“NUR EIN WORT, NUR EINE BITTE”: SETS AND TENSION-DEGREES IN ERNST KRENEK’S FÜNF LIEDER

NACH WORTEN VON FRANZ KAFKA, OP. 82

by

Makoto Mikawa August, 2004

A thesis submitted to the Faculty of the Graduate School of

The State University of New York at Buffalo in partial fulfillment of the requirements for the

degree of

Master of Arts

Department of Music

ACKNOWLEDGEMENTS

First and foremost, I would like to thank Dr. Martha Hyde for her patience,

encouragement, and guidance. Dr. Hyde has provided various analytic approaches that

helped to improve the contents of this thesis, in addition to the enormous amount of time

spent discussing and editing the details. Every comment and suggestion from Dr. Hyde

has guided me to a deeper level of analytic insight and thought. This project would have

never been completed without her support.

I would also like to thank the committee members. Dr. Charles J. Smith, who has

supplied me the analytical technique for the formal and harmonic exploration which has

tremendously broadened my point of view; and Dr. Peter Schmelz, who has taught me

about the valuable historical materials of twentieth-century music. I would also like to

thank Dr. Richard S. Parks (outside reader) for his guidance and advice. Especially,

Chapter 4 would not have been accomplished without Dr. Parks’ thorough comments.

I am also grateful to the music librarians and staff in the Music Library of SUNY

at Buffalo. An immense amount of excellent resources and their quick response to my

requests not only aided in the completion of this project, but also contributed to my

academic development.

I wish to thank my colleagues for sharing precious ideas and resource materials. I

would also like to thank my friends in Germany, Ute Dannhel and Siegfried and Esther

Imhoff, for helping with translations and providing the necessary resources which were

an important part of this thesis.

ii

Finally, I am thankful to my parents for their support and understanding of my

studies.

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

LIST OF MUSICAL EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTER 1 Pitch-Class Set/ Harmony

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Row Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Harmonies Derived from the Primary Series of Fünf Lieder . . . . . . . . . 12

Form of “Nur ein Wort, nur eine Bitte” . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Simultaneity of Motivic and Melodic Pitch-Class Sets . . . . . . . . . . . . . 40

A Pitch-Class Set for the Climax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

CHAPTER 2 Vocal Configuration

The Vocal Line Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Pitch-Space Projection and Voice-Leading . . . . . . . . . . . . . . . . . . . . . . . 50

Transformational Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

CHAPTER 3 Tension-Degrees

Krenek’s “Tension-Degrees” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

“Tension-Degrees” and Pitch-Class Sets . . . . . . . . . . . . . . . . . . . . . . . . . 64

Application of Tension-Degrees to “Nur ein Wort” . . . . . . . . . . . . . . . . 78

iv

Three Primary Tension-Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Three Additional Tension-Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Expansion and Contraction of Tension-Degrees . . . . . . . . . . . . . . . . . . . 87

Tension-Degree and Musical Context . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

CHAPTER 4 Pitch-Class Set Genera

The Application of Forte’s Pitch-Class Set Genera . . . . . . . . . . . . . . . . . 97

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

APPENDIX I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

APPENDIX II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

APPENDIX III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

v

LIST OF MUSICAL EXAMPLES

1.1 Primary row form (P9) and trichordal partitioning with pitch-class sets of Fünf Lieder nach Worten von Franz Kafka, op. 82 (1937/38) . . . 4

1.2 Primary row forms of Krenek’s works composed before and after Fünf Lieder . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 The beginning of the first movement of the Sixth String Quartet, op. 78, mm. 1 – 9 . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 The beginning of the second movement of Zwölf Variationen in drei Sätzen, op. 79, mm. 127 – 30 . . . . . . . . . . . . . 9

1.5 “Dancing Toys” in Twelve Short Piano-Pieces written in the Twelve-Tone Technique, op. 83, mm. 10 – 12 . . . . . . . . . 11

1.6 Segmentation by the row forms of P9 and I9 in “Nur ein Wort, nur eine Bitte” mm.1 – 12 . . . . . . . . . . . . . . . . . . . . . . . 18

1.7 “Nur ein Wort” indicating lines dividing into two sections and fifteen subsections . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.8 Motivic pc sets: 3-2, 4-13, 4-1, 4-21, 4-Z29, and 5-29 . . . . . . . . . . . . . 23

1.9 Segmentation for the introduction in m. 1 (pitch-reduction) . . . . . . . . . 23

1.10 Absence of pc set 3-2: segmentation for the second subsection in the first section, mm. 1 – 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.11 The vocal line with annotated pc set names and circled 3-2 . . . . . . . . . . 26

1.12 Tetrachordal motivic segments with pc set 3-2 motifs, mm. 4 – 5 . . . . . 28

1.13 Four re-ordered hexachords derived from basic partitionings of the series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.14 Pentachords in the introduction, m. 1

a) Segmentation based on sustained pentachords in the introduction . . . 31

b) Representation for verticalized pentachords extracted from the hexachord and trichord as an inclusion . . . . . . . . . . . . . . . . . . . . . 31

vi

1.15 An extracted primary harmony of pc set 5-29 from P9 (ONs 7-B) in the codetta, mm. 9 – 10 . . . . . . . . . . . . . . . . . . . . . . . . 32

1.16 Pitch-class set 5-29 {457A0} in the first and fourth subsections . . . . . . 33

1.17 Occurrence of motivic pc set 5-29 in the second section . . . . . . . . . . . . 34

1.18 “Espressivo” pc set 3-9 {B16} included in 5-29, at the climax, mm. 17 – 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.19 Pitch-class set 3-2 segment (a) in mm. 3 – 5 and 4-13’s (b) in mm. 7 – 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.20 Simultaneous occurrence of pc sets 3-2 and 5-29 (dividable into 3-9 and 2-6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.21 The musical climax and its implication in the second section, mm. 13 – 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.1 Pitch-contour representation for the vocal line in the first section, mm. 1 – 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2 Pitch-contour representation for the vocal line in the second section, mm. 10 – 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3 The marking of the highest and lowest pitches for each segment in the first section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4 Dyad transformation in the first section . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Dyadic voice exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6 Dyad transformation in the second section . . . . . . . . . . . . . . . . . . . . . . . 52

3.1 Trichords containing interval of (a) a perfect fourth and (b) a tritone from examples presented by Krenek in Studies in Counterpoint (1940) 65

3.2 Distribution of pc sets 3-2 (boxed set) and 3-9 (circled set)

a) mm. 3 – 8, first section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

b) mm. 14 – 15, second section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.3 Intersection of tension-degrees 4 (pc set 3-9) and 9 (pc set 3-2) in the codetta, m. 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vii

3.4 Tension-degree 8 (pc set 3-4) and 10 (pc set 3-3) in the second section a) mm. 1 – 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 b) Transformational tension-degrees 8 (pc set 3-4) and 10 (pc set 3-3) . . 84

3.5 Harmonic contrast at the climax, mm. 17 – 18 . . . . . . . . . . . . . . . . . . . . 87

3.6 An example for comparing the same tension-degrees in Studies in Counterpoint (1940) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.7 Comparable tension-degrees in different registral contexts presented by Krenek in Studies in Counterpoint (1940) . . . . . . . . . . . . . 92

3.8 The configurations of pc set 3-2 (tension-degree 9) in the first section, mm. 1 – 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.9 The configurations of pc set 3-2 (tension-degree 9)

in the second section, mm. 12 – 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1 The segmentation of the introduction, m. 1 . . . . . . . . . . . . . . . . . . . . . . 98 4.2 The segmentation of the second subsection,

“Nur ein Wort, nur eine Bitte,” mm. 1 – 3 . . . . . . . . . . . . . . . . . . . . . . 103

4.3 The segmentations of the codetta (m. 9) and coda (mm. 20 – 21) . . . . . 106 4.4 The segmentation of the thirteenth subsection,

“kein Gedanke,” mm. 17 – 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

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LIST OF FIGURES

1.1 Pitch-classes of the primary series for Zwölf Variationen, op. 79, and the intervals between adjacent pitches . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Diagram presenting three kinds of linear harmonies . . . . . . . . . . . . . . . . 12

1.3 Linear harmonies (primary harmonies) . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Representations for invariant sets

a) Trichordal invariant sets between the rows P9 and I9 . . . . . . . . . . . . . 26

b) The ORMAP function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5 Comparable segmentation for pc sets 3-2, 4-13, and 5-29 in primary series, P9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.6 Pentachordal invariance between P9 and I9 . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 Interval vectors of three pc sets derived from the reduction of pitch-contour . . . . . . . . . . . . . . . . . . . . . . 48

2.2 The overall pitch-contour of the first section . . . . . . . . . . . . . . . . . . . . . 50

2.3 Overall dyad transformation in the vocal line . . . . . . . . . . . . . . . . . . . . . 53

2.4 Transformation by I11 in the first section a) A cohesive transformation by I11 in verticalized dyads . . . . . . . . . . . 55 b) A clock-face diagram for representing I11 as an axis . . . . . . . . . . . . . 56

2.5 I11 transformation in the second section

a) Interval 13 dyad group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

b) A clock-face diagram representation for inversion operation by an I11 axis . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.6 I11 transformation of an interval 8 pair in the second section . . . . . . . . . 57

ix

3.1 Classification of tension-degrees and pc sets

a) A diagram for tension-degrees and relevant pc sets . . . . . . . . . . . . . . . 64

b) Classification of tension-degrees in the interval vector . . . . . . . . . . . . 64

3.2 Charts for abstract representation from Example 3.1 (a) and (b)

a) All trichords that contain at least one perfect fourth (5 semitones) . . . 67

b) All trichords that contain one tritone (6 semitones) . . . . . . . . . . . . . . . 67

3.3 Tension-degrees grouping based on Krenek’s classification

a) Three tension-degree groups and pc sets . . . . . . . . . . . . . . . . . . . . . . . 68

b) Three tension-degree groups and pc sets with the interval vectors . . . 68

3.4 A list of the contents of tension-degree factors . . . . . . . . . . . . . . . . . . . 69

3.5 Diagrams for comparable frequency ratios of trichords in CONS group 72

3.6 Diagrams for comparable frequency ratios of trichords in M group . . . . 74

3.7 Diagrams for comparable frequency ratios of trichords in SH group . . . 76

3.8 A table for subdivided tension-degrees . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.9 A table for subdivided tension-degrees of selective pc sets associated with “Nur ein Wort” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.10 Diagram of tension-degrees location in the first three subsections . . . . 85

3.11 Diagram for expansion and contraction of tension-degrees between the second subsection and thirteenth subsection . . . . . . . . . . . . 88

3.12 The numeric value of constituent intervals in each chord in Example 3.6 90

4.1 Forte’s K- & Kh-relation: introduction of “Nur ein Wort” . . . . . . . . . . . 99

4.2 The inclusion diagram of the introduction, m. 1 . . . . . . . . . . . . . . . . . . . 100

4.3 The table of genera and the status quotient (Squo) indices for the introduction, m. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4 The inclusion diagram of the second subsection, mm. 1 – 3 . . . . . . . . . . 104

x

4.5 The table of genera and the Squo indices for the second subsection, mm. 1 – 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.6 The inclusion diagram for the codetta, m. 9 . . . . . . . . . . . . . . . . . . . . . . . 107 4.7 The table of genera and the Squo indices for the codetta, m. 9 . . . . . . . . 108 4.8 The inclusion diagrams of the thirteenth subsection, mm. 17 – 18

a) The inclusion diagram of the first half segmentation . . . . . . . . . . . . . 111 b) The inclusion diagram of the last half segmentation . . . . . . . . . . . . . 112

4.9 The tables of genera and the Squo indices

for two subdivisions in the thirteenth subsection . . . . . . . . . . . . . . . . . . 113 4.10 The genera succession of “Nur ein Wort” . . . . . . . . . . . . . . . . . . . . . . . . 115

xi

ABSTRACT

This thesis explores musical style and form, serial procedures, harmonic features,

and the text setting in the first song “Nur ein Wort, nur eine Bitte” from Ernst Krenek’s

Fünf Lieder nach Worten von Franz Kafka, op. 82. First I explore and compare unique

features between this work and those written just before and after, mainly focusing on the

pitch- and set-class structure of the twelve-tone rows. Second, discovering important

properties of the series, I analyze the details of “Nur ein Wort” by a variety of

contemporary methods. At the same time, my analysis traces how the music enhances

and interprets Kafka’s text throughout the song.

xii

1

CHAPTER 1: Pitch-Class Set/ Harmony

INTRODUCTION

In seeking to discover his own compositional style, Ernst Krenek progressed

through a variety of styles and historical influences. In fact, after initially composing

musical works in a late-romantic style, Krenek’s compositional technique oscillated as

follows: tonal expansion - quasi-expressionist atonality – neo-classicism – neo-

romanticism – the twelve-tone technique – featuring medieval and renaissance musical

components – electronic music – aleatory.

While early in his career, Krenek placed himself against the twelve-tone

technique and the works of Arnold Schoenberg, in the 1930s he adopted this method and

continued using it for the remainder of his life. It is noteworthy that Krenek is the first

and most significant twelve-tone composer not to belong to Schoenberg’s inner circle.

Although he was in contact with Anton Webern and Alban Berg when they had already

composed twelve-tone music, Krenek had not discussed the method with them. Yet,

Krenek carefully studied their and Schoenberg’s compositional techniques and this study

clearly influenced his own compositional style. As I will discuss later, for example, the

symmetrical balance in a row either among the partitioned four triads or between the two

hexachords often creates a mirror configuration typically found in the rows of Webern.

Occasionally, Krenek’s rows may also recall pieces of Berg in which tonal sonorities

intrude in an atonal context or as an ordered segment of a twelve-tone row. However,

Krenek’s twelve-tone music neither simply imitates the serial styles of Berg and Webern,

nor mixes them eclectically. Rather, despite his respect for the music of the Viennese

2

School, Krenek sustains his own lyricism and extraordinary literary sensitivity to text

which deeply informs his twelve-tone works.

Although the maturity of his twelve-tone technique already is evident in Karl V

(1930 – 1933), the first twelve-tone opera, Krenek’s ability to fully realize the structural

potentiality of a row emerged in his later works. As a result, not until ten years after

Krenek adopted the twelve-tone technique did he develop his own unique serial method

which characterized his later works. Described as “rotation series composition,”1 his

method most likely later inspired Igor Stravinsky’s serial procedures. Ironically, works

composed using this technique by Stravinsky are better known than those by Krenek.2

Some of Krenek’s serial works of the late 1930s use a twelve-tone technique that

seems still to be in the shadow of his two monumental pieces, Karl V and Lamentatio

Jeremiae Prophetae (1941), the first work composed by rotational array. However, these

pieces show particular features or tendencies of Krenek’s serial style. In the Sixth String

Quartet (1936) and Zwölf Variationen in drei Sätzen für Klavier (1937), for example,

Krenek’s twelve-tone composition reached its greatest achievement. Nevertheless,

Krenek’s musical style drastically turned more lyrical and self-expressive shortly after

1 It is unclear whether Krenek was the first inventor of rotational series. Although Krenek already

used that technique in Lamintatio Jeremiae Prophetae op. 93 in 1941 (published in 1957), George Perle first met him around that time and presented “his” own method to Krenek. In an interesting article in memory of Krenek, Perle’s statement regarding the origins of the rotational series is vague, as if avoiding unambiguous claim of being its true inventor. See George Perle, “Krenek,” The Musical Quarterly 77/1 (Spring 1993): 145-153. Also, see Joseph N. Straus, “Stravinsky’s ‘Construction of Twelve Verticals’: An Aspect of Harmony in the Serial Music,” Music Theory Spectrum 21/1 (Spring 1999): 43-73.

2 Clare Hogan, “‘Threni’: Stravinsky’s ‘Debt’ to Krenek,” Tempo 141 (June 1982): 22-29. Hogan points out crucial distinctions between Lamintatio Jeremiae Prophetae and Stravinsky’s Threni. She argues convincingly that Krenek’s use of the technique was quite limited and less expansive than Stravinsky’s. “For Krenek, the rotation technique was the solution to a self-imposed problem, namely, to integrate certain principles of the twelve-tone technique with those of ancient modality.”

3

this culmination. Composed during 1937 to 1938, Fünf Lieder nach Worten von Franz

Kafka, op. 82 offers one good example.

From a broad perspective, Fünf Lieder stands at a crucial and subtle

compositional point in his career in comparison both to earlier and later works. As will be

briefly examined below, unlike two other works (the Sixth String Quartet and Zwölf

Variationen in drei Sätzen für Klavier) composed in systematic twelve-tone technique,

Fünf Lieder expresses an extraordinary lyricism, matching and merging music and text

beautifully, as if the composer and poet have nearly the same personality. This feature of

Fünf Lieder is unique among his works of the same decade. One could also describe this

work as a personal expression, like Schoenberg’s early atonal expressionist works,

inspired by the composer’s personal feelings and beliefs.3

3 See Robert Erickson, “Krenek’s Later Music,” The Music Review 9 (1948): 29-44. Although

Erickson does not analyze any structural dimension of Fünf Lieder, he magnificently expresses the idiosyncrasy of the piece: “[T]he Kafka Songs should be mentioned because they contain some of the finest, most expressive music he has written. There are certain striking similarities between the minds of these two men. It is to be expected, in view of what we already know of Krenek’s music, that Kafka’s religious depth, his love of detail, and his view of the world as an infinitely complex organism which is almost insurmountably difficult, would set up sympathetic vibrations in Krenek’s mind. The aphorisms which he uses as texts for the songs are consequently absorbed, yet revealed, with such complete success that the songs are almost overpowering. It is as though either Kafka or Krenek wrote the words and the music – there is no separation between music and text. It may be interesting also, in this connection, to note that the Kafka Songs are by far the most expressionistic music Krenek has written in the twelve-tone technique.”

4

1.1 Row Forms Example 1.1: Primary row form (P9) and trichordal partitioning with pitch-class sets of

Fünf Lieder nach Worten von Franz Kafka, op. 82 (1937-38)4

Order number (ON): 0 1 2 3 4 5 6 7 8 9 A B

Pitch class (pc): 9 8 6 2 1 B 3 0 7 5 4 A

er

In

y fits the text and the musical structure. Before

nalyzing the work, let us look closely at the row itself, comparing it with other rows he

sed before and after Fünf Lieder.

All five songs of Fünf Lieder employ one primary twelve-tone row (shown in

Example 1.1) and its usual transformations through transposition, inversion, retrograde,

and retrograde-inversion. The first pitch of any prime row form is always labeled ord

number (ON) 0 and the last is labeled B (eleven). The pitch classes (pcs) below the row

form in Example 1.1 represent the numeral indication for each pitch instead of note

names C, D, and so forth, regardless of the register; e.g., C = 0, C# = 1 and so forth.

the first song, “Nur ein Wort, nur eine Bitte,” Krenek simply uses the primary row form,

P9, and its inversion, I9, which neatl

a

u

4 The order of pitch registers is from a transcription by Matthias Schmidt, Theorie und Praxis der

Zwölftontechnik: Ernst Krenek und die Reihenkomposition der Wiener Schule (Laaber: Laaber, 1998).

5

Example 1.2: Primary row forms of Krenek’s works composed before and after

a) Sixth String Quartet, op. 78 (1936)5

ON: 0 1 2 3 4 5 6 7 8 9 A B 6-1 6-1

Fünf Lieder

pc: 9 7 8 5 A 6 0 4 B 1 3 2

b) Zwölf Variationen in drei Sätzen für Klavier, op. 79 (1937)6

6-8 6-8

pc: 1 2 4 B 3 6 0 8 5 A 9 7

, op. 83

c) Twelve Short Piano Pieces written in the Twelve-Tone Technique7 (1938)

pc: 5 2 9 A B 0 8 3 6 1 4 7

5 The pitch register is from a transcription by Claudia Maurer Zenck, “A glance at the sketches for Ernst Krenek’s Sixth String Quartet, opus 78,” Newsletter of the Ernst Krenek Archive 5 (Spring-Fall 1995): 1-6, and Erickson (1948).

6 Pitch registers are from a transcription by Schmidt (1998) and Erickson (1948). Incidentally, Erickson transcribes the I6 row form in addition to the primary series P6. In the article, Erickson also points out the row’s symmetrical structure. However, Schmidt presents a more detailed exploration of the primary row form for this piece in terms of trichordal symmetry. (See footnote 13.)

