies of

certain prime forms found In the horizontal plane of the Somers and

Welnzwelg quartets in particular.®

A limitation of the method, alluded to in Begener's comments,

is the ignoring of temporal relationships such as progressions. Further

research about the application of set theory to atonal music ought to

involve an accounting for temporal succession of prime forms, because

it may be possible to establish for a given composer patterns of formal

succession. Perhaps the revelation of such patterns could support

Stravinsky's notion of the polar attraction of sounds (sets, in this

case):

Having reached this point, it is no less Indispensable to obey, not new idols, but the eternal necessity of affirm­ ing the axis of our music and to recognize the existence of certain poles of attraction. Diatonic tonality is only one means of orienting music towards these poles. The function of tonality is completely subordinated to the force of attraction of the pole of sonority. . . .

So our chief concern is not so much what is known as tonality as what one might term the polar attraction of sound, of an Interval, or even of a complex of tones.^

Stravinsky's reference to "the polar attraction of sound, of an inter­ val, or even of a complex of tones" above may be an area for further exploration that is entirely consistent with the concerns of set theory and that may subsume the issue of "tonality versus atonality," as

Stravinsky suggests.

®See pages 121-24. y Igor Stravinsky, Poetics of Music (Cambridge: Harvard Univer­ sity Press, 1947), pp. 35-36. 182

My Investigation of prime form usage In string quartets has made it obvious that further work needs to be pursued In the area of

temporal succession. It appears that a necessary first step would be

to encode the music (whether the quartets chosen here or some other music) Into a computer. Various operations relative to the localized

expression of specific prime forms would then be feasible. Not only might patterns of prime form succession be revealed for a given compo­

sition, but also the identification of specific pitch classes and arrangements of various sets might yield more specific information than

could be obtained by the methodology presented in this study.

The area of perception with respect to both tonality and aton— ality also deserves further exploration. Questions concerning our

ability to perceive mathematically expressed relations (such as Forte's

similarity relations) need to be answered. Paul Griffiths in his re­ view of The Structure of Atonal Music touches upon this question in an

evaluation of Forte's approach.

It is not Forte's purpose to probe historical and aesthetic questions, such as how atonal music of this consistency and complexity came Into being without, it seems, elaborate pre-composltional planning (and, if structures based on Forte's set-complex theory can be composed "intuitively," that might affect our view of the aural perceptibility of such structures).^

In conclusion, the present study has shown that the various

techniques developed for this investigation have provided an effective

analysis of the pitch class and Interval class structure of the selec­

ted quartets. Meaningful differences In prime form usage both within

^Paul Griffiths, "Atonality Analyzed," Musical Times. CXV (October, 1974), p. 848. individual works and between the works of different composers have been observed. Although the horizontal and vertical surveys were purpose­ fully aimed at an elemental level, the results are nevertheless con­ crete and convincing descriptions of stylistic variety in the quartets.

The method is judged to be an effective tool for quantifying elemental aspects of set usage in tonal/non-tonal and atonal music. In this respect the method is one in a series of steps taken in the field of set theory that will ultimately lead us to more satisfactory explana­ tions of that music which requires new perspectives for interpretation. APPENDIX A

Prime Forms and Inversions Notated Conventionally and in Integer Notation: 3-Sets and 4-Sets

P> ______— 7 --- L'~okzr. --- v 0 — y O — ^ — i-— i------L 1 ~----- r bo-e-i* 1 3-1 inv 3-2 Inv

0 ,1,2 0,11,10 0,1,3 0,11,9

I) f --- 1— I7J------7?----- b ----- 1------0 ----- M>»3------k£o 1

3-3 inv 3-4 inv

0,1,4 0,11,8 0,1,5 0,11,7

I------— j p O g - .. 4 3— L - e ------— ------\ .....' 3-5 inv 3-6 Inv

0,1,6 0,11,6 0,2,4 0 ,10,8

184 185

k £ ------1 — - o ------^ 0 ------v - e * 3—7 Inv 3—8 inv

0,2,5 0,10,7 0,2,6 0,10,6

)------L - - ;-n ------i}-0------?-- B------V --,U ... i * o - r-- — 1 '---- 1 3—9 Inv 3—10 Inv