7 See the Preface of Twelve Short Piano-Pieces: Written in the Twelve-Tone Technique (G. Schirmer, 1939).

6

The rows shown in Example 1.2 are the primary ones used in Krenek’s works

composed just before and after Fünf Lieder. The comparison of the structure of (a) and

(b), respectively, reveals that Krenek constructs an “inner symmetry” in both.8 In fact,

Krenek himself admits to the rigidity of his compositional approach with regard to th

two works composed before Fünf Lieder: “My most uncompromising twelve-tone w

of that period, the Sixth String Quartet and the Zwölf Variationen in drei Sätzen für

Klavier, were written in 1936 and 1937.”

e

orks

’s

s

transposition by four semitone et theory, these trichordal

sets are represented as pc set 3-1 and the relationship between the two is as follows:

Pc set 3-1 of ONs 0-2 = X Pc set 3-1 of ONs 9-B = T4I(X)

9 As Zenck and Schmidt argue, Krenek

conspicuous tendency to construct a consistent symmetry among trichordal pc sets i

obvious.10 In the row for the Sixth String Quartet (1936), for example, the first

partitioned trichordal pc set is equivalent to the last trichord through inversion and

s. In terms of the pitch-class s

8 Zenck (1995) and Schmidt (1998) point out this inner symmetrical structure of the Sixth String

Quartet, op. 78, and of Zwölf Variationen in drei Sätzen, op. 79, respectively. Zenck presents the process by which Krenek eventually reaches the final form of this row as the primary series, with a few examples of other rows.

9 Ernst Krenek, “A Composer’s Influence,” Perspectives of New Music 3/1 (Autumn-Winter 1964):

36-41.

10 Neither author uses the term “pitch class” or “pitch-class set” in their statements.

7

Example 1.3: The beginning of the first movement of the Sixth String Quartet, op. 78, mm. 1 – 9

Similarly, the second and third trichords are related by retrograde transposition,

which is represented as RT6. Inevitably, these symmetrical pairs of pc sets are the same:

c set 3-1 and 3-4, respectively. In the beginning of the first movement, for example,

Krenek crete harmonic unit. As Example 1.3

shows, Krenek distributes each trichordal set to respective parts with a different

expression or articulation marking (e.g., sul tasto, sul ponticello, sul A or D, pizz.,

espressivo, flüchtig im Ton. etc.). Thus, one can easily distinguish not only the partitioned

p

clearly makes use of each pc set as a dis

8

trichordal distribution, but also the contrast between the chromatic (pc set 3-1) and

diatonic (3-4) sonorities.11

The row of Example 1.2(b) for Zwölf Variationen in drei Sätzen, op. 79, also

ts the same quasi-mirrored shape; when the second trichordal set 3-11 (tonal triad) is

, the t ird is T1I(X), and when the first trichordal set 3-2 is Y, the fourth is T9I(Y).

Accordingly, the symmetrical balance of partitioned trichords in respective rows is

preserved by a one-to-one m

Another significant aspect of the primary series for Zwölf Variationen is its all-

intervallic row structure. The upper row in Figure 1.1 shows the ordered pitch classes

(pcs) of the primary row form (P1) and the row below represents the intervals (int.)

succession (the number of semitones in mod 12 between adjacent pitch classes). Note that

the int. string contains all possible intervals: from minor second (int. 1) to major seventh

adop

X h

apping. Although there is no equivalent partitioned trichordal

sets between the two rows for the Sixth String Quartet and Zwölf Variationen, both form

trichordal palindromes (see Example 1.2[a] and [b]).

Figure 1.1: Pitch-classes of primary series for Zwölf Variationen, op. 79, and the

intervals between adjacent pitches

11 However, Krenek does not seem to emphasize a particular trichordal set, because of the main

line of the melody annotated as the “Hauptstimme” designation.

9 7 Intervals (int.): 1 2 7 4 3 6 8 9 5 B A

pcs (P1): 1 2 4 B 3 6 0 8 5 A

11 13 12 13 11

int. sum

9

(int. 11). Moreover, sums of intervals standing at symmetrical positions in relation to a

The most distinctive feature of the row for Zwölf Variationen in comparison to

at of the Sixth String Quartet is that the former contains two apparent tonal triads,

beled as pc set 3-11. At first glance, one may assume an eclectic merging of typical

of

ost

Sätzen für Klavier, op. 79, mm. 127 – 30

center axis, int. 6, structure a symmetrical string: 11-13-12-13-11.12

th

la

properties of Webern and Berg’s row forms, but s/he should immediately notice that the

sound of Krenek’s music is quite different. In the beginning of the second movement

Zwölf Variationen, for instance, we find a highly contrapuntal texture that Krenek m

likely learned from his study with Franz Schreker. Here he uses the row in a somewhat

straightforward manner.

Example 1.4: The beginning of the second movement of Zwölf Variationen in drei

The excerpt in Example 1.4 well represents Krenek’s distinct handling of the

The primary series, P1, unfolds in the right-hand with its inversion transposed seve

row.

n

12 At first, Krenek apparently structured the prime series for op. 79 not only as an all-intervallic

row, but also with intervallic symmetry. In this case, the pitches of the order numbers 7 and 10 are exchanged, so that each sum of symmetric intervals projects 12. In Studies in Counterpoint: Based on the Twelve-Tone Technique (New York: G. Schirmer, 1940), Krenek presents a transposed series of op. 79 (P11) as a row form that contains both properties, but its pitches of the order numbers above are exchanged to form the intervallic symmetry. Schmidt (1998) explicitly interprets the purpose of this pitch exchange.

10

semitones, I8, appearing in the left. While P1 flows as a single line with few pitch

repetitions (the only exceptions are E as a sixteenth-note in m. 127 and B as a grace note

in m. 1

m.

y

tonal triad does not

ean a tendency toward tonal restoration. Rather, partial tonal sonority in his music

radically abandon tonality for the purpose of composing something new to create his own

musical expression.14

The primary row form of Fünf Lieder projects a different “inner structure” from

the two row forms discussed above. First, there is no symmetry in the trichordal partitions

n

the first hexachord. Instead, the pattern of ordering the pitch classes in the respective

29), all measures of I8 except for the first consist of far more repetitions and

overlappings, creating a more homophonic texture. Nevertheless, serial disposition in

which one part moves in contrasting motion to another, as in the first two measures,

evokes a conventional contrapuntal texture.13 Among boxed trichord pc sets 3-11 in

Example 1.4, the simple ascending progression of a B-major triad in the right-hand (

128) and an arpeggiated A-major triad repetition in the left-hand (m. 129) strongly

project a diatonic sound. Such direct representation of the tonal triad is largely alien to

the usual serial textures of Schoenberg and Webern; however, it is found more commonl

in Berg’s serial textures. For Krenek, such direct employment of a

m

seems to be a part of the compositional materials. In other words, Krenek did not

in the row, despite the juxtaposition of two of the same pc sets 3-2, ONs 0-2 and 3-5, i

13 Specifically, these measures recall the beginning of Bach’s Invention in E major, although

Bach’s radiating lines move more smoothly.

14 Krenek’s preference for the use of tonal triads even appears in a piece whose primary row form does not contain tonal triad pc set 3-11 as a primary harmony (the details to be explained later). See m. 5 of the first violin part in Example 1.3: Bb-Gb-Db. It is worth noting that only the Gb-major triad string in measure 5 comprises non-successive order number, ONs 4-5-9, in the first “Hauptstimme” (mm. 1 – 7).

11

trichord is related by a five-semitone transposition: T5 (986) = (21B).15 Second, the

maining two trichords in the second hexachord represent different set classes and

conseq

one

D-minor triad ( pc set 3-11 {259})

re

uently do not project a symmetrical pattern for the row’s partitioned trichords.

However, there is one pc set 3-11 (ONs 6-8), which represents a C-minor triad, and it is

preserved again in the row of the next work as a linear harmony.

Example 1.5: “Dancing Toys” in Twelve Short Piano-Pieces written in the Twelve-T

Technique, op. 83, mm. 10 – 12

Ab-major triad (pc set 3-11 {803})

The row for Twelve Short Piano-Pieces written in the Twelve-Tone Technique, op.

3, in Example 1.2(c) no longer displays inner trichordal symmetry, nor even any of the

me pc sets in the basic partitioning. On the other hand, it does contain two triads among

its ordered segments: order numbers (ONs) 0-2 for a D-mi

ma e the tionship of o pc set 3-11’s in op. 79, the order of pitch

lasses in each linear major or minor triad for op. 83 is not the same. However, if we

view the pc set as a unit, these trichords are related by a transformational operation: when

8

sa

nor triad, ONs 5-7 for an Ab-

jor triad. Unlik rela tw

c

15 Parentheses for pitch classes represent “unordered sets” as in order of occurrence or in a chord.

When the set is represented with curly brackets, it means that the pc set is in “normal order” according to Allen Forte’s definition. For these pc sets, (986) and (21B), for example, the normal orders are {689} and

when transposed to the starting of pitch class 0 with ascending order. Accordingly, the prime form of both(986) and (21B) are represented as [012]. See Allen Forte, The Structure of Atonal Music (New Haven:

{B12}, respectively. In addition, square brackets represent “prime form” which represents the normal order

Yale University Press, 1973).

12

the D-minor triad {259} is X, the Ab-major triad is derived by T1I(X), which is the same

operation that is used in op. 79. Analogous to Zwölf Variationen, op. 79, as shown in

Example 1.5, Krenek explicitly distributes linear tonal trichords in a straightforward

manner. Since the musical texture unfolds more simply than in op. 79, a listener likely

ears the stronger projection of a tonal triad in op. 83.

Figure

Figure 1.2 summarizes the number of triads, semitone-dyads, and tritones in the

successive intervals of the respective series. As I mentioned above, all but op. 78 include

one or two triads. The second row in the diagram reveals that Fünf Lieder contains the

reatest number of linear harmonic semitone-dyads, while the third reveals that all but op.

83 include at least one tritone at the center of each row, i.e., ONs 5-6. The properties of

tive from which

tures.

1.2 Ha ieder

Having examined these four primary series somewhat broadly – in particular,

trichordal symmetry and some specific pc sets and intervals between dyads, we shift our

attention to the row of Fünf Lieder. To explore some specific harmonic structures in this

h

1.2: Diagram presenting three kinds of linear harmonies

op. 78 (1936) op. 79 (1937) Fünf Lieder, op. 82 (1937) op. 83 (1938)

3-11 (triad) 0 2 1 2 Semitone-dyad 2 2 4 3 Tritone 1 2 2 0

g

the row, obtained from this diagram, suggest that one important perspec

to analyze Fünf Lieder is how the music exploits these distinctive fea

rmonies Derived from the Primary Series of Fünf L

13

work, Hyde’s (1980) definition of the primary and secondary harmonies of a row is

seful. While the former consists of all pc sets found by linear or contiguous pitches in

series, the latter consists of pc sets formed by pitches that are not contiguous, but

nonethe s are .

Figure 1.3: Linear harmonies (primary harmonies)17

3 4 5 6

9 8 6

u 16

a

les equivalent to pc sets formed by contiguous segments

a) ON: 0 1 2 7 8 9 10 11

2 1 B 3 0 7 5 4 A 3-2 3-2 3-11 3-5 _____ _ _ 3-8 3-6 3-9 3-5 ___ _ _ _

r segments

3-4 3-3 3-2 3-1

b) Cardinality: 3-: 2, 8, 4, 2, 6, 3, 11, 9, 2, 5, 5, 1 4-: Z29, 16, 14, 2, 2, 19, 22, 14, 13, 8, 5, 2 5-: 20, 29, 11, 1, 13, 30, 11, 29, Z12, 6, 9, 13 6-: Z25, Z47, Z36, Z37, 22, 16, Z47, Z25, Z3, Z4, 22, 16

c) Pc sets that occur more than once as linea

2 segments 3 segments Z-related pairs 3-5 3-2 - 4-14 4-2 -

5-11, 5-13, 5-29 - - 6-16, 6-22, 6-Z25, 6-Z47 - 6-Z25/Z47, 6-Z36/Z3, 6-Z37/Z4

16 According to the terminology in Martha Hyde, “The Roots of Form in Schoenberg’s Sketches,”

Journal of Music Theory 24/1 (Spring 1980): 1-36, these are called “primary dimension” or “primary harmonic dimension” and “secondary dimension” or “secondary harmonic dimension.”

17 Ibid.

14

Figure 1.3(a) exhibits all of the possible primary harmonies for pc sets of

cardinality three in the prime series of Fünf Lieder. Regardless of the harmony’s sp

cardinality, the total number of the primary harmonies for each cardinality is always

twelve (see Figure 1.3[b]). Similarly, secondary harmonies also comprise a possible

maximum number of twelve, since it must be equivalent to a primary harmony; however,

as an analytical procedure, only specific pc sets as the secondary harmonies are focused

on in order to examine particular musical features.

ecific

With regard to the row itself, Schmidt’s (1998) thorough analysis reveals

form. As shown by Figure 1.3(a) which

primary harmonies of cardinality three, pc set 3-2 appears the

greates

s

more

Krenek’s specific manner of structuring a row

enumerates the row’s

t number of times (bold type in Figure 1.3[b]), while 3-11 appears only once.18

Schmidt claims that a hexachordal segment derived from two successive trichords, ON

8-10 and 11-1, corresponds to the row’s first hexachord, ON 0-5. However, the pc set of

this primary harmony listed in Figure 1.3(b) reveals that these hexachords are not

identical; 6-Z25 begins with ON 0 and 6-Z3 begins with ON 8 (bold type in the row for

cardinality 6). That is, while their interval vectors are identical, they are not related by

transposition or inversion. Figure 1.3(c) displays the primary harmonies that occur

than once in the row and the 3 pairs of Z-related sets in respective cardinalities. We will

explore which particular pc set(s) in this table play a more significant role than others.

18 Schmidt (1998) maintains that the trichord segments in the first hexachord (ONs 1-3 and 4-6)

structure a symmetric shape – not in terms of mirroring configuration, but in terms of pc set itself; the same trichord figure is set in the last half of the row (ONs 9-11), even though here it assumes asymmetric arrangement.

15

1.3 Form of “Nur ein Wort, nur eine Bitte”

1.3.1 Two Sections

Shortly before his first trip to the United States, Krenek obtained a collection of

Franz Kafka’s fragments, and copied some of them to bring with him.19 Presumably,

Krenek already might have planned to compose a song cycle, joining Kafka’s fragments

with th

d,

lf-

this song cycle reflects Krenek’s personal difficulties under the Nazi

regime nd Krenek’s expressionistic aesthetic may

omple

ft in his notebooks, “Nur ein Wort” shows a

omewhat different writing style from others. While most fragments are likely to be part

e twelve-tone technique.20 Moreover, the period during which Krenek composed

Fünf Lieder precipitated a serious personal crisis both as a composer and as an Austrian

citizen. Labeled a decadent artist by the National Socialism Party in the late 1930s, his

musical activity – like Schoenberg’s and Hindemith’s – was drastically restricted.

Although Krenek did not regard himself as an expressionist composer in his early perio

his radical atonal works (e.g., Symphony No. 2) may have precipitated more intense se

expression. Hence, one may recall a similar “expressive” effect in Fünf Lieder to

Schoenberg’s expressionist atonal pieces – fear, anxiousness, or hopelessness.21 In any

event, the circumstances surrounding Fünf Lieder are complicated; it is quite possible

that the intensity of

. Thus, Kafka’s apocalyptic vision a

c ment one another.

In the bulk of Kafka’s fragments le

s

19 The book was edited by Kafka’s close friend, Max Brod, although Kafka asked him to burn the

fragments in the year before his death. Kafka seems to have believed that the fragments were not worth to publishing.

20 See footnote in Ernst Krenek, Die Amerikanischen Tagebücher 1937-1942 Dokumente aus dem

Exil, ed. Claudia Maurer Zenck (Wien: Böhlan Verlag Gesellschaft, 1992), 21. 21 See review by Hans Nathan, in “Vocal Music,” Notes, 2d Ser., 17/2 (March 1960): 322-23.

16

of a sto tself a composition in prose, which is

possible to regard as a single poem. In fact, repetitive occurrences of words “nur” and

“ein(e) haracteristic of a prose-poem. Thus, an overall

“Nur ei al phenomenon, but rather one’s internal

whisper. As well, “Nur ein Wort” explores two poles: the text of the first half is

characterized by affirmation and that of the second half by negation. Note that the second

stanza opens with “Nein” and successive lines begin with “kein.” The text of the first

song is as follows:

Nur ein Wort, nur eine Bitte,

dass du noch lebst und wartest.

Nein, keine Bitte, nur ein Atmen,

kein Bereitsein, nur ein Gedanke,

just a movement of the air, just a proof,

no breath, just a preparation,

no thought, just quietly sleeping.]

give fo za

draws o moving of the air, and her/his life. However, one word,

ry, “Nur ein Wort” as a fragment shapes i

22

” (or “kein[e]”) strengthen the c

n Wort” seems to reveal not an extern

nur ein Bewegen der Luft, nur ein Beweis,

kein Atmen, nur ein Bereitsein,

kein Gedanke, nur ruhiger Schlaf.

[Just a word, just a request,

that you are still alive and waiting.

No, not a request, just a breath,

no preparation, just a thought, 23

The first line of the first stanza in this poem presents static objects that one might

rth – a word and request (or a request by a word). In contrast, the rest of the stan

n dynamic movement – a

22 Also, the fragments includes dialogues and several single short sentences and phrases. See

Kafka, “Fragmente aus Heften und losen Blättern” in Gesammelte Werke Herausgegeben von Max Brod: Hochzeitsvorbereitungen aug dem Land und andere Prosa aus dem Nachlaß (New York: Schocken Books, 1953), 224-417.

23 My translation.

17

“still” (noch), in the fourth line implies that that life may be near the end. Even though at

first glance the first stanza figures peacefulness, the word “still” immediately leads the

reader t etitive

, the last

If one tries to reconnoiter the writer’s position from his notes and sketches, it is almost understandable that most, and the most important, of his works remained

how th have gone on. . . . [T]hey did not go on but were lost in the darkness of their immense environment, but their meaning immediately goes on in another story which begins to shape itself until it too breaks off somewhere.25

a fragment as “Nur ein Wort, nur eine Bitte,” Krenek might see literary completeness

r an “eternal light which art seeks.”26

Based on Krenek’s appreciation, it is not surprising that he reflects the

in his composition. In fact, one remarkable feature of “Nur ein

ort” in Fünf Lieder is that Krenek’s use of two different row forms, P9 and I9,

analogously portrays the contradiction between the poem’s two stanzas. That is, the

primary series, P9, and its inversion, I9, structure the first and second half of the song

o picture one’s unhealthy or critical condition. In the second stanza, rep

negation emphasizes that one is almost at death’s door. Especially the last negation “no

thought” (kein Gedanke) suggests hopelessness in the face of death. Nevertheless

phrase “just quietly sleeping” (nur ruhiger Schlaf) brings unexpected relief.

In Exploring Music (1966), Krenek expresses a special interest in and affection

for the writings of Kafka and asserts the importance of the fragments at the end of the

chapter “Notes on Kafka’s Collected Works”24:

fragments. But in this case fragmentariness is hardly a question of finding out is or that story Kafka began would

In

o

contradictions of the poem

W

24 Ernst Krenek, Exploring Music, trans. Margaret Shenfield and Geoffrey Skelton (London:

Calder and Boyars, 1966), 123-128.

25 Ibid., 128. 26 Ibid.

18

respectively, a division that corresponds to the structure of Kafka’s poem. This aspect

complements the structure of the two stanzas. Hence, I divide this piece into two sections.

Example 1.6: Segmentation by the row forms of P9 and I9 in “Nur ein Wort, nur eine

Bitte” in Fünf Lieder, mm. 1 – 12

P9

19

Example 1.6: Continued

I9

Example 1.6 shows the first section (mm. 1 – 10) and the initial measures of the

second section of “Nur ein Wort.” The passages annotated in boxes in the excerpt of the

first full page (mm. 1 – 9) above are all derived from prime series P9. Only the last five

pitches in the last occurrence of P9 overlap with the beginning of the second section, with

the first word of the second stanza, “Nein.” Similar to the first, the second section is

derived only from I9, along with the second stanza of the poem, which also continues the

“inversion” of affirmation.

P9

20

Example 1.7: “Nur ein Wort” indicating lines dividing into two sections and fifteen Subsections

“Coda”

“Codetta”

4

7

SS 4SS 6

SS 8

SS 13

SS 15

Subsection (SS) 1 SS 2SS 5

10

13

16

19

SS 3

SS 9 SS 10

SS 12

SS 14

SS 7

SS 11

First section Second section

(SS 3)

21

In contrast to the clear contradiction between the two stanzas of the text, on first

hearing the two sections seem to lack cohesiveness. However, there is an obvious

pseudo-cohesiveness in them that may easily be recognized. Rhythmic patterns in m. 9

and mm. 20 – 21 are quite similar, although the patterns in both hands are exchanged.

Apparently, Krenek set this inverted similarity to present a distinguishable close to each

section. For this reason, I shall call the final two measures of the piece (mm. 20 – 21

“coda.” Also, a shorter closure in m. 9 will be described as the “codetta” (see E

musical

) the

xample

.7).