0,2,7 0,10,5 0,3,6 0,9,6

3-11 Inv 3-12 Inv

0,3,7 . 0,9,5 0,4,8 0,8,4

— k ogJfo — b k O-^-ifO — 1 *—

J 1 h ogljo

4-1 Inv 4-2 inv

0 ,1,2,3 0,11,10,9 0,1,2,4 0,11,10,8 186

__ )------1--- f- -— 7T---- L ■ fl / j. p O o Jrzr — ^ g g y O 5— j—f— — t ---- ... / /j i --- & O x ^ O ---- b a g g e r -

4-3 Inv 4—4 Inv

0,1,3,4 0,11,9,8 0,1,2,5 0,11,10,7

------1— hio^ho— — ^

4-5 Inv 4-6 Inv

0,1,2,6 0,11,10,6 0,1,2,7 0,11,10,5

------^---- t-yj------—— JL --- 0 1 0 a ------— b - 8 ~j- 4— o — r — b q a .

4-7 Inv 4-8 Inv

0,1,4,5 0,11,8,7 0,1,5,6 0,11,7,6 D j _ - ----L ----- 1 2 — ^ - g - g - ... 5 --- ■rb-'Q-h-Q— — b - % a ------irznr— it O % r ...... — ■ Q" 4-9 Inv 4-10 Inv

0,1,6,7 0,11,6,5 0,2,3,5 0,10,9,7 187

---- ) ------, t

7------L / — o------~ g g — ------b h - % ^ k S L p " ------J p g 4-11 Inv 4-12 inv

0 ,1 ,3 ,5 0 ,1 1 ,9 ,7 0 ,2 ,3 ,6 0 ,1 0 ,9 ,6

&r ------, A g g ..... ------^ # 0 ------") In , o 6 ^ ------W it------y ' O g TT" 4-13 inv 4-14 inv

0 ,1 ,3 ,6 0 ,1 1 ,9 ,6 0 ,2 ,3 ,7 0 ,1 0 ,9 ,5

h r ------— ____J " l \ ° n g | ------e.Jf-G§------r ° “2 .____

4-15 inv 4-16 inv

0 ,1 ,4 ,6 0 ,1 1 ,8 ,6 0 ,1 ,5 ,7 0 ,1 1 ,7 ,5

f t

1------1---- Z T 7 ------/ a 0 b ° o 3 ) ------4—— O f ----- V 9 ------r - d ------& 1------£-€K -frO----- h j ^ P

4-17 inv 4-18 inv

0 ,3 ,4 ,7 0 ,9 ,8 ,5 0 ,1 ,4 ,7 0 ,1 1 ,8 ,5 188

------If ----- T-.l., ------! > Y Z ~ .... — b >— .... ° % -4 -JM 4 ------& ------£ ------4 _ g _ o ------1 4-19 Inv 4-20 inv

0 , 1 ,4 ,8 0 ,1 1 ,8 ,4 0 , 1 , 5 ,8 0 ,1 1 ,7 ,4

ft / — 7 i k i n « i L .. r ^ g - 4 * 5 — 4 - e ------£------o J L

4-21 inv 4-22 inv

0 ,2 ,4 ,6 0 , 1 0 , 8,6 0 ,2 ,4 ,7 0 ,1 0 ,8 ,5

Ift J/ ...... / Q ------/ i L a _ J ------— 6 - n — ------b P % 0 ------a ~ 0 ------0 ------• r ^ -e~

4-23 inv 4-24 inv

0 ,2 ,5 ,7 0 ,1 0 ,7 ,5 0 ,2 ,4 ,8 0 ,1 0 ,8 ,4

o j .:...... 6 - o ° ' _4 £V ------g ------4-g-Z T ------1 ------9 ------1 ------4-25 inv 4-26 inv

0 , 2 , 6,8 0 ,1 0 ,6 ,4 0 ,3 .5 ,8 0 ,9 ,7 ,4 « 189

r\ —? j ■■ ------Lii ^ ------, a ------? > g ------: ------j\ b ° - o — f------Lf------g ------P -----& ------——J7----0 ------*

4-27 inv 4-28 Inv

0 ,2 ,5 ,8 0 ,1 0 ,7 ,4 0 ,3 ,6 ,9 0 ,9 ,6 ,3

4-29 inv

0,1,3,7 0,11,9,5 APPENDIX B

3-Sets and 4-Sets In Integer Notation with Interval Vectors

3-Sets

Numerical Prime Interval Name Form Inversion Vector

3-1 0 , 1, 2 (12) 0, 11, 10 (equivalent) 210000 3-2 0, 1, 3 0, 11, 9 111000 3-3 0* 1 , * 0, 11, 8 101100 3-4 0, 1, 5 0, 11, 7 100110