.3.2 Subsections

Having divided the piece into two sections, we further separate each section into

everal subsections mainly based on the text. Dotted lines in Example 1.7 indicate the

ubsectional division, and a bold line marks the song’s two main sections. In the first

ection, the division into subsections is relatively straightforward. Note that the second

nd third subsections contain two phrases of the text, respectively, although a rest in the

from a musical

standpoint. If we subdivide “Nur ein Wort” by each phrase of the text, it apparently

e any

perceivable meaning.

“Nur ein Wort” projects a dialogue between the piano and voice, a dialogue that

in fact begins in the introduction between the left- and right-hands of the piano itself.

While the right hand plays a slurred phrase, the left inserts a fragmented motif that fills in

the gap by a rest or by longer held dyads. One may hear it as an irregular call and

1

1

s

s

s

a

voice line separates them. This separation is, of course, accomplished

would not help to analyze the piece, and the subsection itself would not hav

22

response, or the left-hand echoing the right. Indeed, this technique generates momentum,

and thus an occasional moment of complete silence contributes to the dramatic effect of

the whole. In any event, subsections in the first section result from the articulation of an

intimate textural interaction between the voice and piano.

In contrast, subsections in the second section are partitioned by one verse, with

one exception. Whereas the voice preserves, as in the first section, a patterned form in

which the figure of one verse followed by a rest is repeated, the texture of the piano is

much denser than that of the first section. There are a few rapid gestura

ght hand) and static vertical chords (e.g., m. 15 in the left, m. 16, 17,

forth) that seem in conflict with the voice. These new

charact

ion,

l passages (mm.

11 – 12, 14 in the ri

and 18 in the right, and so

eristics are highly effective in projecting a feeling of anxiety which complements

the text’s negativity. Since the musical texture is dense and, in contrast to the first sect

the momentum intensifies toward the climax, smaller textural partitions characterize the

song’s second section. The process of articulating subsections already conveys the

obvious contradiction in the text, a feature that will be further explored below.

23

Examp

le 1.8: Motivic pc sets: 3-2, 4-13, 4-1, 4-21, 4-Z29, and 5-29

Example 1.9: Segmentation for the introduction in m. 1 (pitch reduction)

* Primary harmony

^ Secondary harmony

24

1.4 Segmentation

A brief introduction by the piano consists of the series P9, plus the first pitches of

e following P9. In Example 1.9, the segmentation in the pitch reduction reveals that

nearly all pc sets represent either primary or secondary harmonies. Of the pc sets above,

ome directly or indirectly serve as motivic harmonies. As discussed in the structure of

re associated not registrally, but rather metrically. Looking first at pc set 3-2 in the bass

does not have phrasal

interruption (see m. 1 in Example 1.8). First, the intervals in each pc set 3-2 segment are

entical: nine semitones ascending between the first pitch to the remaining dyad (as

xample 1.8 and 1.9 show). Second, both share a similar rhythmic pattern, i.e., short-long

of a rhythmic motif. Finally, this same

h contour) derived from

ed primary in the song.

ificant m in later segments.

Pitch class set 3-2 as two motivic triads reappears in reverse order, i.e., ONs 3-5

e third subsection (mm. 4 – 5). In this occurrence, the

rhythmic patterns of th . 4 – 5 in Example 1.8).

urthermore, the pitch-space and contour in respective sets are also identical: while the

rst pc set 3-2 {689} in the introduction recurs an octave higher in m. 5, the second {21B}

reappears two octaves higher in m. 4.

1.4.1 Pitch-Class Set 3-2

th

s

the primary row form for Fünf Lieder, for example, two pc sets 3-2, {689} and {B12},

a

clef, we see similarities to another 3-2 in the treble clef, although it

id

E

note value succession, which enhances the sense

pc set 3-2 motif (repeated in its rhythmic pattern and pitc

partition harmonies, ONs 0-2 and 3-5, effectively appears later

Other sign otivic sets will be identified as they are introduced

followed by ONs 0-2, in th

e two motivic 3-2’s are identical (see mm

F

fi

25

Similar to the introduction, these pc set 3-2 harmonies are inserted to fill the

temporal space, when a vocal line’s pitch sustains by a longer note value (D in m. 4)

a transitional silence between the third and fourth phrasal segments (a eighth rest

followed by “nur ein Beweis” phrase in the vocal line in m. 5). The motivic pc set 3-2

once again occurs in the piano (m. 7), but in a slightly different rhythmic figure from its

previous ones: D as a grace note, instead of a sixteenth (see Example 1.8).

At the same time, the voice projects other pc set 3-2’s as primary harmonies and

excludes those which occur in the piano discussed above. The rhythm and contour shapes

of the two 3-2’s, {457} and {B13} in m. 3 and 6, respectively, are mutually similar, whi

there is little resemblance to those in the piano (see Example 1.8). In the voice, although

pc set 3-2 never structures complete phrases, the set is articulated in almost every

measure up to the end of the vo

and

le

cal line in the first section (F3 with “[war]test”). Thus, the

econd subsection is harmonically isolated from the remaining subsections, lacking any

ic 3-2 (see Example 1.10 below).

m. 1 – 3

s

convincing use of the motiv

Example 1.10: Absence of pc set 3-2: segmentation for the second subsection, m

26

1.4.2 Invariance of Pitch-Class Set 3-2 in the Vocal Line

Example 1.11: The vocal line with annotated pc set names and circled 3-2 3-4 {1 } 5-9 {4568A} 5-2 {23457} 4-2 {689A} 7-9 {B012357} 26

4-13 {457A} 4-8 {8912} 3-8 {A04} 5-Z36 {8B123} 5-20 {459A0}

5-20 {B1267} 3-1 {567} 4-3 {4578}

Figure 1.4: Representations for invariant sets (a) Trichordal invariant sets between the rows P9 and I9

P9: 9 8 6 2 1 B 3 0 7 5 4 A

I9: 9 A 0 4 5 7 3 6 B 1 2 8 (b) The ORMAP function27

ORMAP (P9) <0 1 2 3 4 5 6 7 8 9 A B> ORMAP (I9) <0 B 7 A 9 8 6 2 5 4 3 1>

pshire: Frog Peak Music, 27 Robert D. Morris, Class Notes for Atonal Music Theory (New Ham

1991).

27

Analysis of the vocal line alone shows that the pc sets of partitioned segments

ary. Example 1.11 shows only the vocal line with pc sets indicated for partitioned

segments. The first system contains the first section, and the remaining systems contain

e second. Notice that partitioning the vocal line into segments differs from dividing it

to subsections: segmentation is based on each phrase of the text that is set off by rest(s).

mong

nding

ses

tween

f) a larger

e

by

ates

v

th

in

A these segments, some contain ordered forms of pc set 3-2. Yet, while there are a

number of partitioned trichords, none projects 3-2. Circled trichordal segments indicate

pc set 3-2 as a primary harmony which always appears as either the beginning or e

of respective phrases. It is interesting that more than half of the song’s thirteen phra

unfold pc set 3-2 as a primary harmony. The invariant relationship of pc set 3-2 be

P9 and I9 shown in Figure 1.4 further suggests that Krenek intended this trichord to

assume harmonic importance.

1.4.3 Tetrachordal Motif

Although we have rather broadly categorized pc set 3-2 into two types of

configuration, not all of their occurrences function as independent motifs. In the

introduction, it is possible to interpret the first 3-2 motif as part of (or a subset o

tetrachordal motif pc set 4-13; here pc set 4-13 represents a secondary harmony. Th

rhythm and contour of this motif is repeated successively in mm. 4 – 5, now framed

rests and articulated as independent motives. The first measure in Example 1.8 illustr

three prominent kinds of motives: boxed pc set 3-2, circled 4-13.

28

Example 1.12: Tetrachordal motivic segments with pc set 3-2 motifs, mm. 4 – 5

{B012} + {2689} + {9B13} = 8-13 {689B0123}

When we pay close attention to the tetrachordal motivic sets, as shown in

Example 1.12, both pc set 4-1 and 4-Z29 contain 3-2 as a projected subset. While the

remaining tetrachordal motif 4-21 in the left-hand mimics rhythmically both 4-1 and 4-

29, it switches their initial pitch and dyad; moreover, 4-21 does not contain 3-2.

ccordingly, the piano in the third subsection creates the striking impression of

rojecting a series of repeating motives, even though the voice simultaneously seems to

roject dissimilar motives. More interestingly, although each tetrachordal motivic set in

e third subsection differs (only 4-Z29 is a primary harmony), the aggregate of the three

egments (4-1 {B012}, 4-21 {9B13}, and 4-Z29 {2689}) represents pc set 8-13

689B0123}, which is the abstract complement of the motivic 4-13 heard at the

beginning of the piece. Even if Krenek was not conscious of this complementary

harmonic association, he may well have grasped the initial partitioning of the row into the

complementary sets, 4-13 and 8-13.

Z

A

p

p

th

s

{

29

1.4.5 Hexachords and Pitch-Class Set 5-29

xample 1.13: Four re-ordered hexachords derived from basic partitionings of the series

P9: a: : 6-Z47 {3457A0}

E

6-Z25 {689B12} b

ON: 2 1 0 5 4 3 A 9 8 B 7 6

9: c: 6-Z25 {4579A0} d: 6-Z47 {B12368} d’ I

ON: 3 4 5 0 1 2 6 7 B 8 9 A

e

of I9 forms part of an F-major or D-minor scale and contains five invariant

the

nly alien pitc ny of the keys above. In comparison with the

row forms of two earlier pieces in Example 1.2a and b, the relationship between

rienta

modernist perspective to one that cultivated historical materials. Krenek in fact

Before examining the function of pc set 5-29, let us look at an interesting featur

of the hexachords derived from the basic partitioning of P9 and I9. Example 1.13 displays

four hexachords, re-ordered to project segments of possible tonal scales. The first

hexachord of P9 represents six pitches of A-major or F#-minor and comprises five

identical pitches with the last hexachord of I9: F#-G#-B-C#-D. Similarly, the last

hexachord

pitches in the first hexachord of P9: E-F-G-Bb-C. One remaining pitch, Eb (or D#), is

o h that does not belong to a

partitioned hexachords of Fünf Lieder shifts from pc set equivalence to a modal

o tion. This discrepancy suggests a compositional turn from a non-tonal or

30

enthusiastically studied medieval and Renaissance music later in his life. One can

interpret such a tendency in “Nur ein Wort;” for example, for the most part Krenek

notates pc 3 as Eb and not D#, except for one instance in the second subsection (see

Example 1.7). (Other features of this subsection may be problematic, as will be discussed

below.) Moreover, except in the coda (mm. 20 – 21), no sharp is used as an accidental in

the last section. In other words, Krenek may have heard the last hexachord of I9 as a scale

based not on sharps but on flats (hexachord labeled d’ in Example 1.13). However, it is

also possible that Krenek was more sensitive to the features of the rows’ primary

hexachords, as was clearly the case in later pieces such as Lamentatio Jeremiae

Prophetae. In any event, the introduction does seem to juxtapose pseudo-bitonal (or

modal) sonorities.

1.4.6 Pitch-Class Set 3-11 and 2-5 Division from 5-29

Two particular pentachords in the introduction, clearly projected by sustained

pitches, late become an important structural component. Both chords represent pc set 5-

29, despite the fact that the hexachord to which each belongs differs, 6-Z25 and 6-Z47,

respectively (see Example 1.14[a] below).

31

Example 1.14: Pentachords in the introduction, m. 1

a) Segmentation based on sustained pentachords in the introduction

6-Z25 6-Z47

b) Representation for verticalized pentachords extracted from the hexachord and trichord as an inclusion

5-29{68B12} ⊃ 3-11 {B26} 5-29 {457A0} ⊃ 3-11 {047}

As shown in Example 1.14(b), the first chord consists of G#-B-D-F#-C# which

an be arran d as a alf-di inishe d with an added eleventh; similarly,

the second chord consists of C-E-G-Bb-F which can be rearranged as a seventh chord

th. Furthermore, each chord includes a triad that is directly projected

jor triad in the codetta (m. 9 – 10) of the

irst section, and the B-minor triad is projected in the coda (mm. 20 – 21) of the second.

In other words, pc set 3-11 as a subset of 5-29 is clearly reproduced at the ending of each

section.

B-minor C-major

5-29 {68B12} 5-29 {457A0}

c re ge h m d seventh chor

with an added eleven

in the codetta and coda, respectively; the C-ma

f

32

Example 1.15: An extracted primary harmony of pc set 5-29 from P9 (ONs 7-B) in the codetta, mm. 9 – 10

2-5 {5A}

0 7 5 4 A (ONs: 7 – B of P

3-11 {047} = C-major

9)

: the

h sections, we can explore pc set 5-

29 from a different angle. Appearing as a vertical or sustained harmony in the

introduction, 5-29 plays a “musically” significant role in the first section. In mm. 5 – 6,

the forth subsection, Krenek annotates “cantabile espressivo,” surprisingly not in the

Example 1.15 confirms the pitch distribution from the last five notes, ONs 7-B,

extracted from the P9 series of the codetta; here 5-29 is projected by two subsets

tonal triad set 3-11 appearing as a vertical C-major chord in the left-hand (a secondary

harmony, ONs 7, 8, and A) and the registrally and metrically distant dyad {5A}, 2-5.

Pitch class set 5-29 in the coda is represented identically, even though the second section

uses the inverted row form, I9. At this point, the tonal triad, pc set 3-11, forms a B-minor

triad.

1.4.7 Pitch-Class Set 3-9 and 2-6 Division from 5-29

Having verified that the two pentachordal sets in the introduction are primary

harmonies intimately connected to the endings of bot

33

voice, b

xample 1.16: Pitch-class set 5-29 {457A0} in the first and fourth subsections.

ut in the right hand of the piano. This simple short phrase, indeed, is derived from

the second 5-29 segment in the introduction.

E

3-9 {570}

m. 1 m. 6 2-6 {4A}

ets but

xample 1.16, a dotted quarter note, is equally partitioned into

5-29 {457A0}

Example 1.16 illustrates a further segmentation of pc set 5-29 {457A0} as it

appears in the introduction and in m. 6. In the introduction, pc set 5-29 is partitioned

registrally into pc set 3-9 and 2-6. This pentachordal harmony is transformed in the

“cantabile espressivo” phrase as a linear presentation of the same partitioned subs

without transposition. In other words, the nearly vertical harmony in m. 1 is now

represented as a complete “cantabile” melodic line. Although the melody projects a

single short phrase without clearly partitioned subsets, it is important to recognize this

structural connection, for it recurs in another repetition of pc set 5-29 in mm. 11 – 12 and

m. 14 (see Example 1.17). Here, the note value of the vertical trichord 3-9 in the first

instance of 5-29 (m. 1) in E

34

three eighth notes in the second instance (m. 6). Then the tritone dyad 2-6 is rhythmically

reversed.28

Example 1.17: Occurrence of motivic pc set 5-29 in the second section

5-29

mm. 11 – 12 m. 14 {28} b.

ctio (mm. 1 – 12 , the “ rase of pc set 5-29

recurs in the piano. Note that the phra 3-9 {B16} and 2-6

{28} as discrete partitions, as shown by Example 1.17a. Furthermore, the passage

suggests a rhythm

a. 3-9 {B16} 2-6

In the eighth subse n 1 ) cantabile” ph

sing clearly articulates pc sets

ic similarity to the original “cantabile” phrase (m. 6): three eighth notes

of a boxed pc set 3-9 in the second excerpt (m. 6) in Example 1.16 are now equally

reduced in note value to thirty-second notes of the phrase in mm. 11 – 12 (see the left

excerpt in Example 1.17). Similarly, a dotted eighth and an eighth in the circled pc set 2-

6 in m. 6 (the right excerpt in Example 1.16) are reduced to a dotted eighth and sixteenth

notes, respectively, in the left excerpt of Example 1.17.

28 In this case, a sixteenth rest followed by pc set 2-6 in the first 5-29 (m. 1, the left excerpt in

Example 1.16) is counted and merged into E, so that the note value of E and that of Bbcan compensate.

in the second 5-29

35

A difference between the “contractions”29 of the note value between the two pc

set 3-9’s and two 2-6’s, respectively, suggests Krenek’s harmonic and metrical

manipulation. The “contraction” rate of pc set 3-9 in comparison to that in mm. 6 and 11

yields one-quarter: a thirty-second note value results from dividing the eighth note by

ur. On the other hand, the “contraction” rate of note value between the two occurrences

f pc set 2-6 (the same measures as above) yields one-half: a dotted eighth is derived

from dividing a dotted quarter by two, and a sixteenth is a half note value of an eighth. It

is interesting that Krenek treats pc set 5-29 both as a phrase (e.g., “cantabile espressivo”

in m. 6) and as a superset that contains pc sets 3-9 and 2-6 as harmonies. This enables

him to juxtapose two different materials (e.g., vertical harmonies in m. 1 in Example 1.16

or short passages in mm. 11 – 12 in Example 1.17).

In m. 14, as shown at the right excerpt in Example 1.17, a similar passage appears

again in a lower register, but its rhythmic similarity is slightly altered: the last pitch Ab in

pc set 5-29 has twice the note value as the previous one in m. 12: Ab in Example 1.17b.

Example 1.18: “Espressivo” pc set 3-9 {B16} included in 5-29, at the climax,

fo

o

mm. 17 – 18

3-9 {B16}

he Music of Edgard Varèse,” Music Theory Spectrum 3 (Spring 1981): 1-

5, Jonathan W. Bernard defines “contraction” as a term to present a transformational process of which one pitch-space at a point is diminished at another point. Here I use this term to apply for comparison of two different reductive rates in note value.

5-29 {68B12} 29 In “Pitch/Register in t

2

36

Krenek once again annotates “espressivo,” but without “cantabile,” in the p

right-hand at the song’s climax in m. 17 (Example 1.18). Because of the phrasing slur

immediately perceive that this term for musical expression, notated between the staves,

designates only the three pitches in the right-hand. Indeed, this trichordal set {B16} is th

first three pitches of pc set 5-29; as before, 3-9 is followed by the dyad 2-6 {28}.

Furthermore, the note values of the three pitches of 3-9 are again equivalent, if we regard

the eighth rest as a part of the value of the preceding Db. Each note value, then, can be

represented as a dotted eighth-note. To sum up, pc set 5-29 in the prime series of Fünf

Lieder comprises the diatonic trichordal set 3-9 and the tritone dyadic set 2-6; above

the former contributes to enhancing the expressiveness of the piece, either by itself or by

conjoining with t

iano’s

, we

e

all,

he latter.

in primary series, P9

3-2 3-2 5-29

P9: 9 8 6

1.4.8 Summary of Pitch-Class Set 5-29

A key to understanding the complexity of this song is to appreciate Krenek’s

comprehensive use of pc set 5-29 as a structural motive. Having examined two features

of 5-29 in terms of different partitioned subsets, we finally can explore a somewhat

concealed inclusion property.

Figure 1.5: Comparable segmentation for pc sets 3-2, 4-13, and 5-29

2 1 B 3 0 7 5 4 A

3-2

4-13 4-13

37

As the final five-note segment of the series, as shown in Figure 1.5 to the right

the bold vertical line, pc set 5-29 comprises th

of

e prominent linear motivic pc sets, 3-2 and

-13. The remaining seven-note row segment (to the left of the bold line) also contains

sets 3-2 and

al

4-13)

4

these same subsets, but not all as linear segments. Here the two motivic

tetrachordal motif set 4-13 from the introduction are indicated. Pc set 5-29 from the fin

five-note segment of the series, then, encompasses potential motivic pc sets (3-2 and

as primary harmonies (see Figure 1.5).

Example 1.19: Pitch-class set 3-2 segment (a) in mm. 3 – 5 and 4-13’s (b) in mm. 7 – 8

(a) 3-2 {457} (b) 4-13 {457A}

lve a dyadic partner set in the

1.19(a) shows pc set 3-2 {457}, whose pitch classes are identical to that

in the f

are also subsets of the same pc set 5-29 in Figure 1.5. However, these two 4-13 no longer

form a rhythmic motif. Thus, pc sets 3-2 and 4-13 as subsets of 5-29 do not project a

sonority of 5-29 to the extent that they project 3-9 or 3-11 separately. Accordingly, two

Nevertheless, the pc set 3-2 does not always invo

music. Example

inal five-note segment of the series in Figure 1.5. Unlike the examples for pc set

3-9 or 3-11 used as a subset discussed above, neither a proper 5-29 nor a dyad partner

(presumably 2-2 {A0}) of the 3-2 are present. Two pc sets 4-13 boxed in Example 1.19(b)

38

kinds of partitionings from pc set 5-29 combining pc sets 3-9 and 2-6 and combining 3-

11 and 2-5, function as an important source for motive and harmonic structure.

In the introduction, at first glance, pc set 3-2 appears to be the smallest motivic

unit that may assume a more prominent role, since this pc set functions not only as a

motif that consists of a dyad and single pitch combination, but also as a part of a single

linear shape. However, as demonstrated above, pc set 5-29 as a primary harmony actually

plays a more important role “musically” than does pc set 3-2.

he order numbers of both of the first two dyads are A and B. As discussed above, these

armony which occurs

ithin a single melodic line (see Example 1.16 and 1.17). In contrast, the tritone dyad

ore often appears as a vertical harmony in the second section than in the first; in fact, it

pproximately twic is difference

strengthens the contrast between the song’s two sections not only in musical effect, but

also in the text.

The secondary harmonic tritone A – Eb appears less often as a projected harmony

resented at the

1.4.9 Tritone Dyad

As Schmidt (1998) points out, a tritone dyadic harmony creates a contrast

between the two sections of “Nur ein Wort.” In the whole piece, there are only three

different tritones that function as discrete principal harmonies: the primary harmonies E –

Bb and D – Ab derived from the respective series, and the secondary harmony A – Eb.

T

pc sets 2-6 occasionally conjoin with 3-9 to form 5-29 as a linear h

w

m

appears a e as often in the second than in the first. Th

than the other two tritone dyads. Nonetheless, this tritone dyad exhibits an interesting

feature when we look at the piece as a whole. This secondary harmony is p

39

very beginning of the piece as the tetrachordal motif pc set 4-13, but here the tritone does

not sou ns,

that it

olariz

nd simultaneously. The pitch classes 3 and 9 appear in the first and last positio

respectively, in the small motif. That is, a particular feature of this dyadic set 2-6 is

does not occur as a simultaneous sonority, as do the other two (E – Bb and D – Ab).

Rather, it is used as the outline of the tetrachordal motif by marking its first and last

pitches. What is further intriguing about the use of pc set 2-6 {39} is that its polarized

placement is vastly expanded in the song’s second section. An isolation of these two

single pitches (A and Eb) in fact appears as the first and last pitches of the second section

(the former with “Nein” in m. 10, the latter with “Schlaf” in m. 20). Presented as a

p ation in the small motif at the beginning of the piece, pc set 2-6 {39} is further

expanded and isolated in the second section.

1.4.10 Summary of Pentachordal Invariant Sets and Tritone Dyad

In Twelve-Tone Tonality (1996), George Perle, a pupil of Krenek, presents his

own definition for determining a unique tonality, based on invariance between two

different rows and the dyad harmonies derived vertically from them.30 In Gretchen

Foley’s exhaustive study of Perle’s theory and musical output, the author discusses the

“axis-dyad” that categorizes three kinds of tonal definition, based on Perle’s assertion.31

The “three tonalities” are identified by the sum of the dyad aggregate.32

1996).

30 George Perle, Twelve-Tone Tonality, Second Edition (Berkeley: University of California Press,

32 Ibid., 49-51.

31 Gretchen Foley, “Pitch and Interval Structures in George Perle’s Theory of Twelve-Tone Tonality” (Ph.D. diss., University of Western Ontario, 1998).

40

Figure 1.6: Pentachordal invariance between P9 and I9

a b

P9: 9 8 6 2 1 B 3 0 7 5 4 A

I9: 9 A 0 4 5 7 3 6 B 1 2 8

c d

The model of the representation in Figure 1.6 is from a diagram in Foley’s study.

nction, but merely apply this diagram to analyze

of

o be

ek flexibly employs the first 5-29, a and c, either with or without

e firs

d

t in

rates a notable contrast between the two sections.

Here I do not refer to the “axis-dyad” fu

the invariant pentachordal set 5-29 that Krenek extracts from the two row forms P9 and I9.

As Figure 1.6 shows, pitch classes 3 and 9 literally stand as the “axes” for two forms

pc set 5-29: a = d and b = c. However, these invariant unordered segments seem not t

exploited in the piece. Rather, as we have already observed, two pc sets 5-29 to the right

of the rows (b and d in Figure 1.6) apparently function to present a crucial musical

feature. In contrast, Kren

th t pitch 9. Interestingly, the positions in which the pitch classes of the “dyad-axis”

occur preserve a tritone distance that may imply the idea of polarization or isolation.

1.5 Simultaneity of Motivic and Melodic Pitch-Class Sets

One of the complexities in the second section is that motivic pc set 3-2 and

melodic set 5-29 are concurrently projected in the same subsection, but have not appeare

in the first section. Of course, such a texture contributes to making the musical contex

the second section denser. This illust

41

Example 1.20: Simultaneous occurrence of pc sets 3-2 and 5-29 (dividable into 3-9 and 2- 6)

(a) mm. 11 – 12. (b) mm. 14 – 15.

5-29 (⊃ 3-9 + 2-6)

3-2 3-2 3-2 3-9 2-6

Shortly after the beginning of the second section, the first simultaneity of the

principal pc sets occurs in the piano. Although there is a slight temporal difference

between the beginning pitch of the righ

3-9 2-6 5-29

5-29

t- and left-hand, as shown in Example 1.20(a), a

otivic pc set 3-2 in the bass clef enters while 5-29 in the treble is still unfolding. The

otivic pc set 3-2 appears with the reversed contour of the one already discussed above.

he note value of each dyad in the motif is shorter than that in the first section, but the

etrical interval between a single note and a dyad in the pc set is identical to motivic 3-2

the first section: a sixteenth-note interval (compare with pc set 3-2 in Example 1.12).

The structure of the next simultaneity which enhances the intensification and

ensity of the piano’s musical texture is more complicated. For pc set 5-29 in the right-

and is no longer separated into pc sets 3-9 and 2-6; rather, it shapes a contiguous

scending passage. During the ascending pentachordal passage, successive two fragments

f motivic pc set 3-2 appear almost simultaneously in the left-hand. It may seem difficult

m

m

T

m

in

d

h

a

o

42

to hear the two pc set 3-2, because these are not quite independent motives, but part of

assages, respectively (see Example 1.20[b]). However, the rhythmic configuration of

ese two pc set 3-2 is quite similar to the previous motivic occurrence in the left-hand of

xample 1.20(a). Furthermore, soon after the last motivic 3-2 in Example 1.20(b), pc set

-29 recurs partitioned into pc sets 3-9 and 2-6 as the vertical harmonies. This condensed

ggregation of the prominent characteristic pc sets 3-2, 3-9, 2-6, and 5-29 again reveals

renek’s preference for particular fragmental harmonies. Above all, the last pc set 3-9

comprised of Gb-Cb-Db e second section. In other

ords, we hear a “pure” vertical chord of pc set 3-9 at the end of the passage in mm. 14 –

5 after a somewhat complicated simultaneity of pc sets 3-2 and 5-29. As in the first

section, the diatonic harmony of pc set 3-9 features prominently in the second section as

well. Remember that pc set 3-9 is only one occurrence in the twelve primary harmonies

of cardinality three in the prime series, whereas pc set 3-2 occurs three times (see Figure

1.3[b]). A most interesting feature in comparing these two pc sets is that in “Nur ein

Wort” pc set 3-9 plays as significant a role as does pc set 3-2.

1.6 A Pitch-Class Set for the Climax

The song reaches its climax near the end of m. 17. At the moment of the strongest

dynamics, f, the simultaneous attack point of the voice and right hand enhances and

intensifies the effect of the climax. However, the “espressivo” fragment in the piano

seems to block or diminish the climax since its dynamics immediately diminish to p. And

as we shall explore below, the climactic G in the voice does not represent the highest

pitch of the vocal line.

p

th

E

5

a

K

mphasizes its harmonic features in the

w

1

43

With the loudest dynamics in the entire song, pc set 3-1 {567} is not among the

principal materials that structures the frame of this song. Also, pc set 3-1 {567} is not one

that is included in 5-29 as a primary harmony (see Figure 1.5). Even in compariso

another pc set 3-1 fragment in the voice in m. 5, which is a primary harmony, 3-1

climax is nonetheless a secondary harmony. That is, this climactic 3-1 is less associated

with the song’s linear structure and its principal pc sets. However, such isolation of th

secondary harmony 3-1 serves to furth

n to

at the

e

er intensify the climax, even though the piano

ems not to adequately support it.

xample 1.21: The musical climax and its implication in the second section,

se

E mm. 13 – 18

3-1 {567}

Chromatic pc set 3-1 {567} at the climax in mm. 17 – 18 accompanies “kein” as

the last negation in the text. Again, there is only one 3-1 that constitutes an entire phrase

(although 3-1 {89A} in m. 5 is quite audible as a chromatic stepwise progression, it

climax

44

remains part of the tetrachordal phrase set). The single climactic pc set 3-1 in the vocal

line, in fact, is already prepared a few measures before in the piano’s right hand (m. 13).

Despite conjoining with the tritone dyad to form a phrase, the extracted 3-1 here presents

almost an identical shape to the culminating phrase in the voice (see Example 1.21).

preparation seems to affect a listener’s musical experience at the climax almost as if the

climax had been prematurely echoed.

This

45

CHAPTER 2: Vocal Configuration 2.1 The Vocal Line Contour

Despite the use of only two row forms, the row’s primary hexachord is not

binatorial, and therefore “Nur ein Wort” cannot exploit the combinatorial property in

the formation of aggregates. Moreover, because each series is assigned to its respective

sections, the choice of pitches for the voice part is quite limited. Under such a situation, it

ons for the vocal

ne. As we have already demonstrated, a few particular pc sets, e.g., pc set 3-2,

characterize the linear harmonic structure of the voice. But it is worth noting that the

contour of these recurring vocal motives often varies. Furthermore, since neither the pc

sets nor the number of pitches in respective phrases is consistent, the interpretation of

musical contour in the voice requires the flexible application of various contour measures

and relations.

Rudimentary application of Morris’s theory for musical contour proves useful to

capture an overview of the configuration of the voice.33 In terms of the overall analysis of

pitch contour, the abstract consequence that is obtained is analogous to a contour

“background” of the musical work. For Morris’s theory, such a “background” is

presented by both signs and numbers as “contour intervals.” Although Morris’s contour

eory is also applicable to other domains, such as dynamics and density, here I shall

apply it only to linear pitch co

com

is worth exploring how Krenek creates the specific contour configurati

li

re

th

ntour.

33 Robert D. Morris, “New Directions in the eory and Analysis of Musical Contour,” Music

heory Spectrum 15/2 (Autumn 1993): 205-228.

ThT

46

Example 2.1: Pitch-contour representation for the vocal line in the first section,

mm. 1 – 8

3-4[015] 5-9[01246] 5-2[01235] 4-2[0124] 7-9[0123468]

INT: <− +> <− − + −> <− − + − + +> <− − +> <− − + − + −> CC: <1 0 2> <3 2 1 4 0> <2 1 0 3 0 2 4> <2 1 0 3> <4 3 1 6 2 5 0>

<2 1 0 3> <2 1 0 3 ↓ ↓ ↓ ↓

> <2 1 0 3> <2 1 0 3>

parates two adjacent phrases. Note that as the strings of INT

dicate, the beginning motion of all phrases descends. Moreover, all but the first

e us to imagine a more specific kind of association: notice that the contour

of the fourth segment <2103> (bold type) is identical to the first four motions in the third

xt row interprets the segment <2103> on a more background level,

associating the first four pitches in respective segments with the same contour <2103>.

Having

ow

that the contour of the third, forth, and fifth segment contains one or two occurrence of pc

set 3-2. Also, the contour of these pc sets 3-2 is either <103> or <210>. In reference to

↓ ↓ ↓ ↓ 3-2{457} {245} 3-2{689} 3-2{B12} Example 2.1 shows the voice part in the first section with contour strings by two

kinds of signs, “+” and “–” (INT): “+” represents ascending motion; “–” represents

descending motion; and the numbers in hairpin brackets represent contour-class (CC).34

Since here we focus only on the pitch contour of the vocal line, the segmentation simply

derives from the rest that se

in

segment contin ve in the same direction. The numerical representations of vocal ue to mo

line’s CC enabl

segment. The ne

discovered an identical contour among the four segments, we discover an

interesting feature of the constituent pc sets. The bottom two rows in Example 2.1 sh

34 These notations appear in Morris (1993). Ibid., 206.

47

voice contour in the first section and the progression of the phrasal segments, the

harmonic relationship between contour and the use of pc set 3-2 is tightly controlled. On

the other hand, the first two segments of the voice line in the first section (pc set 3-4 and

5-9) do not share the common harmonic feature, pc set 3-2, of the remaining segments.

Above all, it seems difficult to find reasonable pitch contour connections between th

first phrase of the vocal line, pc set 3-4, and the remaining four in Example 2.1. In fact,

pc set 3-4 as the beg

e

inning phrasal segment has yet to be discussed among the prominent

of pc set 3-4 resulting from the

nalysis of pitch contour suggests that it may be an anomaly in the context of the entire

song.

xample 2.2: Pitch-contour representation for the vocal line in the second section, mm. 10 – 21

harmonies in my analysis. An irregularity of this instance

a

E

4-13[0136] 4-8[0156] 3-8[026] 5-Z36[01247] 5-20[01568]

INT: <+ − +> <+ + − +> <+ −> <+ + − +> <+ + − +> CC: <1 2 0 3> <1 2 3 0 4> <1 2 0> <1 3 4 0 2> <1 2 3 0 4> ↓ ↓ ↓ <1 2 0 3> <1 2 0 3> ↓ <1 2 0 3> ↓ ↓ ↓ 3-5{45A} 2-6 {28} 3-8 {A04} 3-8 {A04

}

5-3[01245]

5-20[01568] 3-1[012]

INT: <− − + +> <+ −> <− + + −>

CC: <4 3 0 1 2> <1 2 0> <3 0 1 4 2>

48

The pitch contours of the second section differ markedly from those of the first.

As Example 2.2 indicates, whereas the first half of the second section remains somewhat

onsistent in the pitch contour relationships among its phrase segments, the last half does

presentation of pc set 4-8 and 5-

0 is identical: <+ + – +> and <12304>. Partial contour subsets, <+ – +> with <1203>

become fixed among pc sets 4-13, 4-8, and 5-20. In addition, <+ – > with <120> is

common to all the segments except for pc set 5-Z36. The pc set obtained by this

the sets, we do find a distinctive intervallic feature that connects them.

Figure 2.1: Interval vectors of three pc sets derived from the reduction of pitch-contour

3-8 [010101]

The interval vectors in Figure 2.1 reveal that the three pc sets, 3-5, 2-6, and 3-8,

since the dyad is a tritone itself, 3-5 and 3-8 represent two of the three trichords that

contain ic6 out of the twelve possible trichords.35 Among the pitch classes of these sets

form in the second section, I (see Example 2.2). On the other hand, the possible tritone

c

not. In the first half, the interval contour and numeral re

2

reduction procedure varies, in comparison to the first section. Yet when closely analyzing

Pc set name Interval vector 3-5 [100011] 2-6 [000001]

are related by a single tritone interval class (ic) 6. While pc set 2-6 contains only ic6,

derived from the reduction contour, only 2-6 {28} is a primary harmony of the prime row

9

among the remaining pc sets 3-5 {45A} and 3-8 {A04} is 2-6 {4A} as a secondary

35 The third is 3-10, which does not appear in my segmentation for this piece.

49

harmony.36 Remember that Krenek uses more than twice the number of vertical tritone

chords in the second section as in the first. Analysis of pitch contour and its reduc

shows that half of the phra

tion

sal segments of the vocal line in the second section contains a

itone set, while no tritone set is included in pc set 3-2 as a final result from the reduction

ontour in the first section.37

xample 2.3: The marking of the highest and lowest pitches for each segment in the first section

tr

c

E

tive.

.

ile the

es to maintain equality and balance between the music

nd Kafka’s text.

Now, let us look at the pitch contour in the first section from a larger perspec

In comparing phrasal segments in the first section, there is a remarkable feature that

Krenek seems to have manipulated consciously. In Example 2.3, the circled notes

indicate the highest pitches and boxed notes indicate the lowest in each partitioned phrase

Every noun and the verb (lebst) take the highest pitch in respective segments, wh

lowest pitches accompany another verb (wartest) and either a noun or an article (ein).

The consequence of this large range structure of pitch contour reveals how Krenek

deliberately joins words and pitch

a

P9.

37 There are two phrasal segments that include a tritone set in the voice of the first section: pc set second 5-20. om pitch

contour of each phrasal segment.

36 Tritone pc set 2-6 {4A}, however, is a primary harmony in

5-9 and 7-9 (see Example 2.1). Also, there are two in the second section: pc set 5-Z36 and theHowever, these particular pc sets are irrelevant to the examination of shared pc sets derived fr

50

Figure 2.2: The overall pitch-contour of the first section

The highest group The lowest group INT: <+ − + −> = <+ − + −>

We have expanded the application of pitch contour analysis from a single phrase

line. Yet this outline is

sufficient to provide a precise view of how the voice is shaped in the first section. Thus,

y other internal structures.

xamp 2.4: Dyad transformation in the first section

CC: <0 4 2 3 1> ↔ <1 3 2 4 0>

The interval contour (INT) and register (CC) measure a structural similarity and a

palindromic pattern, respectively. Figure 2.2 compares two groups that join the highest

and lowest pitches, respectively, from each segment. Although the interval contour is

identical, <+ – + –>, the contour class unfolds in retrograde. This suggests further

analysis by a different method, since these slowly unfolding structures would be difficult

to perceive. Therefore, we need to apply other analytic approaches to the vocal line.

2.2 Pitch-Space Projection and Voice-Leading

to an entire section to obtain a broader outline of the vocal

in

we must examine the vocal line using a different approach, so that we can see whether or

not the voice’s overall motion is controlled b

E le

T9 T9

seg. 1 seg. 2 seg. 3 seg. 4 seg. 5

51

Each vertical dyad chord in Example 2.4 represents the highest and lowest pi

of the respective segments (seg.) in the first section. Since this section consists of five

phrases, the number of dyad chords is five. The number to the right of each square

bracket indicates the interval in semitones between the two pitches. The interval of the

first dyad, for example, is eleven semitones. All dyads except for the first have a partner

whose interval is identical, and each dyad in a pair is adjacent to on

tches

e another. According

to Jona

r

e

he more

e correct register, the numbers in the string when the F is placed an octave

igher drastically change to <03142>. However, the INT string representation is

than Bernard’s terminology (1981), a transformation in which an interval value is

preserved when one element – normally a dyad or a triad – moves to another register o

to other pitch classes is called “projection.”38 In the first section, the “projection” is

straightforward: the interval of fourteen-semitone of the second dyad projects to the

following dyad by T9 or three semitones lower. Similarly, the ten-semitone interval of th

forth dyad transforms to the fifth again by T9 (the actual pitch register of the lower note,

F, in the last dyad is an octave lower). In terms of pitch contour, the representation of F

by placing it an octave higher may be inappropriate. In this case, it seems that t

important issue is a change of the “interval contour” string: while CC yields <13240>

when F is in th

h

maintained as <+ – + –>.

38 Bernard, “Pitch/Register in the Music of Edgard Varèse,” 9-10.

52

Example 2.5: Dyadic voice exchange

When observing this d rent angle, we find that

two pairs of dyad in the first s 5, interact via a voice

exchange, as illustrated in Example 2.5. Interestingly, only four pitches (Eb, F, F#, G#)

represent the boundaries of pitch contour among these dyads. In addition, the resultant set

found by these boundary pitches forms pc set 4-10 {3568} (prime form [0235]) which is

a symmetrical pc set. Thus, except for seg. 1, specific pitches structure the outer

boundary pitches of the voice in the first section. However, the first phrase seem

little connection to those that follow. Yet, we will see that this initial “outsider” phrase

reappears in the second section.

Example 2.6: Dyad transformation in the second section climax

yad succession from a slightly diffe

ection; seg. 2 and 4, seg. 3 and

s to have

seg. 6 seg. 7 seg. 8 seg. 9 seg. 10 seg. 11 seg. 12 seg. 13

53

The dyad transformation in the second section is slightly more complicated than

that in the first, not only because it comprises more dyads whose intervals are more

aried, but also because the intervals of some dyads connect with those that appear earlier

the first section. The most striking “projection” in the second section is an interval

irteen (int. 13) dyad that eventually reaches the climax (seg. 12) through an ascending

ansformation: seg. 7-10-12. Clearly it serves as the fundamental framework of the

ther approximately

ree-fourths of the section. In this transformation, the dyads of int. 8 that are inserted

etween each ascending int. 13 dyads move in the opposite direction (see seg. 8-11 in

xample 2.6).

igure 2.3: Overall dyad transformation in the vocal line

t.: 11 14 14 10 10 14 13 8 11 13 8 13 15

itte” “Bitte”

eg.:

Figure 2.3 summarizes the association among dyad progressions for the entire

song. A double vertical line marks the two sections, and the brackets indicate properties

of identical int. Left as an “outsider” in the first section, int. 11 dyad (seg. 1) finds its

partner in the second section (seg. 9). Accordingly, this long-distance connection is

balanced by the fact that each occurs only once in respective sections. Note that this dyad

moves without transposition: D-C# to D-Db, that is, pc {12} T0 transposition in the

identical register. Also, another “long-distance” pair, int. 14 dyads (seg. 2 and seg. 6), is

v

in

th

tr

second section, since the int. 13 dyad occurs most often and ties toge

th

b

E

F In “B S 1 2 3 4 5 6 7 8 9 10 11 12 13

54

connected with the same text, “Bitte.” As a result, dyad “projection” over the entire

shows that the las

song

t dyad, int. 15 (seg. 13), is the only real outsider.