3-5 0, 1 , 6 0, 11, 6 100011 3-6 0, 2, 4 0, 10, 8 (equivalent) 020100 3—7 0, 2, 5 0, 10, 7 011010 3-8 0 , 2 , 6 0, 10, 6 010101

3-9 ' 0, 2, 7 0, 10, 5 010020 3-10 0, 3, 6 0, 9, 6 (equivalent) 002001 3-11 0, 3, 7 0, 9, 5 001110 3-12 0, 4, 8 0, 4 (equivalent) 000300

190 191

4-Sets

Numerical Prime Interval Name Form Inversion Vector

4-1 0 1. 2 , 3 (12) 0, 11. 10, 9 (equivalent) 321000 4-2 0 1, 2 , 4 0 , 11. 10, 8 221100 4-3 0 1, 3, 4 0 , 11. 9, 8 (equivalent) 212100 4-4 0 1, 2 , 5 0 , 11. 10, 7 211110

4-5 0 1, 2 , 6 0 , 11, 10, 6 210111 4-6 0 1, 2 , 7 0 , 11. 10, 5 210021 4-7 0 1, 4, 5 Of 11. 8, 7 (equivalent) 201210 4-8 0 1, 5, 6 Of 11, 7, 6 (equivalent) 200121

4-9 0 1, 8, 7 0 , 11. 6, 5 (equivalent) 200022 4-10 0 2 , 3, 5 0 , 10, 9, 7 (equivalent) 122010 4-11 0 1, 3, 5 0 , 11. 9, 7 121110 4-12 0 2 , 3, 6 0 , 10, 9, 6 112101

4-13 0 1 , 3, 6 o, 11. 9, 6 112011 4-14 0 2 , 3, 7 o* 10, 9, 5 111120 4-Z15 0 4, 6 o. 11, 8 , 6 111111 4-16 0 1, 5, 7 0, 11. 7, 5 110121 4-17 0 3, 4, 7 0 , 9, 8 , 5 (equivalent) 102210 4-18 0 1, 4, 7 0 , 11, 8 , 5 102111 4-19 0 1, 4, 8 o, 11, 8 , 4 101310 4-20 0 1, 5, 8 0, 11. 7, 4 101220

4-21 0 2 , 4, 6 o, 10, 8, 6 (equivalent) 030201 4-22 0 2 , 4, 7 o, 10, 8 , 5 021120 4-23 0 2 , 5, 7 o, 10, 7, 5 (equivalent) 02 1030 4-24 0 2 , 4, 8 o, 10, 8 , 4 020201

4-25 0 2 , 6, 8 o» 10, 8, 4 (equivalent) 020202 4*?26 0 3, 5, 8 o. 9, 7, 4 (equivalent) 012120 4-27 0 2 , 5, 8 o, 10, 7, 4 012111 4-28 0 3, 6, 9 0, 9, 8, 3 (equivalent) 004002

4-Z29 0, 1, 3, 7 0 , 11, 9, 5 111111 APPENDIX C

All Prime Forms Identified by their Integer Intervals

3-Sets

Intervals Intervals NameName

3-1 3-3 3-2 3-7 3-3 3-10 3-4 3-11 3-5 3-4

3-2 3-8 3-6 3-11 3-7 3-12 3-8 3-5 3-9 3-9

192 4-Sets

Name Intervals

1 4-1 3 2 4-2 4 3 4—4 1 4 4-5 2 5 4-6 3

1 4-3 1 2 4-11 2 3 4-13 1 4 4-Z29 2

1 4-7 3 2 4-Z15 1 3 4—18 2 4 4-19 3

1 4-8 1 2 4-16 2 3 4-20 3

1 4-9

1 4-2 1 2 4-10 2 3 4-12 1 4 4-14 2

1 4-11 1 2 4-21 1 194

5-Sets

Intervals Name Intervals Name

1 , 1 , 1 , 1 5-1 1 , 2 , 4, 1 5-20 1 , 1 , 1 , 2 5-2 1 , 3, 1 , 1 5-6 1 , 1 , 1 * 3 5-4 1 , 3, 1 , 2 5-Z18 1 , 1 , 1 , 4 5-5 1 , 3, 1 , 3 5-21

1 . 1 . 2 , 1 5-3 3, 2, 1 5-19 1 , 1 , 2 , 2 5-9 1 . 3, 2 , 2 5-30 1 -J 1 , 2 , 3 5-Z36 1 , 3, 2 , 3 5-32 1 , 1 , 2 , 4 5-13 1 . ^0 £ 3, 1 5-22

1 , 1 , 3, 1 5-6 1 , 4, 1 . 1 5-7 1 , 3, 2 5-14 1 , 4, 2 , 1 5-20 1 , 1 , 3, 3 5-Z38 2 , 1 , 1 . 1 5-2

1 . 1 , 4, 1 5-7 2, 1 , 1 . 2 5-8 1 , 1 , 4, 2 5-15 2 , 1 , 1 , 3 5-11

1 , 2, 1 , 1 5-3 2 , 1 , 2 , 1 5-10 2, 1 , 2 , 2 5-23 1 , 2 1 , 2 5-10 2 , 1 . 2 , 3 5-25 2 1 , 3 5-16 2 , 1 , 3, 1 5-Z18 1 , 2 , 1 , 4 5-Z17 2 1 1 » 3, 2 5-28 l t 2 , 2 , 1 5-Z12 2 v 2 , 1 » 1 5-9 1 , 2 , 2 , 2 5-24 2 , 2 , 1 , 2 5-23 1 , 2 , 2 , 3 5-27 2, 2 , 1 , 3 5-26

1 , 2 , 3, 1 5-19 2, 2 , 2 , 1 5-24 1 , 2 , 3, 2 5-29 2, 2 , 2 , 2 5-33 2 , 3, 3 5-31 2 , 2 , 2 v 3 5-34

2 , 2 , 3, 1 5-30 2 , 2 , 3, 2 5-34

2, 3, 1 . 1 5-14 2, 3, 1 , 2 5-23

2, 3, 2 , 1 5-29 2 , 3, 2 , 2 5-35 195

5-Sets (Continued)

Intervals Name Intervals Name

5-15 5-31

5-4 5-5 5-11 5-Z37

5-16 5-Z17 5-26

5-21 5-13

5-Z36 5-25

5-27 5-34

5-32

5-Z38 196

6-Sets

Intervals Name Intervals Name

1* 1 . 1, 1, 1 6-1 1 2 , 1, It 1 6-Z3 1. It It 2 6-2 1 2, 1, 1 , 2 6-Z10 1, 1, It 1, 3 6-Z36 1 2, It It 3 6-14 1 , 1, It 1, 4 6-Z37 1 2 , It 2 , 1 6-Z13

1 , 1 , It 2 , 1 6-Z3 1 2 , 1, 2 , 2 6-Z24 1. 1 . It 2 , 2 6-9 1 2 , 1, 2 , 3 6-27 1 , 1, 1 , 2 , 3 6-Z40 1 2 , It 3, 1 6-Z19

1* 1 » It 3, 1 6-5 1 2 , It 3, 2 6-Z49 1, 1 , 1 , 3, 2 6-Z41 1 2 , 2 , 1, 1 6-Z12 1. 1, 1, 3, 3 6-Z42 1 2, 2 , 1 , 2 6-Z25

1 * 1. 1 , 4, 1 6-Z38 1 2 , 2 , 1 , 3 6-Z28

1, 1, 2 , 1, 1 6-Z4 1 2 , 2 , 2 , 1 6-Z26 1, 1, 2 , It 2 6-Z11 1 2, 2 , 2 , 2 6-34 1, 1, 2 , 1, 3 6-15 1 2 , 2 , 3, 1 6-31

1* It 2 , 2 , 1 6-Z12 1 2, 3, 1 , 1 6-18 It 1, 2 , 2 , 2 6-22 1 2 , 3, It 2 6-30 1, 1, 2 , 2 , 3 6-Z46 1 2 , 3, 2 , 1 6-Z29

It It 2 , 3, 1 6-Z17 1 3, 1, 1 , 1 6-5 1, 1, 2 , 3, 2 6-Z47 1 3, It 1, 2 6-16

1, 1, 3, 1, 1 6-Z6 1 3, It 2 , 1 6-Z19 It It 3, 1, 2 6-Z43 1 3, 1 , 3, 1 6-20 1 , 1, 3, It 3 6-Z44 1 3, 2 , 1* 1 ’ 6-Z17