In summary, dyad transformation in the vocal part reveals an asymmetrical and

ontradictory structure between the two sections. For instance, the second section has

ore phrasal segments than the first. Regarding the sections’ internal structure, while the

rojective transformations in the first section remain consistent through the identical

anspositions and voice exchanges, that of the second consists of cross-transformations

nd a phrasal “outsider.” From a broader perspective, the second section is more complex

an the first. Nonetheless, the two sections are structurally associated by two types of

invariant transformation: int. . 1 to seg. 9) and a text

transformation of “Bitte.” No the only common word between the

o sections.

.3 Transformational Axis

We have affirmed some differences and a few connecting features between the

ong’s two sections. Although many pitches in respective phrasal segments in the vocal

ne are left out of the previous analysis, examination at the dyadic level reveals an

triguing, concealed structure. Hence, assuming that the respective dyads extracted from

e pitch contour boundary of each phrasal segment may also be units for further study, I

hall explore these dyads through another transformational procedure.

c

m

p

tr

a

th

11 T0 transformation (seg

t surprisingly, “Bitte” is

tw

2

s

li

in

th

s

55

Figure 2.4: Transformation by I11 in the first section

a) A cohesive transformation in verticalized dyads

seg. 2 seg. 3 seg. 4 seg. 5

F F#

I11 I11 I11

Straus’s recent discu n another aspect of dyad

ansformation in the vocal line. The diagram in Figure 2.4(a) shows the voice leading of

dyads extracted from seg. 2 through seg. 5. All pitches in the four dyads project a “crisp”

inversional relationship. Furthermore, the inversion operation is unified by I11. This

suggests that it may be advantageous to portray the consistency of the dyadic progression

in the image of a clock-face.

G# Eb F# Eb G# F

ssion of voice leading sheds light o

tr

39 40

39 In Joseph N. Straus, “Uniformity, Balance, and Smoothness in Atonal Voice Leading,” Music

Theory Spectrum 25/2 (Fall 2003): 305-352; the author defines “crisp” voice leading when each member of a set transforms to each member of another with a one-to-one mapping in the same transformation. Thus,

onto F in seg. 2 by the same transformation, which is “crisp.” If it were D in seg. 2 instead of Eb,

transformation “fuzzy.”

for instance, G# in seg. 1 maps onto Eb in seg. 2 by I11 transformation, and likewise, F# in seg. 1 maps

inversional transformation between the pitch and G# in seg. 1 yields I10. In such a case, Straus calls the

40 I11 means a transformational operation by which a pitch is first inverted around pc 0, and then transposed by eleven semitones. For instance, when G# (pc 8) in seg. 1 is simply inverted, the operation projects E (pc 4). Then, eleven-semitone transposition from E projects Eb (pc 3) in mod 12. More easily, the sum of two pitch class numbers in mod 12 is equal to the inversional operation; namely, G# (pc 8) + Eb (pc 3) = 11, likewise, F# (pc 6) + F (pc 5) = 11. Whereas the inversional operation is useful and applicable to post-tonal music, it may be controversial, especially in terms of aural perception. Compared with the transpositional operation that is perceivable to the listener because of literal transposition, the inversion operation seems more abstract because it is difficult to immediately perceive an inversional relationship.

56

Figure 2.4: b) A clock-face diagram for representing I11 as an axis

As the next analytical stage, we ca

dyads. In this procedure, a pitch is defined

and G#-F# are the same. Under this defini

diagram representing pitch classes in the p

dotted line. As the dotted line in Figure 2.

inversion axis I11 (= I ): specifically, F#

similarly G# (pc 8) is inverted and maps o

the axis consists of the actual pitches in th

type in Figure 2.4(a). This type of inversio

transformation.

b

F F#

F#

n explore

as a pitc

tion, Figu

revious e

4(b) show

(pc 6) is

nto Eb (p

e dyad pr

nal opera

F

the a

h clas

re 2.4

xamp

s, dy

inver

c 3) (

ogres

tion a

F

E

G#

I = I11 F#

xis of inversion between the two

s; thus, for example, dyad F#-G#

(b) illustrates a clock-face

le with an axis indicated by a

ad F#-G# maps onto Eb-F by

ted and maps onto F (pc 5) and

see footnote 35). In this instance,

sion, as indicated by the bold

lways yields a symmetric

57

Figure 2.5: I11 transformation in the second section

(a) Interval 13 dyad group

seg. 7 seg. 10 seg. 12

Ab E Gb

A F

I11

(b) A clock-face diagram representation for inversion operation by an I11 axis

In the int. 13 transformations, an inversion operation for seg. 10 dyad to another

at the climax, seg. 12, is accomplished by I11 around the same axis. Similar to the

representation of Figure 2.5(a), the pitches that constitute an axis are actual notes in the

music. In the same way, the same I11 axis operates for the transformation of the int. 8

dyads between seg. 8 and seg. 11, as shown in Figure 2.6 below.

Figure 2.6: I11 transformation of an interval 8 pair in the second section seg. 8 seg. 11

CCb

I11

G

GbF

E

G

I = I11 Gb F

G E

58

CHAPTER 3: Tension-Degrees

nek’s “Tension-Degrees”

Pitch-class set theory and contour theory have revealed remarkable features in

3.1 Kre

that create both consistency and contrast, particularly among those pc

sets that represent characteristic sonorities and in their manipulation and distribution.

Above all, the pitch contour theory has clarified the musical features that seem most

easily perceived. In his analysis of Schoenberg’s Piano Piece, op. 19, no. 4, Morris (1993)

asserts that “a combination of contour and pc analysis can help us understand continuity

and coherence in a composition.” In fact, his study shows that some significant

harmonic pc sets represent features that constitute the “continuity and coherence” that

enable clearer perception of harmonic and transformational structures in the piece.

However, Morris fails to show how a pc set, derived from pitch contour theory, produces

the musical effects in different configurations or registers. In other words, although the

“continuity and coherence” are well discussed by exploring the connection among the

contours and particular pc sets, further study of the musical characteristics of each

contour “in the same pc set” remains.

with Pc

space” in which a single pc set unfolds various intervallic and registral combinations.

Here, Morris’s point of view seems to accommodate better the likely limits of human

perception. I agree with his fundamental claim that “the study of an atonal composition

“Nur ein Wort”

41

Morris’s later article, “Equivalence and Similarity in Pitch and Their Interaction

set Theory” (1995), draws closer to the contextual core derived from “pitch-

41 Robert D. Morris, “New Directions in the Theory and Analysis of Musical Contour,” Music

Theory Spectrum, 15/2 (Autumn 1993): 205-28.

59

conside

usical realizations as pitches in various categories of time.”42 Based on this notion,

Morris generalizes a pitch set-class in terms of a pitch spatial measurement denoted as

PSC and develops PCINT (pitch-class interval equivalence) to categorize interval

equivalence. Furthermore, he establishes a FB-class (figured-bass class) to clarify the

intervals between the lowest pitch and those above. In his analysis, these methods are

especially useful for measuring the transformation and similarity of particular pitch-class

sets or collections. What is striking in Morris’s theory is that it enables an analyst to more

clearly grasp the process by which specific musical materials based on registral spacing

constitute the larger structure of a work.

since Krenek had his own classification of intervallic relationships in a chord which

appears in his text Studies in Counterpoint. Consequently, one can further categorize

respective dyadic and triadic intervals within an octave into several groups according to

his definition. Clearly Krenek’s method (1940) is rather rough, perhaps even

unsophisticated or obscure, in comparison with current music theories. However, because

Fünf Lieder was composed around the same time Krenek published his text, it may be

worth exploring his unique system of classifying sonorities.

Around the time of composing Fünf Lieder, Krenek’s notion of how “tonality”

and “atonality” differ is rather subtle, even though he does attempt to precisely clarify

rs the relations between the underlying pitch-class materials and entities and their

m

For “Nur ein Wort,” such a transformational analysis seems less fully applicable

43

42 Robert D. Morris, “Equivalence and Similarity in Pitch and Their Interaction with PCSet

Theory,” Journal of Music Theory, 39/2 (Autumn 1995): 207-43.

43 Ernst Krenek, Studies in Counterpoint: Based on the Twelve-Tone Technique (New York: G. Schirmer, 1940).

60

this difference.44 One representative quotation from Studies in Counterpoint (1940)

suggests that he was still distinguishing between consonance and dissonance: “as the

harmonic relationships of tonal music play no part in atonal music, terms such as ‘major

third,’ ‘perfect fourth,’ ‘diminished fifth,’ and so on, have no meaning.” In “Three-Part

Writing,” Krenek categorizes the “tension-degrees” that “depend on what kinds of

intervals are formed by the constituent tones of the different chords.”45

For the term “tension-degrees,” Krenek’s basic concept is to measure the degree

of tension of a vertical chord derived from a contrapuntal texture by use of the twelve-

tone technique. However, a concept of degree of tension from a slightly different

perspective from Krene

k’s is established by Hindemith in The Craft of Musical

Compo l,

to understand, as the most striking innovations seem to have been introduced into

multitonal and complicated chords . . . drew general attention to what was going on; and many investigations were devoted to the derivation, classification, and evaluation of chords, from Schoenberg’s Harmonielehre (Treatise on Harmony) to Hindemith’s Unterweisung im Tonsatz (Instruction in Composition).46

sition: Book I, Theoretical Part (Unterweisung im Tonsatz: Theoretischer Tei

published in 1937). Thus, it seems likely that Krenek derived the idea of tension-degrees

from Hindemith’s method. In fact, Krenek refers to it under the title of “Attempts to

Create a New Order” in Music Here and Now (1939):

The theoretical approach was primarily limited to harmonic research. This is easy

the field of the material, the harmonies or chords. The sudden emergence of

44 Ernst Krenek, Music Here and Now, trans. Barthold Fles (New York: W. W. Norton, 1939). 45 Krenek, Studies in Counterpoint, 19-25.

46 Krenek, Music Here and Now, 154. The translation of the title into English is different from

that of actual publication in English. The title in English for Unterweisung im Tonsatz in the parentheses seems Krenek’s own translation. While the first publication of Unterweisung im Tonsatz in English translation is 1945, Krenek’s statement above was already published in English translation in 1939.

61

Briefly, Hindemith’s concept of “harmonic tension” derives from measuring the

frequency ratios of an interval(s) in a chord and measures the relationship among a

succession of chords, typically resulting in “harmonic crescendo and diminuendo” a

“harmonic fluctuation.”

nd

harmon

om

,

ine the degree of tension. He argues:

Combination tones are a type of acoustic specter of secondary rank, distinguishable from overtones by being supposedly heard when two real tones

ore, they appear not over, but under the real tone, and so can be referred to as “undertones.” They are believed to be responsible for the tension degree of the interval that is actually heard. Besides

tones” can at best furnish only a physical explanation of the quality of a harmony;

seems that Krenek is arguing not about Hindemith’s mathematical approach, but about

the l m or de ng a cha ic n ly p ceived by

the listener’s ear. Krenek continues to assert the uselessness of H ith’s conception

as tio l creativity: “in no case can us how that harmony

should be applied sit [. s ly e

47 That is, an analyst has to delineate a layer of successive

ies on a somewhat larger level; in addition, one needs to determine the tonal

center in the overall harmony. In contrast, Krenek’s concept of tension-degrees seeks to

measure degrees of tension in the process of contrapuntal composition that derives fr

the twelve-tone technique.

In Music Here and Now, Krenek severely criticizes two features of Hindemith’s

method. First, for the tension degree in relation to “combination tones,” Krenek questions

whether an “undertone,” which is far more difficult to recognize than the actual sound

can determ

are played simultaneously. . . . Furtherm

being even less audible than overtones, the “under48

It

theoretica aterial f termini musical racterist ot direct er

indem

an applica n to musica they tell

in compo ion. That, . .] depend exclusive on creativ

47 Paul Hindemith, Craft of Musical Composition: Book I, Theoretical Part, trans. Arthur Mend l e

(New York: Associated Music Publishers, 1945). 48 Krenek, Music Here and Now, 199-200.

62

imagin material.”49 Second,

s Hindemith’s method is structured on the basis of tonal theory, Krenek seems to regard

as out-of-date. Krenek argues that

In spite of Hindemith’s wish to pave the way to a progressive type of music, his theory necessarily results in favoring the basic facts of tonality – the major triad and simple harmonic rela e old tonal music.50

At any rate, in Studies in Counterpoint, Krenek claims that “harmonic

progressions in atonal music occur preponderantly in the area inhabited by dissonances of

a higher degree of tension, and that the lower tensions of consonances have practically no

part in the scheme.” But Krenek also implies that the above definition does not restrict

musical vocabulary: “Nonetheless, an extraordinary variety of nuances is found even

within these limits.”

Before classifying tension-degrees, Krenek first defines three types of intervals

between two pitches: one “consonance” and two dissonances of different kinds – “mild”

and “sharp” (or “lower” and “higher” tensions, respectively). According to Krenek’s

definition, consonant intervals include major- and minor-thirds and sixths and perfect

fourths and fifths, mild dissonances include major seconds and minor sevenths, and sharp

dissonances contain minor seconds and major sevenths. However, the flexibility of the

remaining two intervallic definitions that Krenek provides may be confusing. First,

Krenek identifies the interval of a perfect fourth, interval-class (ic) 5, as either a

consonance or a dissonance which “depends on the context.” Krenek explains that if a

ation or artistic intention, never on the physical nature of the

a

it

tionships – exactly as in th

51

52

49 Ibid., 200. 50 Ibid.

52 Krenek, Studies in Counterpoint, 7-8.

63

five-semitone dyad is preceded by a sharp dissonance it is consonant, but if it is prec

by a consonance or an octave extended interval of the

eded

perfect fourth, seventeen semitones,

is then dissonant.53

intervals, Krenek labels it as a “neutral interv

dyad does not, by itself, have a firm character of either consonance or dissonance. In

Chapter VII, “Three-Part Writing,” Krenek specifies more clearly the intervallic

flexibility of a tritone dyad with almost the same treatment as for the interval of a perfect

urth. By his definition, the characteristic determination of these two inter lways

influence exerted by the additional third tone.”54 Thus, for the intervals

of a pe

r.

on

it

Second, because the interval of a tritone divides an octave into two identical

al.” In other words, an interval of a tritone

fo vals a

depends on “the

rfect fourth and a tritone, Krenek merely assigns them a secondary role, implying

that the intervals of seconds, thirds, sixths, and sevenths have more distinctive and

definite characteristics. We will see more details of how these intervals interrelate late

Six types of tension-degrees are classified by a combination of three dyadic

intervals based on the definitions above. Since Krenek maintains that the “classificati

of chords does not involve any evaluation of their conformity to traditional ideas of

‘beauty’ or ‘ugliness,’” one of his compositional interests in producing a sonority by the

53 Krenek does not specify the special treatment of the perfect fourth dyad. In the context of his

consonant or dissonant. Krenek may be referring to it from a perspective the pitch space; in tw

tension (Ex. 27), whereas in the following passage the fourth (represented by an eleventh) appears dissonant (Ex. 28):”

description, the interval of the preceding dyad determines whether the perfect fourth dyad is either o examples,

he states: “the character of the fourth is rather consonant, because of the preceding intervals of higher

Thus, for the treatment of a perfect fourth, contrapuntal styles of the medieval and Renaissance eras better reflect Krenek’s musical concept. In fact, Krenek recommends the study of strict counterpoint “as a prerequisite” in the Introduction, ibid., ix.

54 Ibid., 20.

64

twelve-tone method is to determine the “degree of tension” in a trichord, which he

regards as the fundamental unit needed to create “atonal” music. Thus, to further examine

renek’s music in the late 1930s, it may be helpful to engage his concept of tension-

egree.

.2 “Tension-Degrees” and Pitch-Class Sets

e 2 4 5 6

K

d

3 Figure 3.1: Classification of tension-degrees and pc sets (a) A diagram for tension-degrees and relevant pc sets

Degre 1 3 Interva

tel

nts c co

2 mn

sh. 1 . 1 1

m. sh.

lic 3 con

ons. 2 m.

ns. 1

1 cons..

2 co s. 1

consm. sh.

1 2

Possset/

iblprime

form

- ] 3-12 [048]

3-7 [ 25] 4] [03-4 [015]

3-2 [013] 3-1 [ e pc 3 11 [037 0 3-6 [02 3-3 14] 012]

Interval vector

[001110] [000300]

[011010] [020100] [101100] [100110]

[111000] [210000]

cons. = consonance = major, minor 3rds and 6ths, perfect 5th = ic: 3, 4, 7, 8, 9 55

d nc aj arp a m d a = 1

(b) Classification of tension-de terval vector

[ 0 0 0 0 0 0 ] sh. m. neutral

m. = mild issona e = m or 2nd nd minor 7th = ic: 2, 10 sh. = sha disson nce = inor 2n and m jor 7th ic: 1, 1

grees in an in

cons.

55

interval of a perfect fifth in his discussion of consonance and dissonance, he specifies its consonance and dissonance in Chapter III, Two-Part Writing, in Krenek, Studies in Counterpoint, 7, with the following

Although Krenek does not mention in Three-Part Writing (chapter VII) the category for the

example:

65

A tension-degree by Krenek’s definition is categorized by the contents of three

intervals in a trichord formed within an octave. Figure 3.1(a) tabulates the classificati

of six types of tension-degrees.

on

is,

to the bottom: the third row indicates possible pc sets that fulfill the condition

f each degree, with each prime form described in square brackets. Finally, the bottom

terval vector

sho in the second row. Following Krenek’s

fifth digits of an interval

ild” (m.) dissonant

terval, and of the first indicates a “sharp” (sh.) dissonant interval.58 Regarding ic6

(tritone), it is labeled as “neutral” (see Figure 3.1[b]).

xamp e 3.1: richor s cont ning a perf tritone rom examples presented by Krenek in Studi 1940)

(a)

56 The number in the upper row designates the degree of

tension; degree 1 indicates the least tension, whereas degree 6 indicates the most. That

each column towards the right gradually increases in tension. The second row displays all

the intervals contained in each degree. The abbreviations and respective intervallic

contents are shown below the table.57 Based on Krenek’s categorization, I have added

two rows

o

row shows the interval vector of each pc set. The representation of the in

ws the correspondence to the interval contents

definition, the interval-class (ic) entries in the third, fourth, and

vector indicate consonance (cons.); that of the second indicates a “m

in

E l T d ai n interval of (a) a ect fourth and (b) a

f es in Counterpoint (

56

The contents of the table are based on the examples Krenek presents in ibid., 19-20.

parentheses of this sentence and Figure 3.1.

57 These three abbreviations are according to Krenek, ibid., 20.

58 To avoid confusion, I shall describe these three categorizations in italic type, as those in the

66

(b)

Having confirmed the consistency of Krenek’s general classification of tension-

degrees, we focus on a more obscure part of his definition: characteristics of a trichord

that contain the interval of a perfect fourth, tritone, or both. The two examples above

appear in Krenek’s Studies in Counterpoint. The upper example shows trichords that

contain one or two intervals of a perfect fourth, while the lower shows trichords that

contain an interval of a tritone. In these examples, we can easily find a simple intervallic

hierarchy in each trichord. For example, the first trichord in Example 3.1(b) contains a

(C-Db), a perfect fourth (Db-Gb), and a tritone (C-Gb). Since the interva

sh.

l of a tritone is

arded as “neutral” and that of a perfect fourth has nearly the same

th and

r. Therefore, the tension-

egree f this t chord

always reg

characteristic, a minor-second as “sharp” seems required to classify the dissonance as

“sharp.” The second to the last trichord in Example 3.1(a) contains two perfect fourths

(C-F and F-Bb), but the additional interval between C and Bb is a minor-seven

should be entered in the second column of the interval vecto

d o ri is defined as “mild.”

67

Figure

3.2: Charts for abstract representation from Example 3.1(a) and (b)

(a) All trichords that contain at least one perfect fourth (5 semitones)

Trichord C-Db-F C-D-F C-Eb-F C-E-F C-F-Gb C-F-G C-F-Ab C-F-A C-F-Bb C-F-B

Intervallic sh. cons. m. cons. m. sh. cons. sh. dim. m. cons. cons. cons. m. 4th. sh. dim. contents cons. cons. cons. Degree sh. m. m. sh. sh. m. cons. cons. m. sh.

Pc set 3-4 3-7 3-7 3-4 3-5 3-9 3-11 3-11 3-9 3-5

Interval vector

[101100] [011010] [100011] [010020] [001110]

(b) All trichords that contain one tritone (6 semitones)

Trichord C-Db-Gb C-D-Gb C-Eb-Gb C-Fb-Gb C-F-Gb C-F#-G C-Gb-Ab C-Gb- C-Gb-Bb C-Gb-Cb A

Intervallic contents

sh. 4th. m. cons. cons. cons.

m. cons. sh. 4th. sh. cons. m. cons. cons. cons.

m. cons. sh. 4th.