It It 3, 2 , 1 6-18 1 3, 2 , 1, 2 6-Z50 It It 3, 2 , 2 6-Z48 1 3, 2 , 2 , 1 6-31

1 , 1 , 4, ^■t 1 6-7 1 . 4, 1, ^■t 1 6-Z38 197

6-Sets (Continued)

Intervals Name Intervals Name

2 1 , 1 , 1 , 1 6-2 2 , 3, 2, 1 , 1 6-Z47 2 1 , 1 , 1 . 2 6-8 3, 1. 1> 1 . 1 6-Z36 2 1 , 1, 1 , 3 6-Z39 3. 1 . 1, 1. 2 6-Z39

2 1 , lf 2 , 1 6-Z10 3, 1, 1. 2 , 1 6-14 2 1 , 1 , 2 , 2 6-21 3, I. 2 , 1. 1 6-15 2 1, 2 , 3 6-Z45 3, 1* 2 , 2 , 1 6-Z28

2 1 * 1 , 3, 1 6-16 3, 1 . 3, 1. 1 6-Z44

2 2 , 1 . 1 6-Z11 3, 2 , 1, 1 , 1 6-Z40 2 2 , 1, 2 6-Z23 3, 2 , 1. 1 . 2 6-Z45

2 1 , 2 , 2 , 1 6-Z25 3. 2 , 1. 2 , 1 6-27 2 1* 2 , 2 , 2 6-33 3, 2, 2, 1. 1 6-Z46

2 1 . 2 , 3, 1 6-Z50 3, 3, 1. 1, 1 6-Z42

2 1 . 3, 1 . 1 6-Z43 4, 1. 1. 1. 1 6-Z37

2 1 . 3, 2 , 1 6-30 2 2 , 1 , 1 6-9 2 2 , 1 , 1 , 2 6-21

2 2 , 1 , 2 , 1 6-Z24 2 2 , 1 , 2 , 2 6-32

2 2 , 2 , 1 , 1 6-22 2 2 , 2 , 1 , 2 6-33

2 2 , 2 » 2 , 1 6-34 2 2 , 2 , 2 , 2 6-35

2 2 , 3, 1, 1 6-Z48

2 3, 1, 1 , 1 6-Z41

2 3, 1 , 2 , 1 6-Z49 198

7-Sets

Intervals Name Intervals Name

1, 1, l t 1, 1, 1 7-1 1, 1 2, 2, 2, 1 7-30 1, 1, 1, 1, It 2 7-2 It 1 2, 2, 2, 2 7-33 1, 1, 1, 1, 1. 3 7-3 1, 1 2, 3, 1, 1 7-20

1* 1. 1* 1, 2, 1 7-4 It 1 3, 1, 1, 1 7-7 1, 1, 1, 1, 2, 2 7-9 It 1 3, It 2, 1 7-22 1» 1# It 1, 2, 3 7-10 1, 1 3, 2, 1, 1 7-20

It It It 1, 3t 1 7-6 It 2 1, !• 1, 1 7-4 1, 1, 1, 1, 3, 2 7-Z12 1, 2 1, 1, 1, 2 7-11

1, 1, 1, 2, 1 y 1 7-5 1, 2 1, 1, 2, 1 7-Z37 It It It 2, 1, 2 7-Z36 1, 2 1, 1, 2, 2 7-26 It It It 2, 1, 3 7-16 1, 2 1, 2, 1, 1 7-Z38

1, 1, 1, 2, 2, 1 7-14 1, 2 1, 2, 1, 2 7-31 It It It 2, 2, 2 7-24 1» 2 It 2, 2, 1 7-32

It It It 2, 3, 1 7-Z18 It 2 1, 2, 2, 2 7-34

It 1* It 3, 1, 1 7-7 It 2 1, 3, 1, 1 7-22 It It It 3, 1. 2 7-19 1, 2 2, 1, 1, 1 7-14

It It 2, 1, 1, 1 7-5 1, 2 2, 1, 1, 2 7-28 1, 1, 2, 1, It 2 7-13 It 2 2, It 2 , 1 7-32 It It 2, It 1. 3 7-Z17 1, 2 2, It 2 , 2 7-35

It It 2, 1, 2 » 1 7-Z38 1, 2 2, 2, It 1 7-30 1» 1, 2, l t 2, 2 7-27 It 3 1, It 1, 1 7-6

1, 1, 2, It 3. 1 7-21 It 3 1, 2, 1, 1 7-21

1, 1, 2, 2, It 1 7-15 It 3 2, 1, 1, 1 7-Z18 1, 1, 2, 2, It 2 7-29 2, 1 1, 1, 1, 1 7-2

2, 1 It It It 2 7-8

2, 1 1, 1, 2, 1 7-11 2, 1 1, It 2, 2 7-23 199

7-Sets (Continued)

,Intervals Name

2 1, 1 7-13 2 1, 2 7-25

2 2 , 1 7-28

2 1 1, 1 7-Z36 2 1 1, 2 7-25

2 1 2, 1 7-31

2 2 1, 1 7-29 2 1 1 , 1 7-19

2 1 1. 1 7-9 2 1 1, 2 7-23

2 1 2, 1 7-26 2 2 1, 1 7-27

2 2 2, 1 7-35 2 1 1, 1 7-24

2 1 2, 1 7-34

2 2 1, 1 7-33 2 1 1, 1 7-Z12

3 1 1, 1 7-3

3 2 1, 1 7-Z17

3 1 1, 1 7-16

3 1 1, 1 7-10 200

8-Sets

Intervals Name Intervals Name

1. 1. 1, 1, If 1 8-1 1, 1 2, 1, 2, 1, 2 8-27 1* 1, 1, .1. 1, 1, 2 8-2 1, 1 2, 1, 2, 2, 1 8-26 1, 1, 1, 1, 1, 1, 3 8-3 1, 1 2, 2f 1, 1, 1 8-16

1. If 1, 2, 1 8-4 1, 1 2, 2, 1, 1, 2 8-25 1, 1, 1, 1, l t 2, 2 8-11 1, 1 3, 1> 1, 1, 1 8-8

1* 1. 1. 1, 1, 3, 1 8-7 If 2 1. 1. If If 1 8-4

1, 1, 1, 1,2, 1, 1 8-5 1, 2 1, 1, 1, If 2 8-12 1* 1* 1. 1 , 2, If 2 8-13 If 2 1. 1, 1, 2, 1 8-17

1, 1, 1, 1, 2, 2» 1 8-Z15 1, 2 1. 1. 2, 1 # 1 8-19 1, 1, 1, 1, 2, 2, 2 8-21 If 2 If 2 f 1, 1, 1 8-18 00 00 CM 1» 1» 1. 1, 3, 1, 1 8-8 1, 2 1, 2, 1, 2, 1 1

1, 1, 1, 2, If If 1 8-6 1, 2 2, 1, 1, 1, 1 8-Z15 1* 1. 1. 2, 1, 1, 2 8-Z29 If 2 2, 1, 2, 1, 1 8-26

1# If If 2, 1, 2, 1 8-18 1, 3 1, 1, If 1, 1 8-7 If If If 2, If 2, 2 8-22 2, 1 1, 1, 1, If 1 8-2

If If If 2, 2, If 1 8-16 2, 1 1. 1. It 1, 2 8-10 If If If 2, 2, 1, 2 8-23

1, 1, 1, 3, 1, 1, 1 8-9 2, 1 1. 1. 1, 2, 1 8-12

It It 2, 1, 1, 1, 1 8-5 2, 1 1. 1. 2, 1, 1 8-14 I, 1, 2, 1, If If 2 8-14

I, 1, 2, 1, 1, 2, 1 8-19 If If 2, 1, 1, 2. 2 8-24

I, 1. 2, 1. 2, 1. 1 8-20 201

8-Sets (Continued)

Intervals Name

2, 1. 1. 2, 1. 1, 1 8-Z29

2, 1. 1, 2, 2, 1, 1 8-25

2, 1, 2, It 1. It 1 8-13

2, 1. 2, 1, 2, It 1 8-27

2, 1, 2, 2, 1, 1, 1 8-23

2, 2, 1. 1. 1. It 1 8-11

2, 2, 1. 1, 2, It 1 8—24

2, 2, It 2, 1. 1, 1 8-22

2, 2, 2, 1, 1. It 1 8-21

3, 1. 1. 1, 1. 1, 1 8-3 202

9-Sets

Intervals

9-1 9-2

9-3 9-6

9-4 9-7

9-5 9-8

9-10

9-5 9-9

9-11

9-4

9-12

9-3

9-11

9-10

9-2

9-9

9-8

9-7

9-6 SELECT BIBLIOGRAPHY

References

Beckwith, John. "Canada.” Dictionary of Contemporary Music. Edited by John Vinton. New York: E. P. Dutton and Co., Inc., 1974, pp. 119-24.