Degree sh. m. cons. m. sh. sh. m. cons. m. sh.

Pc set 3-5 3-8 3-10 3-8 3-5 3-5 3-8 3-10 3-8 3-5

Interval vector

[100011] [010101] [002001]

ds in

w

l

ee are

Two tables in Figure 3.2 show the interval contents, tension-degree defined by

Krenek, pc set, and interval vector59 of eight of the possible twelve trichords. The pitch

contents of each column correspond to respective trichords in Figure 3.2. All trichor

Figure 3.2(a) contain at least one perfect fourth, and thus they are omitted from the ro

of intervallic contents. Similarly, the interval of a tritone is omitted in Figure 3.2(b).

Unlike the previous table in Figure 3.1, the row for “Degree” displays not the numera

indication, but the three types of Krenek’s tension-degree. With the representation of

additional trichords that all contain a “flexible” interval, all pc sets of cardinality thr

presented. However, the tables in Figure 3.2 suggest more complexity than those in

59 Since there are two or more identical pc sets in each table, I describe only one interval vector for

each respective trichord.

68

Figure n-

egrees.

Figure 3.3: Tension-degrees grouping based on Krenek’s classification

(a) Three tension-degree groups and pc sets

Consonance group (CONS) : 3-10, 3-11, 3-12

Sharp dissonance group (SH): 3-1, 3-2, 3-3, 3-4, 3-5

-11 [001110] -7 [011010] -2 [111000]

-9 [010020] -4 [100110]

tter

contains at least one sh. interval belongs to a sharp dissonance group as SH, even though

it also contains either m. or cons. interval or an interval of a perfect fourth or tritone.

Based on the pc set distribution in each group, shown in Figure 3.3(a), a list of the

interval vectors illustrated in Figure 3.3(b) reveals the properties of the intervallic

3.1(a) since the intervallic contents vary, even though there are only three tensio

d

Mild dissonance group (M) : 3-6, 3-7, 3-8, 3-9

(b) Three tension-degree groups and pc sets with the interval vectors

CONS: 3-10 [002001] M: 3-6[020100] SH: 3-1 [210000]

-12 [000300] -8 [010101] -3 [101100]

-5 [100011]

To clarify Krenek’s more detailed categorization, let us classify pc sets of

cardinality three into three broader groups. The criteria for grouping pc sets are as

follows: a pc set that contains two or three cons. intervals, regardless of the existence of

an interval of a perfect forth or a tritone, belongs to the consonant group, denoted as

CONS. A pc set that contains an m. interval and excludes any sh. interval belongs to the

mild dissonance group, denoted as M. Also, in this broader grouping, it does not ma

whether a pc set contains an interval of a perfect fourth or a tritone. Finally, a pc set that

69

contents. The entries of ic for the pc sets in CONS stand in all cons. position (see Fi

3.1[b]): ic3, 4, or 5, except for pc set 3-10 which contains an ic6 “neutral” entry. For M,

while none of the pc sets contains an ic

gure

1 entry, there is at least one entry in ic2, which is a

mild”

-11 – cons., cons., cons. or perfect fourth (4 )

M: 3-6 – m., m., cons.

-8 – m., cons., tt.

SH: 3-1 – sh., sh., m.

-3 – sh., cons., cons.

-5 – sh., tt., 4 or cons. (5 )

Having classified the twelve pc sets of cardinality 3 into three broader categories

of tension-degree, we face more difficulty in determining the degree of tension by a

umeral representation within each group. For instance, if we look at pc sets in CONS,

ere is an ambiguity in the tension-degree contents in 3-11; the third degree is either

cons. or a perfect fourth (4th) (see Figure 3.4). Inevitably, the third cons. should be an

inversion of a perfect fourth, a perfect fifth. That is, the combination of intervallic

contents depends on whether pc set 3-11 (major or minor triad) figures the root position

“ dissonance. In contrast, all of pc sets in SH entries contain at least one ic1, which

is “sharp.” Similar to CONS, both M and SH have a pc set that contains an ic6 entry.

Interestingly, each group comprises the successive pc set name, and it seems that the

lower the number of the pc set label, the higher is the tension-degree.

Figure 3.4: A list for the contents of tension-degree factors

CONS: 3-10 – cons., cons., tritone (tt.) th

-12 – cons., cons., cons.

-7 – m., cons., cons. or 4th

-9 – m., 4th, 4th or cons. (5th)

-2 – sh., m., cons.

-4 – sh., cons., 4th

th th

n

th

70

or two inversions, or , in a conventional description for a chord. On the other hand, the

interva opt

the me

atio -de

sonor d n to h

labe efin of conson d dissonance come fro raditional tonal

musical theory. If we assume that pc set 3-12 is more consonant than 3-11 because of its

fixed three cons., we find that Krenek’s concept of tension-degrees contradicts our

general

B ust

Although his comparison between a triad (presumably regardless of major or minor) and

a diminished seventh chord seems imprecise because of their differing number of pitches,

Krenek detects more tension in a diminished chord with a tritone than in a major or minor

triad. To one familiar with Western music, Krenek’s sense of intervallic tension is quite

natural, regardless of whether it is in the context of tonal or post-tonal music. Thus, as an

llic contents of pc set 3-10 and 3-12 do not have such an ion. In other words,

mber(s) of the tension-degree contents of pc set 3-10 and 3-12 are fixed.

Krenek’s unique categoriz n of tension grees reveals that he considers the

ity of a trichor ot as a nal function, but as t e sound itself – even though his

ling and basic d ition ance an m t

tonal harmonic sense that an augmented triad (3-12) is more likely to create

tension in the resolution of a stable harmony, such as a major or minor triad (3-11).

Krenek’s statements with regard to consonance and dissonance in his earlier

publication may confuse a reader. Whereas Krenek defines a diminished triad and a

major or minor triad as a consonance without a particular differentiation of tension-

degree, he asserts a distinction between major or minor and diminished sonorities in

Music Here and Now, published in 1939:

Experimentation through hearing convinces us that the chord of E Flat-C-D-reveals a higher degree of tension than the chord of E-C-D-B Flat; but one madmit that the difference is not so spectacular as the distinction between a triad and a diminished seventh chord.60

60 Krenek, Music Here and Now, 148.

6 6 3 4

71

overall sound, one also should hear more tension in a diminished triad than in a major

minor triad. In fact, the frequency ratio of a tritone, 32:45, psycho-acoustically creates th

effect of far more tension than any other intervals of cons. and a perfect fourth.

Krenek’s basic idea of distinguishing between consonance and dissonance seems

to be based on octave equivalences and the conventional definition of an interval between

two pitches within a single octave.

or

e

e.

egrees should be interpretable by “vibration-ratios” and “combination-tones.”62 In other

the

thetic concepts”63 that might rely on context; nonetheless he published several books

n his mpos u . Ther

ssigni relat ot su he

s both to grasp more concretely the acoustic differences among the twelve trichordal pc

r understand Krenek’s dramatic use of specific sets in “Nur ein Wort.”

While the representation listed in Figure 3.4 is still too abstract t usically apply the

61 If so, it is quite possible to assume that Krenek was

also familiar with frequency ratios. Nonetheless, Krenek’s ambiguous classification of a

perfect fourth and a tritone does not fit our usual concept of consonance and dissonanc

Apparently, Krenek’s basic notion of the distinction between consonance and dissonance

is contained within his musical aesthetics, even though he clearly believes that tension-

d

words, far more important to Krenek than systematizing a rigorous method was

“aes

o co itional techniq e efore, given Krenek’s emphasis on context in

a ng ive tension, it is n rprising that did not quantify the precise degree

of tension among all twelve trichords.

However, further classification of tension-degrees is necessary, for it will enable

u

sets and to bette

o m

61 Ibid., 7. 62 Ibid.

63 Ibid.

72

degrees of tension, we need to explore the potential of subdivision into the individual

tension-degrees by relying partly on a physiological aspect, using the frequency ratios

just intonation. Although there have been numerous investigations to clearly distinguish

consonance and dissonance, producing many consistent and convincing theories, there

seems even today no standard theory in terms of psychophysics.

of

more credible theory related to frequency

ratios, f et

te

Pc set name/ letter notation frequency ratios (fr.) fr. of trichord LCM

400

64 In any event, the

frequency ratio derived from two tones is one of the most fundamental features used to

measure relative dissonance. However, since my primary interest is neither to discuss

such a psychophysical issue nor to investigate a

requency ratios here will only serve to clarify the harmonic tension of each pc s

in Krenek’s classification. Hence, my application of frequency ratios should be qui

simple, for I simply adopt the most common ratios in just intonation.

Figure 3.5: Diagrams for comparable frequency ratios of trichords in CONS group

(a)65

66

3-11 [C-Eb-G] (minor) C-Eb = 5:6 Eb-G = 4:5 10:12:15 → 60

3-12 [C-E-G#] (augment) C-E = 4:5 E-Ab = 4:5 16:20:25 → 3-10 [C-Eb-Gb] (diminish) C-Eb = 5:6 Eb-Gb = 5:6 25:30:36 → 900

64 In Experiments on Tone Perception (Soesterberg: Institute for Perception RVO-TNO, 1966), 2,

Reinier Plomp asserts that in the conflicting problem between the analyses of frequency and periodicit

tone perception.”

y “neither frequency analysis nor periodicity analysis gives a satisfactory explanation of all basic aspects of

uch a

major and minor triads. See also Hermann L. F. von Helmholtz, On the Sensations of Tone as a

66 To avoid complication, the third ratio is eliminated; e.g., the ratio of C-G in pc set 3-11 is

omitted from the table.

65 For pitch-class contents of pc set 3-11, I omit that of a major triad [C-E-G], which yields the ratio of a smaller integer than that of the minor triad, as shown in Figure 3.5. In tonal music, there is, because of the ratios, a hierarchical relation between these two kinds of triads. However, employing straditional theory does not fit Krenek’s notion of tension-degrees which does not differentiate between

Physiological Basis for the Theory of Music, 2d ed. (1885), reprinted by Dover Publications (New York, 1954).

73

(b)

Figure 3.5 shows three frequency ratios in each trichord within the CONS group

The letter notation in square brackets corresponds to the prime form in pc set theory; e.g.,

3-11 [C-Eb-G] is iden

CONS group LCM Degree within CONS Tension-degree among 12 pc sets 3-11 (minor) 60 cons. 1 1 3-12 (augment) 400 cons. 2 2 3-10 (diminish) 900 cons. 3 3

.

tical to pitch-classes [037]. To preserve to some extent the original

concep of

hus, pc set 3-11 [C-Eb-G] is more consonant than the remaining sets; that is, it has the

a

er degree of tension by virtue of the value of its LCM. Notice that although Krenek

t tone h

fluences its tension-degree. Consequently, we can label these three pc sets, respectively,

ith a distinctive numeral degree; tension-degree (among twelve pc sets) 1: 3-11 (major

2: 3-12 (augmented triad), and 3: 3-10 (diminished triad). Even if we

tion of Krenek’s tension-degrees, we focus at this point only on the complexity

the frequency ratio to measure the degree of tension. The simpler the ratio, the more

consonant the trichord.67 To make the ratios easily comparable, I present a number for the

least common multiple (LCM) derived from the three numbers of each frequency ratio

(see the right column in Figure 3.5[a]). In comparing the LCM’s, we posit that the

smaller the number, the more consonant is the trichord. As indicated in Figure 3.5(a), the

frequency ratio and its LCM of pc set 3-11 are the simplest among the CONS group.

T

least degree of tension. In comparing pc sets 3-10 and 3-12, the former represents

high

defines the interval of a tritone as “neutral,” in pc set 3-10 [C-Eb-Gb] the ri eavily

in

w

or minor triad),

67 Edward M. Burns and W. Dixon Ward, “Intervals, Scales, and Tuning,” in The Psychology of

Music, ed. Diana Deutsch (New York: Academic Press, 1982), 255.

74

define e sion o

ay lab 1 (pc d

re 3.6: Diagrams for comparable frequency ratios of trichords in M group

(a) Pc set nam tter no frequency ratios (fr.) fr. of trichord LCM

the degree of t n nly within the CONS group, the order is identical, and one

m el them as cons. set 3-11), cons. 2 (3-12), an cons. 3 (3-10), respectively.

Figu

e/ le tation 3-6 [C-D-E] C-D = 8:9 D-E = 8:9 64:72:81 → 5184 3-7 [C-D-F] C-D = 8:9 D-F = 5:6 40:45:54 → 1080 1440

(b)

M group LCM Degree within M Tension-degree in the whole

3-8 [C-D-F#] C-D = 8:9 D-F# = 4:5 32:36:45 →

3-9 [C-D-G] C-D = 8:9 D-G = 3:4 8:9:12 → 72

3-9 72 1 4 3-7 1080 2 5 3-6 1440 3 6 3-8 5184 4 7

According to the LCMs and the frequency ratios of the trichords in the M group

3-9 [C-D-G] (a whole step and a

perfect fourth) is the “m and 3-8 o a

third) 68 is th harpest. ication of the CONS group, a tritone interval

intensifies the degree of 8 [C-D-F#] ) and an interval of a perfect

fourth produces a less dissonant trichord (pc sets that ive-sem interva in

(as shown in Figure 3.6[a]), it is obvious that pc set

ildest” dissonance [C-D-F#] (a wh le step and major

e s Similar to the classif

tension (i.e., pc set 3-

contain a f itone l

68 The mes are of each pc

se na based on the prime form set.

75

respective p ms: [C-D-G .6(b) s rizes the

bdivided degrees of tension in the M group.

At this point, my categorization differs somewhat from that of Krenek’s. If we

combine these two groups, the value of LCM is less consistent in determining the

individual degrees of tension in comparison with Krenek’s classification. In fact, the

LCM value of pc set 3-9, 72, is considerably smaller than that of two pc sets in the CONS

group: 3-12 = 400 and 3-11 = 900. Indeed, this is an inconsistency between tension-

degrees in Krenek’s classification and in my method that must be solved.

Here the frequency ratio is used only to clarify more precisely Krenek’s concept

of tension-degrees. In other words, the psycho-physical aspect is not the primary

discussion in Krenek’s conception. If so, how should we conceptualize a method that

accommodates both Krenek’s categorization and mine? One may ignore the psycho-

acoustical aspect, since it would create a quite flexible method that enables the analyst to

apply it to a musical work without considering twelve kinds of classified tension-degrees.

This perspective seems to fit Krenek’s fundamental concept of constructing tension-

degrees. In fact, although he discusses a more mathematical approach in music in Music

ere and Now (1939),69 his main argument for creating a musical work concerns

musicality.” On the other hand, one may conclude that twelve clearly categorized

tension-degrees from a psycho-acoustical standpoint, derived from a more mathematical

procedure, are both more important and reliable than the concept of Krenek’s tension-

degrees. This notion seems to correct and expand Krenek’s conception, so that one can

better analyze a musical piece in detail. Indeed, the conflict between the two points of

rime for 3-7 [C-D-F] and 3-9 ]). Figure 3 umma

su

H

“ 70

Krenek, Music Here and Now, 192-217.

70

69

Krenek, Studies in Counterpoint, 21.

76

view is reflected by the inconsistency derived from the value of the LCM calculated f

the frequency ratios.

Both perspectives in the previous paragraph can be persuasive. However, my

purpose for further classifying Krenek’s tension-degrees is to examine not the sound of

the frequency’s vibration, but instead Krenek’s hearing and interpretation of the sonori

If we restrict the a

rom

ty.

ssignment of the degree of tension to pc sets only on the basis of the

alue of the LCM, we no longer need Krenek’s concept of tension-degrees. Thus, I

mploy s

3-1 [C-C#-D] C-C# = 15:16 C#-D = 15:16 225:240:256 → 57600

C-Db-Eb] C-Db = 15:16 Db-Eb = 8:9 15:16:18 → 720

3-3 [C-C#-E] C-C# = 15:16 C#-E = 5:6 75:80:96 → 2400

3-5 [C-C#-F#] C-C# = 15:16 C#-F# = 3:4 45:48:64 → 2880 (b)

group ithin SH T ree in t le

v

e an eclectic method that accepts inconsistency: the value of the LCM determine

the degree of tension only within a single group (CONS, M, and SH, respectively).71

Figure 3.7: Diagrams for comparable frequency ratios of trichords in SH group

(a)

Pc set name/ letter notation frequency ratios (fr.) fr. of trichord LCM

3-2 [

3-4 [C-Db-F] C-Db = 15:16 Db-F = 4:5 15:16:20 → 240

SH LCM Degree w ension-deg he who3-4 1 8 240 3-2 720 2 9 3-3 2400 3 10 3-5 2880 4 11 3-1 57600 5 12

71 Tension-degrees ordered by the value of the LCM are indicated in Appendix II.

77

Finally, let us verify the subdividable tension-degrees in the SH group. In

comparing the frequency ratios of pc set 3-2 [C-Db-Eb] and 3-4 [C-Db-F], 3-2 at first

glance appears less dissonant, since it contains a slightly smaller integer of the trichordal

ratio than that of 3-4 (15:16:18 for 3-2 versus 15:16:20 for 3-4). However, the value of

the LCM of pc set 3-4 (240) is smaller than that of 3-2 (720) (see Figure 3.7[a]). This fact

suggests not only the simplest total ratio for pc set 3-4 among the SH group, but also the

ast dissonant. The table in Figure 3.7(b) summarizes the results of subdivided tension-

egrees within the SH group.

notation

le

d

Figure 3.8: A table for subdivided tension-degrees

Tension-degree Pc set name/letter Interval vector Category Group

1 3-11 [C-Eb-G] [001110] cons. 1 2 3-12 [C-E-G#] [000300] cons. 2 3 3-10 [C-Eb-Gb] [002001] cons. 3

CONS.

4 3-9 [C-D-G] [010020] m. 1 5 3-7 [C-D-F] [011010] m. 2 6 3-6 [C-D-E] [020100] m. 3 7 3-8 [C-D-F#] [010101] m. 4

M.

8 3-4 [C-Db-F] [100110] sh. 1 9 3-2 [C-Db-Eb] [111000] sh. 2 10 3-3 [C-C#-E] [101100] sh. 3 11 3-5 [C-C#-F#] [100011] sh. 4 12 3-1 [C-C#-D] [210000] sh. 5

SH.

The table in Figure 3.8 summarizes the subdivided tension-degrees of pc sets of

e extent

k suggests in regard to the neutralities of the intervals of a

perfect fourth and a tritone. In the first two groups based on Krenek’s categorization of

tension-degrees – which I have denoted as CONS and M, respectively – a pc set that

cardinality three. This result reveals an interesting aspect that coincides to som

with the concept that Krene

78

contains an entry of ic6 (tritone) in the interval vector becomes the highest degree of

nsion within each group; specifically, pc set 3-10 [002001] = cons. 3 and 3-8 [010101]

m. 4. Furthermore, a pc set(s) that contains one or two entries of ic5, indicating the

clusion of a perfect fourth, represents the lesser degrees of tension in these two larger

roups. However, the only pc set that comprises both the entries of ic5 and ic6 in the

ers of SH group, 3-5, is neither the most subdivided

tension-degree nor the least. It is of interest that some points examined by more physical

means can support parts of Krenek’s classification.

3.3 Application of Tension-Degrees to “Nur ein Wort” Although we have classified twelve levels of tension-degrees, according to

Krenek’s concepts, another question remains as to whether these are of use in the analysis

of “Nur ein Wort.” First, in Studies in Counterpoint, Krenek conceives of tension-degrees

as most likely applied to a vertical simultaneity that derives from contrapuntal lines.

Accordingly, all of the trichordal examples for tension-degrees that Krenek uses are

vertical chords. Therefore, one probably should not apply Krenek’s tension-degrees to

linear pc sets in the vocal part or in the melodic lines of the piano. Second and more

importantly, Krenek’s compositional procedure in “Nur ein Wort” is less systematic than

that of former works (e.g., the Sixth String Quartet and Zwölf Variationen) and the later

work, Twelve Short Piano Pieces Written in the Twelve-Tone Technique, op. 83, which

illustrates didactically how the twelve-tone technique can be used; thus, it is a kind of

“practice” piece.

te

=

in

g

interval vector among the memb

79

However, the unique musical feature of Fünf Lieder, which realizes Krenek’s

emerging of the twelve-tone technique, does not necessarily negate the analytic rele

of tension-degrees. In fact, as we have seen, there are a few particularly significant

trichordal harmonies that influen

vance

ce the song’s structure. Furthermore, many of these

ichordal sets are used to portray dramatic moments in the poetic text. Thus, tension-

degrees can rev ow Krenek coordinates the interaction between text

nd music.

igure 3.9: A table for subdivided tension-degrees of selective pc sets associated with “Nur ein Wort”

Tension-degree Pc set name/letter notation Interval vector Category

tr

eal to some extent h

a

F

1 3-11 [C-Eb-G] [001110] cons. 1 4 3-9 [C-D-G] [010020] m. 1 9 3-2 [C-Db-Eb] [111000] sh. 2 8 3-4 [C-Db-F] [100110] sh. 1 10 3-3 [C-C#-E] [101100] sh. 4 12 3-1 [C-C#-D] [210000] sh. 5

We have discussed the relevant pc sets through the analytic segmentation. The

first three rows in Figure 3.9 display the pc sets that frequently occur throughout “Nur ein

Wort.” In addition, the three less frequently occurring pc sets appear in the bottom three

rows of the table: 3-3 and 3-4 which appear only in the second subsection, and 3-1 whic

appears at the climax. Particularly interesting, as shown in the right column in Figure 3.9,

is that these temporarily occurring pc sets all belong to the SH group.

h

80

3.4 Three Primary Tension-Degrees

reveal that pc set 3-11, which occurs as a

nly in

e)

s

may cultivate a milder dissonance, although most of those pc sets

ppear not as simultaneities with the one part, but rather as a small segment of one part

xample 3.2: Distribution of pc sets 3-2 (boxed set) and 3-9 (circled set)

a) mm. 3 – 8, first section

The upper three rows in Figure 3.9

vertical harmony in the codetta and coda, represents the least tension-degree, not o

the twelve degrees, but also in the CONS group, cons. 1. Similarly, pc set 3-9 which

appears mainly at melodically expressive moments is located at the least degree in the M

(mild dissonance) group. Only pc set 3-2, which occurs most frequently in the work as a

motivic harmony, represents the second least tension degree in the SH (sharp dissonanc

group. From a broader perspective, the distribution of subdivided tension-degrees implie

that “Nur ein Wort”

a

that comprises only part of the complete texture.

E

2

9

3-9

2

3-

3-

3-2

3-2

3-2

3-2

3-

81

b) mm. 14 – 15, second section

However, when l

the premise that milder d

Among the three most pr

two exhibit a subtle relat

tension-degrees are 9 and

proximity – occasionally

compare the contextual d

sections. Example 3.2(b)

appear either overlapping

in the first section they n

we listen only to the tens

tension in Example 3.2(b

Krenek’s premise that th

textures and musical con

2

9

3-

ooking closely at the detai

issonance dominates “Nur

imary trichords identified

ionship. Whereas the appe

4, occur separately in the

even overlapping – in the

ensity in terms of pc sets 3

shows how in the song’s s

or in immediate successi

ever overlap or appear in i

ion-degrees of these two p

) than in Example 3.2(a). I

e perception of tension-deg

text.

9

3-2 2

3-

ls

e

ab

ar

fi

se

-2

e

on

m

c

n

r

3-

of distribut

in Wort” m

ove – 3-2,

ance of the

rst section,

cond. Exam

and 3-9 b

cond sectio

, while Exa

mediate suc

sets, we hea

deed, this e

ees necessa

2

3- 3-

ion of pc sets 3-2 and 3-9,

ay appear questionable.

3-9, and 3-11, the first

se two pc sets, whose

they occur in closer

ples 3.2(a) and 3.2(b)

etween parts of both

n these two pc sets

mple 3.2(a) shows how

cession. That is, even if

r far more musical

xample illustrates

rily depends on specific

82

Example 3.3: Intersection of tension-degrees 4 (pc set 3-9) and 9 (pc set 3-2)

in the codetta, m. 9

Tension-degree 9: pc set 3-2 {689}

Tension-degree 9: pc set 3-2 {B12} Tension-degree 4: pc set 3-9 {B16}

ction as an example. Example 3.3 shows that two pc sets with tension-degrees of 9

tersec

f

e

c

In the codetta and coda, we encounter overlapping sets that mingle the tension-

degrees of 4 and 9 (pc sets 3-9 and 3-2, respectively). Since the pitches that form the

structural pc sets in the coda are merely inverted in the codetta, I will use only the latter

se

in t with pitches that form pc set 3-2 which has a tension-degree of 4; the end point

of two horizontal pc set 3-2’s vertically forms pc set 3-9. One can interpret the relative

tension-degree in one of two ways: either as a decrease in the tension-degree because o

the accentuated vertical 3-9 chord, or as an increase of the tension-degree because of the

two horizontal 3-2’s. Since either hearing is plausible, the aural perception of the more

important harmonic trichord is unpredictable. We only can say that in this measure of th

codetta, the “mingled” tension-degrees are drastically softened by the following chord, p

set 3-11, with its tension-degree of 1 (piano’s left hand).

83

3.5 Three Additional Tension-Degrees

Among the three additional tension-degrees in Figure 3.9, the number and

placement of pc set 3-1 with a tension-degree of 12, the highest degree of tension,

musically appropriate; pc set 3-1 occurs much less often than the three primary tension-

degrees and is reserved for emphasis at the climax in the last half of the second section

(mm. 17 – 18). On the other hand, the remaining two degrees – which occur even less

than degree 12 (pc set 3-1) – appear in only one subsection in a somewhat odd place

terms of the text.

Example 3.4: Tension-degrees 8 (pc set 3-4) and 10 (pc set 3-3) in the second subsection

a) mm. 1 – 3

seems

in

Tension-degree 8: pc set 3-4 {126}

Tension-degree 10: pc set 3-3 {B03}

84

b) Transformational tension-degrees 8 (pc set 3-4) and 10 (pc set 3-3)

(mm. 1 – 3) in the first section and

arks the trichordal sets that are either tension-degree 8 or 10 (pc sets 3-4 and 3-3,

spectively). In fact, all of boxed sets in Example 3.4(a) are quite audible since each

sounds independently of any other pitches. Furthermore, as Example 3.4(b) illustrates,

the harmonies of these respective sets are explicitly reproduced by non-transpositional

transformations within the subsection. While only D (pc 2) in the treble clef of Example

3.4(b) moves an octave higher while the remaining two pitches remain fixed (C# and F#:

pcs 1 and 6), all pitches in the bass clef move, resulting in an equivalent trichord that

spans a wider range. Each transformed trichord is represented as a complete vertical

harmony. In addition to the fact that these degrees of tension occur only in the second

subsection, these obvious transformations of pc sets 3-3 and 3-4 emphasize the intense

alienation portrayed in the musical context of the second subsection of the song. This

feature, of course, complements the structural use of pitch-contour discussed above (see

Example 2.1).

Tension-degree 8: pc set 3-4 {126}

Tension-degree 10: pc set 3-3 {B03}

Example 3.4(a) shows the second subsection

m

re

85

The above observations support the view that the second subsection of the first

section cultivates harmonic contrast in relation to the entirety of “Nur ein Wort.”

Nevertheless, this second subsection does not convey contextual isolation or the

impression of a new or inappropriate sonority. In terms of tension-degrees, degree 8 (pc

set 3-4) is one rank less in tension than degree 9 (3-2) which oc

curs most frequently,

hile degree 10 (3-3) is one rank higher in tension than degree 9; moreover, sets with

igure 3.10: Diagram of tension-degrees location in the first three subsections

. 1) 2 (mm. 1 – 3) (mm. 3 – 5)

Tension-degree: 9 9 4 8 10 8 10 9 9 9 9

itch-cl 3-9 3-3 4 3 2 3-2 3-2 3-2 Pitch cl 7}{689}{B12}{689}

R. V. L. V.& R. R.& L. V. L. R. R.

V = the vocal line R = right hand of the piano

L = left hand of the piano

) in

ty

the

o

9 in

w

these tension-degrees are never mingled or overlapped in any of the other subsections.

F

Subsection: 1 (introduction, m 3

P -ass set: 3-2 3-2 3-4 3 -3 3-asses: {689}{B12}{570} {126}{B13} {126} {B13} {45

Part: R. L.

The diagram in Figure 3.10 confirms that the unique tension-degrees (8 and 10

the second subsection are inserted between the first and third subsections whose majori

of trichords represents tension-degrees of 9. At first glance, the diagram suggests

isolation of the second subsection because of the contrast of tension-degrees in relation t

subsections 1 and 3. However, the difference in tension-degrees between the second and

third subsections is quite small – only one degree higher or lower. Since the average of

degrees in each is identical – (8+10+8+10)/4 = 9 in the second and (9+9+9+9)/4 =

86

the third, it may appear that there is almost no difference of musical tension between the

second and third subsections. This, however, is not the case. Even though the degree of

tension helps to analyze the details of this passage, it is merely one of several tools that

can be used for analyzing the music. In other words, tension-degrees may not be the most

important means for determining the expressive affect of the music; rather, it is a

supportive tool to be used in conjunction with other analytical approaches. If one

examines the succession of pitch-class sets shown in Figure 3.10, as well as how they are

constructed from the piano’s right-hand, left-hand, and the vocal line, it becomes clear

that the musical context in the second subsection differs significantly from that of the firs

and third subsections. Most importantly, the joining of lines to create s

t

tructural sets

ons

ctively)

le

tch-distribution, and the

cceed in focusing the listener’s attention on the entrance of the vocal line.

es appear in the second subsection, the

whole clearly is structured by a number of musical features.

occurs only with the last occurrences of pc sets 3-3 and 3-4, forming complete vertical

trichords which create the strongly contrasting harmonic effect. Both of the last

occurrences of pc sets 3-3 and 3-4 comprise pitches from two of the three parts, i.e., both

hands of the piano (3-4) and the vocal line and the piano’s right-hand (3-3). Moreover,

these two vertical chords project the loudest dynamics (mp) in the first three subsecti

(see excerpt of the first section in Example 1.7).

In conclusion, the tension-degrees of 8 and 10 (pc sets 3-4 and 3-3, respe

in the second subsection to some extent do contribute to making this moment

expressively unique, but the musical effect is created in conjunction with far more subt

means. Specifically, the idiosyncratic trichordal harmonies, the pi

dynamics all su

Therefore, although distinctive tension-degre

87

3.6 Expansion and Contraction of Tension-degree

Althou the musical uniqueness of the second subsection

which consists of several distinctive features, we have not sufficiently discussed

armonic combination. Let us look again at Example 3.4(b). As we discussed above,

ly

occurs .2 and -

c

ombination of tension emphasizes the harmonic discontinuity – e.g., the second

that

gh we have examined

h

while the combination of pc sets 3-2 and 3-9, with tension-degrees of 9 and 4, frequent

(see Examples 3 3.3), that of 3 3 and 3-4, with tension-degrees of 10 and 8,

occurs only in the second subsection.72 The abrupt occurrence of such a unique harmoni

c

subsection’s harmonic contrast with the first and third subsections (see Figure 3.10) –

should strongly attract the listener’s attention. Thus, in order to create harmonic

discontinuity, we must locate a unique pc set, with its associated tension-degree, or a

unique combination of pc sets with their associated tension-degrees.

Example 3.5: Harmonic contrast at the climax, mm. 17 – 18

Tension-degree 12: pc set 3-1 {567}

Tension-degree 4: pc set 3-9 {B16} 72 The reader should notice that use of my application of tension-degrees to a piece differ from

both the Krenek’s use of tension-degrees and of Hindemith’s harmonic tension. While my application of tension-degrees is available to horizontal and vertical harmonies, their methods refer only to vertical harmony at a exact moment. Thus, simultaneity of two or more tension-degrees or harmonic tensions cannot be seen in their applications.

88

As we have analyzed in Chapter 1, pc set 3-1 in the vocal line at the song’s clim

is one of the less often occurrences in the entire piece. Furthermore, the climax relies on

the highest tension-degree of 12, pc set 3-1. Hence, one may easily perceive harmon

disjunction at the climax in the thirteenth subsection, when compared to the harmonies in

subsections immediately before and after. My emphasis here is on the harmonic

combination at the climactic point, rather than on the harmonic discontinuity, since whil

pc set 3-1 (with the highest tension-degree) is sung in the vocal line, pc set 3-9 with a f

lower tension-degree of

ax

ic

e

ar

4 is posited below (see Example 3.5).

igure 3.11: Diagram for expansion and contraction of tension-degrees between the second subsection and thirteenth subsection

ubsection (SS): 2 (mm. 1-3) 13 (mm. 17-18)

ocal line Tension-degree: 8 12

Piano ension-degree: 10 4

Pitch-class set: 3-3 3-9 Difference in combination of 2 8 tension-degrees

Although there is no direct connection between the second subsection and the

climactic subsection, we can compare these subsections in terms of the difference in

tension derived from each combination of tension-degree. The diagram in Figure 3.11

represents the tension-degrees and pc sets appearing in the second and thirteenth

subsections. Each occurrence of tension-degree and pc set is presented in two layers by

F

S V

Pitch-class set: 3-4 3-1

T

89

dividing the vocal line and the piano. A degree of difference in a combination of tension-

degrees can be indicated by a number by subtracting from the higher tension-degree to

the lower tension-degree in each subsection: (tension-degree of 10 in the piano) –

(tension-degree of 8 in the vocal line) = 2 in the second subsection and (tension-degree of

12 in the vocal line) – (tension-degree of 4 in the piano) = 8 in the thirteenth subsection.

In the diagram of Figure 3.11, we find a cont sting otion of tension-degrees between

the two subsections – expansion and contraction. While the vocal line’s tension-degree in

the second subsection moves to a higher tension-degree in the thirteenth subsection, the

motion of the piano’s tension-degrees between the two sections is contrary to that of the

vocal line. Although the separated expansion and contraction of tension-degrees should

not imply a direct connection between the two subsections, their contrasting motion of

tension-degrees that contain the most idiosyncratic harmonic combination suggests that

harmonic discontinuities intensify specific points in “Nur ein Wort.”

The difference in combination of tension-degrees in the thirteenth subsection is

far greater than that of the second subsection (see Figure 3.11). In other words, the degree

of difference in combination of tension-degrees between the two subsections projects an

expansion. This creates a unique harmonic texture at the climactic point. Furthermore, the

musical context at the climax enhances the sonority: while the texture in the second

subsection is somewhat pointillistic, suggesting a pseudo-call-and-response between the

vocal line and the piano, that of the climax is a simultaneity. When hearing the

simultaneity of extreme tension in the vocal line and a calm flowing with expressiveness

ra m

73

73 One may hear a Webernesque sound especially at the first occurrence of pc set 3-3 and 3-4 (i.e.,

the first passage of “Nur ein Wort” and the following trichord in the piano’s left hand [mm. 1 – 2]).

90

at the climactic point, one may recognize the sound derived from two rather polarized

lines which produce markedly distinct kinds of musical tension.

3.7 Tension-Degree and Musical Context

Example 3.6: An example for comparing the same tension-degrees in Studies in Counterpoint (1940)

Figure 3.12: The numeric value of constituent intervals in each chord in Example 3.6

3-1 [B-C-Db] B-C = 15:16, C-Db = 15:16 → 225:240:256 LCM = 57600

C-B = 8:15, B-Db = 8:9 → 64:120:135 8640

C-Db = 15:32, Db-B = 5:9 → 75:160:288 7200

Despite his exhaustive classification of the tension-degrees, Krenek does not

consistently exploit this property in his compositions. In his Studies in Counterpoint

940), Krenek points out that the wider the intervals are in a trichord, the greater the

egree of tension when compared to an equivalent trichord that spans a smaller register.

or the brief example shown in Example 3.6, Krenek explains that “under … equal

onditions of dynamics, instrumentation, etc., the last of these three chords will most

robably appear as the sharpest.”74 This is Krenek’s generalization of tension-degree,

ierarchy within a tension-degree (a pc set) that is derived from different intervals by

(1

d

F

c

p

based more on aural perception than on any mathematical measurement. Thus, a

h

74 Krenek, Studies in Counterpoint, 20.

91

various pitch dispositions does not match the order of numerical value calculated by

frequency ratios (compare Example 3.6 and Figure 3.12). Consequently, it appears that

Krenek does not consider register as a crucially important factor that should be included

under “equal conditions.” In the case of the first chord in Example 3.6, for instance, the

ay be heightened or softened by the specific register in which the

itches are presented. This is one of the least consistent points in Krenek’s concept of

nsion-degrees. At the same time, this concept (which seems insufficient or incomplete)

s

degree of tension m

p

te

suggests that it is one available measurement that may be applied depending on the

musical context. Since Krenek maintains that “the student must be guided in this field by

his musicality, taste, and imagination,”75 his notion of tension-degrees does not

systematize a compositional or theoretical technique. Rather, Krenek asserts that one’

“musicality” should be influenced, but not necessarily dictated by any one theoretical

premise or system.76

Ibid., 21.

In fact, Krenek clearly distinguishes the terms “system” and “technique,” regarding the former as the material of tonal music and the latter as of twelve-tone music (Music Here and Now, 125): “For the term ‘system’ applies exclusively to a primitive and fundamental division of material; and in so far as tonal music was never limited to the seven tones of the diatonic scale, tonality always used the ‘twelve-tone

system’ has nothing to do with a ‘system’ in the sense of an original premusical organization of mate

75

76 Krenek seems to dislike the term “system,” especially in the context of twelve-tone composition.

system.’ The technique which some modern composers use within the enclosure of the ‘twelve-tone

rial. And, for that reason, it should not be called the ‘twelve-tone system,’ but rather the ‘twelve-tone technique’ (Italics by Krenek).”

92

Example 3.7: Comparable tension-degrees in different registral contexts presented by 77

Krenek in Studies in Counterpoint (1940)

tension-degree: 12 6

of

ainst

the

d

at is,

y in the

rsus

pc set/ pcs: 3-1 {B01} 3-6 {024}

According to Krenek’s view, musical context may alter a chord’s degree

tension. Example 3.6, which Krenek uses in his text to illustrate this point, argues ag

the concept that tension-degree can depend on musical context. Krenek maintains that

chord at the right in Example 3.7 presents a “sharper” degree of tension than that on the

left, even though according to his classification it should be “milder.” The annotate

subdivided tension-degrees at the bottom of Example 3.7 show the contradiction. Th

although tension degree 6 (pc set 3-6) on the right of Example 3.7 is located nearl

middle of the twelve tension-degrees, the listener will likely perceive it as a higher

tension-degree than the “sharpest” tension-degree of 12 (pc set 3-1) on the left. This is

because of the specific musical context – specifically, the expanded register, the louder

dynamics, the shaper articulation, and the brighter instrumentation. Thus, representing

two chords with different tension-degrees, as shown in Example 3.7, Krenek asks the

reader to imagine the aural effects of the musical context: pp versus ff, con sordino ve

77 Krenek, Studies in Counterpoint, 21.

93

accent, string versus wood wind and brass instruments, and registral and intervallic

differences.

ics,

cterizes

f

nsion-degree of 9, for instance, comprehensively affect the overall piece. To illustrate

is point, I shall select some contextual features among these conditions and apply them

nly to pc set 3-2 (tension-degree 9).

Krenek’s belief that tension-degrees explain relative harmonic dissonance is

important if we analyze a musical work using pitch-class set theory. In fact, his view that

musical context works in conjunction with tension-degrees to affect dissonance (see

Example 3.7) warns us against simply applying one analytic method to define a work’s

aesthetic effect; rather, various analytic approaches mat be needed to fully understand the

work.