Benjamin, William E. Review of The Structure of Atonal Music by Allen Forte. Perspective of New Music. XIII. No. 1 (Fall-Winter. 1974), pp. 170-90.

Boretz, Benjamin. "Babbitt, Milton." Dictionary of Contemporary Music. Edited by John Vinton. New York: E. P. Dutton and Co., Inc., 1974.

. Review of Serial Composition and Atonality by George Perle. Perspectives of New Music. I, No. 2 (Spring, 1963), pp. 125—36.

Browne, Richmond. Review of The Structure of Atonal Music by Allen Forte. Journal of Music Theory, XVIII, No. 2 (Fall. 1974), pp. 390-415.

Canadian Broadcasting Corporation. Thirty-Four Biographies of Canadian Composers. Reprint of 1964 ed. St. Clair Shores, Michigan: Scholarly Press, 1972.

Cherney, Brian. Harry Somers. Toronto: University of Toronto Press, 1975.

Chrlsman, Richard. "Describing Structural Aspects of PItch-Sets Using Successive-Interval Arrays." Journal of Music Theory, XXI, No. 1 (Spring, 1977), pp. 1-28.

Forte, Allen. "A Theory of Set-Complexes for Music." Journal of Music Theory. XVIII, No. 2 (Winter, 1964), pp. 136-83.

. "Context and Continuity In an Atonal Work: A Set-Theoretic Approach." Perspectives of New Music. I, No. 2 (Spring, 1963), pp. 72-82.

. The Structure of Atonal Music. New Haven: Yale University Press, 1973.

203 204

Griffiths, Paul. "Atonality Analysed.” Musical Times. CXV (October, 1974), pp. 848-49.

Grout, Donald Jay. A History of Western Music. New York: W. W. Norton & Co., 1973.

Hanson, Howard. Harmonic Materials of Mode m Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts, Inc., 1960.

Kallmann, Helmut. "Canada." Harvard Dictionary of Music. Edited by Willi Apel. Second edition, revised and enlarged. Cambridge: The Belknap Press of Harvard University Press, 1969, pp. 122- 24.

______, editor. Catalogue of Canadian Composers. Revised and en­ larged edition. St. Clair Shores, Michigan: Scholarly Press, 1972 (reprint of 1952 ed.).

Lewln, David. "Forte’s Interval Vector, My Interval Function, and Regener's Common-Note Function." Journal of Music Theory. XXI, No. 2 (Fall, 1977), pp. 194-237.

MacMillan, Keith. "Canada." The Encyclopedia Americana; International Edition, II. New York: Americana Corporation, 1971, p. 437.

McNeal, Horace Pitman, Jr. "An Analytical Investigation of Igor Stravinsky’s Threni." Unpublished M.A. thesis, University of Georgia, 1974.

Newlln, Dika. Review of Serial Composition and Atonality by George Perle. Music Library Association Notes. XIX, No. 3 (June, 1962), p. 434.

Perle, George. Serial Composition and Atonality: An Introduction to the Music of Schoenberg. Berg, and Webern. Fourth edition, revised. Berkeley: University of California Press, 1977.

Regener, Eric. "On Allen Forte's Theory of Chords." Perspectives of New Music. XIII, No. 1 (Fall-Winter, 1974), pp. 191-212.

Sentleri, Alfred Richard. "A Method for the Specification of Style Change In Music: A Computer-Aided Study of Selected Venetian Sacred Compositions from the Time of Gabrieli to the Time of Vivaldi." Unpublished Ph.D. thesis, The Ohio State Unlveristy, 1978.

Slonlmsky, Nicolas. Review of Harmonic Materials of Modern Music by Howard Hanson. Music Library Association Notes. XVIII, No. 3 (June, 1961), pp. 415-16. Stravinsky, Xgor. Poetics of Music* Cambridge: Harvard University Press, 1947.

Vinton, John. "Pitch Class." Dictionary of Contemporary Music. New York: E. P. Dutton and Co., Inc., 1974, p. 577.

Musical Scores

Champagne, Claude. Quatuor 5. Cordes (1956). Toronto: Berandol, 1974.

Coulthard, Jean. String Quartet No. 2 (Threnody). Unpublished, 1953.

Somers, Harry. String Quartet No. 3. Unpublished, 1959.

Weinzweig, John. String Quartet No. 3. Unpublished, 1962.