We have found that pc set 3-2 (tension-degree 9) appears most frequently in “Nur

ein Wort” and that it contributes to the organization of the work. However, each specific

musical context of pc set 3-2 always differs, e.g., “the position of the intervals, dynam

instrumentation,”78 register, articulation, and even note value and rhythm. Under such

circumstances, how can one derive a unified description which accurately chara

the work? In fact, it may create unresolvable confusion if one re-examines how pc sets o

te

th

o

78 Ibid., 20.

94

Example 3.8: The configurations of pc set 3-2 (tension-degree 9) in the first section,

mm. 1 – 9

Measure: 1 3 4 5 6 7 8 9

E

song’s f

occurren

the pian

latter se

(j and k)

pc set 3-

section

dynamic

measure

assumin

distribut

differen

importa

through

a b

xample 3.8 illus

irst section – mo

ces are primary

o, the former unf

ts forth many wh

. In other words,

2 (degree 9) in th

(d in mm. 3 – 4),

s; the very small

7. Moreover, de

g that it would pr

ion of tension-de

ce, of course, is r

ntly, far more occ

out the section. A

c

trates

tivic u

harmo

olds s

ose in

by K

e pia

remar

range

spite

obab

grees

eleva

urren

lthou

d

the mo

nit, sma

nies. In

mooth d

tervals

renek’s

no in ge

kably i

articul

the abse

ly be pe

in both

nt to the

ces of t

gh there

e f g h i j

st conspicuous occurrences of pc set 3-2 in

ll passage, and portion of melodic line; all

comparing the occurrences in the vocal lin

escending lines (annotated c and g), while

are more widely spaced, except for the last

definition, the degree of tension derived on

neral is higher than that of the vocal line. T

t seems not increase the tension because of

ated pp sustains a sense of constant tranqui

nce of dynamic marking in the vocal line a

rformed a bit louder than the piano, the unb

parts compensate one another. Therefore, t

difference in instrumentation. But more

he set in the piano preserve moderate tensio

is a set of the highest degree of tension in

k

the

of these

e a

the

two sets

ly

nd in

from

his

the

lity until

nd

alanced

he

n

this

95

motivic units (a, b, e, f, and h) in Example 3.8, whose dynamics are almost always

marked pp, disturb neither the smoothness of the vocal line nor the overall tranquil effect

of the first section.

xample 3.9: The configuration of pc set 3-2 (tension-degree 9) configuration in the second section, mm. 12 – 20

easure:12 14 15 17 19 20

E

M

In contrast to the somewhat “milder” tension-degree in the first sectio

the texture’s small dynamic range, the tension-degree in the second section sh

configurations with wider dynamic ranges, a feature that should create

tension-degree in the second section than in the first. First, the configuration o

motivic sets l, m, n, and o is reversed from that presented in the first section. A

these four motivic sets, the last two n and o in m. 14 occur with a denser textu

of the first two appearances in m. 12 (see Examples 1.7 and 3.9), even though

dynamics are softer than the sets l and m. More importantly, the simultaneity

trichoardal set that occurs twice (q and r in Example 3.9) strongly contributes

intensifying the degree of tension. Above all, of the two vertical trichordal set

l m n op q

r t

s

varied

u

n

o

a

f

m

re

t

of

to

s,

v

created by

ws more

“sharper”

the

ong

than that

he

the

q no

96

doubt projects the highest tension-degree by Krenek’s definition: the widest intervals

within pc sets 3-2 that appear in the overall section; moreover, it is also attached

loudest dyna

to the

mics, mf, in the section. With the following vertical trichordal harmony (r),

ese simultaneities function to prepare the climax that arrives in m. 17. From a broader

perspective, the tension-degree of 9 in the context of specific intervallic textures and

dynamics, sp ical m to ow creates the climax

of the piece. This feature also illustrates how the musical conte ontra

ns

co sion rene onc f tension-degrees can accommodate to some

extent all twelve trichords by creating a broader classification of twelve tension-degrees.

y exploring further classifications of tension-degree based on either Krenek’s or my

ethod

ons

sion-

y

79

80

th

ecif ly in m. 14 17, sh s an intensification that

xt c sts between the

two sectio .

In nclu , K k’s c ept o

B

m , one can accommodate specific contexts by defining more than twelve degrees.

Register is also an issue which deserves consideration. Although Krenek hardly menti

it in Studies in Counterpoint, its incorporation into the tension-degree suggests the

possibility of significantly more subdivisions. Nonetheless, even positing twelve ten

degrees introduces the problem of how to calculate tension-degrees using the frequenc

ratios of a trichord. In any case, it is best to use the concept of tension-degrees as a

secondary or supportive analytic tool, since musical tension can always vary depending

on musical contexts that differ in their “condition of intervals, dynamics, [. . .]

instrumentation” and register, and the listener’s “musicality, taste, and imagination.”

79 See Appendix II.

80 Krenek, Studies in Counterpoint, 20.

97

CHAPTER 4: Pitch-Class Set Genera

4.1 The Application of Forte’s Pitch-Class Set Genera

The theory of pitch-class set genera is useful to further verify the characteristics

of prominent harmonies. The theory explores more concrete musical features of a piece

than elementary pitch-class set theory. Each of the two standard theories for pitch-class

set genera, Allen Forte (1988) and Richard Parks (1988), have advantages and

disadvantages depending on the specific musical style. Although I do not discuss the

etails of each theory, it is necessary to explain their major differences in relation to the

nalysis of “Nur ein Wort.”

Briefly speaking, Fort f pi gns structural

harmonies to one (or more) of the twelve

“progenitor(s),” whereas Parks’ theory requires no such assignment; in other words,

every pc set can itself be a pri un c ral” set.81 For the

analysis of “Nur ein Wort,” I shall mainly apply Forte’s theory of genera, since the

majorit

e to

82

d

a

e’s theory o tch-class set genera assi

trichordal pc sets, which is called the

mary it, which he calls a “ ynosu

y of structural pc sets are trichords (i.e., progenitors), and each of these is labeled

according to Forte’s terminology. However, I will partly adopt Parks’ method in order to

make the strongly connected pc sets in each subsection more comprehensible. Based on

the segmentation of respective subsections, the analytical procedure requires on

derive an “inclusion diagram” which involves identifying for each subsection the subset

81 Richard S. Parks, “Pitch-Class Set Genera: My Theory, Forte’s Theory,” Music Analysis, 17/ii

(July 1998): 206-26. Parks explicitly addresses the advantages and disadvantages of both theories. Richard C. Pye argues for the advantages of Parks’ theory in a recent article, “The Construction and Interpretation for Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, Music Theory Spectrum 25/2 (Fall 2003): 243-274.

82 Parks (1988).

98

or superset relation from the pc set complex, to assign each pc set to the table of genera

classified by Forte, and finally, to define the most influential genus (or genera) by

alculating the “status quotient.”83

set

enera to a few subsections of “Nur ein Wort:” the introduction (first subsection), the

second subsection, the codetta and coda (sixth and the last subsections), and the climax

(thirteenth subse tion). I t er e ov rall s cc sion of genera and explore its

relation to the no able stru l f tu s th hav be n identif egmentations,

K- & Kh-relatio charts, i io di am and the tables of genera and the status

quotient indices, which are not presented here, appear in the Appendix III.

4.1.1 Introducti

The s s f t int duc on

c

In this paper, I only give the results of applying the theory of pitch-class

g

c hen d iv the e u es

t ctura ea re at e e ied. Other s

n nclus n a rg s,

on

Example 4.1:

egmentation o he ro ti , m. 1

^4-23 {57A0}

83 The calculation of the “status quotient (Squo)” is presented in Allen Forte, “Pitch-Class Set

Genera and the Origin of Modern Harmonic Species,” Journal of Music Theory 32/2 (Autumn 1988): 232.

99

The piano’s short introduction presents the prime series and the first two no

its repetition. Example 4.1 shows a pitch reduction of the introduction with the relevant

sets identified. Some sets represent the primary partitionings of the row, for example

29/7-29 (see Figure 1.3), while others do not. Also, complement pairs of these segmented

sets frequently occur as harmonic segments.

tes of

5-

Figure 4.1: Forte’s K- & Kh-relation: introduction of “Nur ein Wort”

3-2

3-9

4-13

4-14

4-22

4-23

7-10 5-10

7-11 5-11

7-29 5-29

84 Note that several pc sets that we have

already discussed (e.g., 3-2, 3-9, and 5-29) are included, suggesting that the introduction

plays an important formal role. We shall see that applying genera theory defines more

precisely the strength of pc set-relations among the song’s prominent structural

harmonies.

4-13 Kh K Kh K Kh 4-14 Kh Kh K Kh Kh 4-22 K Kh K Kh K 4-23 K Kh K Kh

7-10/5-10 Kh K Kh K K 7-11/5-11 Kh Kh K Kh Kh K 7-29/5-29 Kh Kh Kh Kh K Kh

7-29/5-29: #Kh = 5, #K = 1 7-11/5-11: #Kh = 4, #K = 2

Forte’s K- & Kh-relation which is somewhat abstract,

within a single gure 4.1 shows the K- & Kh-

relations for the song’s introduction: all of the relevant pc sets are delineated in the

I first make use of Allen

but does prove useful in summarizing relations among all of the structural harmonies

subsection.85 The table displayed in Fi

84 A pair of pc sets 5-29/7-29 occurs twice as a linear partitioning of the row: 8621B 30754A9 and

0754A 98621B (see Figure 1.3).

85 Forte, The Structure of Atonal Music, 93-177.

100

segmen

K-

h-

oes

t, but

-

nt

d to the right

nd left hands in the piano, respectively. That is, even if we choose to ignore the nexus

/7-10.87

the introduction, m. 1

7-10 7-29 7-11

4-13 4-14 4-22 4-23

3-9

tation shown in Example 4.1. The basic concept of the K- & Kh-relation is as

follows: if a pc set is included in another set “or” its complement it is indicated as a

relation; if a pc set is included in another set “and” its complement it is indicated as a K

relation. Thus, inevitably, the Kh-relation more strongly associates two pc sets than d

the K-relation. For instance, pc set 4-23 [0257] does not include 3-2 [013] as a subse

it is included in its complement pc set 8-23 [0123578A]; accordingly, pc sets 4-23 and 3

2 are related by the K-relation. On the other hand, both pc set 4-23 and its compleme

include 3-9 [027] as a subset; thus, 4-23 and 3-9 are related by the Kh-relation.86

The “nexus” sets for this initial subsection is 5-29/7-29, which reveals that

Krenek projects the most connected sets as vertical harmonies rather than as individual

contrapuntal lines. See, for example, that pc sets 5-10 and 7-10 are assigne

a

sets, pc sets 5-11/7-11 have more K- & Kh-relations than do pc sets 5-10

Figure 4.2: The inclusion diagram of

5-10 5-29 5-11

3-2

86 The detailed definition of the method is explained in “The set complex K” and “The

bcomplex Kh” in Forte (1973), 93-100.

87 Pc sets 5-11/7-11 occur twice as a linear segments of row, 621B3

su

0754A98 and 30754 A98621B.

101

The inclusion diagram in Figure 4.2 is based on K- and Kh-relations among all of

the candidates indicated in the table of Figure 4.1. This kind of diagram is more useful

visualizing the specific relations of the set complex than the table used in Figure 4.1. And

it also has the advantage of eliminating the weaker pc set-connection, which is

determined by a pitch reduction as shown in Example 4.1. Thus, we can immediately

verify which particular pc sets dominate each subsection as an important harmony(ies).

Hence, I will omit the K- & Kh- complex table and show only the inclusion diagram in

later examples.

for

igure 4.3: The table of genera and the status quotient (Squo) indices88 for the

introduction, m. 189

Set Name: G7 G10 G11 3-2 √

F

3-9 √ 4-13 √

4-23 √ √ 5-29 √ √ Count 9 4 3 G7 G G Total . sets: 45 4 Status uotient ( nd

G7 0.099 r a-dia) G o -tonal) G

4-14 √ 4-22 √

5-10 5-11

√

√√

s: 5 11

10

no

1

29

Q Squo) I ices (ch om

10 0.08111

(at nal 0.192 a

(di )

88 The computing form a by Richar Park Western Ontario and

the State University of New Yo B

89 The tables indicated in this paper are all reduced matrices based on the five rules that Forte defines; see Forte (1988): 234-235.

at w s provided d s, the University ofrk at uffalo.

102

The status quotient (Squo) indices of Forte’s genera explicitly demonstrate the

armon

n

is

h ic species of the introduction.90 Genus 11 (dia) dominates this subsection. Pitch-

class sets 5-10/7-10 do not belong to the genus.91 Also, according to the Squo, Genus 11

whose progenitor is pc set 3-9 is more influential in the introduction than Genus 7

(chroma-dia) whose progenitor is 3-2, even though pc set 3-2 appears twice in successio

as a motivic statement. At this point, we notice a somewhat concealed feature that a

tetrachord formed by two sustained dyads in the respective motivic trichords (pc set 3-2

{689} and {B12}) contains two different occurrences of 3-9, {B16} and {681}. These in

turn form 4-23 {68B1}, as shown in the segmentation of Example 4.1. Moreover, there

one more occurrence of 4-23, {57A0}, which already includes the directly presented 3-9

{570} (see Example 4.1): the gap of Squo values between Genera 7 and 11 (0.099 and

0.192, respectively) verifies that pc set 3-9 (or 4-23) may be potentially the most

prominent harmony in the introduction (see Figure 4.3).

90 “The Status Quotient index is a further step toward refining and interpreting the matrix structure

r the analysis of actual compositions….” Ibid., 232.

a “type” to each genus based on that of the twelve “progenitors”: genus (G)1 = , G3 = diminished, G4 = augmented, G5 = chromatic (chroma), G6 = semi

hromatic (semichroma), G7 = chromatic-diatonic (chroma-dia), G8 = atonal, G9 = atonal-tonal, G10 = atonal-tonal, G11 = diatonic (dia), and G12 = diatonic-tonal (dia-tonal). Ibid., 200-202.

fo

91 Forte assignstonal, G2 = whole-tonea

c

103

4.1.2 The Second Subsection

2: The segmentations of the second sub-section of the first section, “Nur ein

Wort, nur eine Bitte,” mm. 1 – 3

Example 4.

The second subsection employs a strikingly unique harmonization, as discussed in

the previous chapter. The application to pitch set genera reveals a similar harmonic

uniqueness in a more concrete way. The pitch reduction in Example 4.2 shows a likely

segmentation of relevant harmonic subsets. The texture suggests a call and response

between the voice and piano, which sounds rather prosaic. Here, the texture largely

obscures the ordering of the series. However, the segmentation reveals two significant

complement-related sets: 5-9/7-9 and 5-26/7-26. While the former derives from the

connection between the second fragmental phrase in the voice (“nur eine Bitte”) and the

piano, the latter comes from the vocal line coupled to the first half of the piano. But the

more interesting point is how the “progenitors,” specifically highlighted in the discussion

of the tension-degrees, influence the musical features of this subsection.

104

Figure 4.4: The inclusion diagram of the second subsection, mm. 1 – 3

7-9 7-26

5 5-26

At first glance, the inclusion diagram of Figure 4.4 seems to indicate pc set 6-Z36

as the nexus set. In fact, the diagram shows 6-Z36 as a central linkage or intersection

between the complementary sets of 5-9/7-9 and 5-26/7-26. However, pc sets 5-9/7-9 have

an equal number of Kh-relations as does 6-Z36, while the complementary sets 5-26/7-26

have only one. When we compare pc sets 5-9/7-9 and 6-Z36, the former seems the

more appropriate candidate for the nexus set, since the dialogue between the voice’s 5-9

and piano’s 7-9 is musically more perceptible than 6-Z36 which spans only the first half

of the sub-section. Curiously, despite the fact that pc set 5-9/7-9 is the nexus set, 5-26 has

direct inclusion relations to both “progenitors” (3-3 and 3-4) (see Figure 4.4).

6-Z36

-1 5-9

4-2

3-3 3-4

92

See Appendix III. 92

105

Figure 4.5: The table of genera and the Squo indices for the second subsection, mm. 1 – 3

Se e: G6 G8 3- √

t Nam G5 3 √

3- √ 4- 5- 5- 5- √ 6- 6 √ Co 4 G

tal s 4

uotie t ( o) ce

4 2 √ √ 1 √ √ 9 √ √ 26 √ Z3 √ √

unts: 7 4 6

G5 G6 8 To no. set : 29 45 1

Status Q n Squ Indi s G5 0.197 (chr ma o ) G6 0. 019 (semichroma)

. 9 (ato al) ion in

s

at the

n G5, because a progenitor pc set 3-3 and a significant complement-

related

he three

G8 0 13 n

The genera indications in the table and Squo indices for the second subsect

Figure 4.5 show a somewhat complicated structure. First, in terms of Forte’s genera, the

harmonic identity of this sub-section differs from that of the introduction; genera specie

of the three greatest Squo values are G5 (chroma), G6 (semichroma), and G8 (atonal),

while those in the introduction are G7, G10, and G11 (compare with Figure 4.3). Second,

the indices indicate that the greatest Squo is Genus 5 (chroma), despite the fact th

largest number of the count is Genus 6. However, G6 seems to be more appropriate as the

primary genus tha

set 5-26/7-26 (the second half of the voice and piano’s right hand) belong not to

G5, but to G6. Finally, the most unexpected result from the table in Figure 4.5 is that it

suggests a possible contradiction between the marked entries in each column (genus) and

the order of the size of Squo value. Notice that Genus 8 (atonal), the only one genus that

contains both progenitors (pc set 3-3 and 3-4), has the least Squo value among t

106

genera indicated in the table. It seems that this troubling contradiction can be resolved by

considering the harmonic characteristics of this subsection from a less-detailed or larger

perspective. In fact, G6 is the most comprehensive and well balanced genus, which is not

only located in the middle of the Squo value, but also lacks only one pc set’s participation

(3-4). That is, not pc sets of the progenitors, but rather the harmony itself predominates

the musical texture in the second subsection. In any case, the first two sub-sections move

from G11 (dia) to G6 (semichroma).

4.1.3 The Codetta

in

Example 4.3: The segmentations of the codetta (m. 9) and coda (mm. 20 – 21)

) Codetta. a

b) Coda.

107

After the ascending and descending passages in the piano (m. 8), the first sec

concludes with the codetta. The pitch reduction in Example 4.3(a) displays how the

texture of this sub-section is denser than those preceding it. This is because it projects a

complete form of the row without any additional pitches. Therefore, we find many sets

derived from the basic or primary partitions shown in Figure 1.3: pc sets 3-2, 3-11, 4-1

5-29, and 7-29. The two pitch reductions indicated in Example 4.3 demonstrate that th

structural pc sets in the coda (below)

tion

3,

e

are exactly identical to those in the codetta (above).

These s n.93

3-2 3-5 3-9 3-11

ections, then, are reciprocally associated by a simple inverted configuratio

Thus, one reason why this subsection sounds like a codetta is because it uses the same

sets and repeats the relations that structure the song’s coda (the final two measures of the

piece). For convenience, we will focus here only on the codetta, but one should

remember that its harmonic structure is strongly connected to that of the coda.

Figure 4.6: The inclusion diagram for the codetta, m. 9

7-29 7-Z36

6-Z47 5-Z36

4-13 4-14 4-22 4-27

93 Pitch-class set 5-32 in the coda is the only unique harmony between the two subsections,

although it appears shortly after the codetta (m. 10). It may be that the pentachordal harmony (5-32 {A0347}) belongs to the codetta rather than to the beginning of the second section. Or perhaps these sections are meant to overlap. At any rate, it is obvious that the chord in the coda (pc set 5-32) functions as a closure for “Nur ein Wort.”

108

The inclusion diagram shown in Figure 4.6 verifies how highly connected the

are in this su

sets

b-section which functions as a short “codetta” to the first section of the piece.

also demonstrates that the harmonic relationship among the pc sets in this subsection is

am

exp t vertical and horizontal segmentations;

the former in bold lines, while he diagram shows that all

ichordal pc sets are potentially associated with 5-29, which is the complement of 7-29,

even though 5-29 does not appear in the code ore, it demonstrates a rather

complicated, interwoven connection among three “progenitors” (which we have

mary sets in the previous chapter) and other pc sets (especially

trachordal sets). Yet, in seeking to identify the primary trichordal harmonies, status

quotien the inclusion diagram in Figure 4.6.

Figure 4.7: The table of genera and the Squo indices for the codetta, m. 9

Set Name: G1 G7 G10 G11 G12 √

It

structured by the distribution of one twelve-tone series. Furthermore, the diagr

lici ly depicts the pc sets’ concatenation in both

the latter in dotted lines. T

tr

tta. Furtherm

identified as pri

te

t (Squo) indices will prove more useful than

3-2 3-5 √ 3-9 √ 3-11 √ √ 4-14 √ 4-22 √ √ 4-27 √ (5-29) √ √ √ √ √ [7-29]

G10 0.089

4-13 √ √

5-Z36 √ √ √ √ 6-Z47 √ √ √ √ √ Counts: 11 5 5 4 5 6 G1 G7 G10 G11 G12 Total no. sets: 63 45 41 29 45 Status Quotient (Squo) Indices

G1 0.072 G7 0.101

G11 0.157 (dia) G12 0.121 (dia-tonal)

109

Let us first look at the half upper of the table displayed in Figure 4.7. The

complement pc set of 5-29, pc set 7-29, includes all of the genera entries presented in the

table. This implies that pc set 5-29 has multiple dimensions of harmonic species in th

context of the entries of trichordal sets. Specifically, whereas respective trichordal se

belong to quite different genus species from on another, the “ghost” pc set 5-29 contains

all of them. The Squo value evinces a hierarchical genera order with prominent

progenitors in the codetta: G11 > G12 > G7 > G10 > G1; that is, pc set 3-9 > 3-11 > 3-2

e

ts

d

cal chordal

t 3-

n

e the

est Squo of 0.157, is the same as that of the

troduction. This maintains a harmonic analogue between the introduction, the codetta,

nic

ature in “Nur ein Wort.”

(> 3-11) > 3-5.

Among the first three pc sets, which were analyzed above, two of the first an

second highest value of Squo indices preserve a subtle relation in the actual music. In

both pitch reductions of the codetta and coda shown in Example 4.3, the verti

progression (pc set 3-9 to 3-11) appears in the same hierarchical order of Squo indices,

and evokes a conventional tonal harmonic concatenation: a triad containing an

appoggiatura to a tonic triad. However, from a “metrical” perspective, the tonal allusion

is contradicted by the result from genera Squo indices: the metrical value of both pc se

9 harmonies in the codetta and coda is less than that of following 3-11’s, respectively. I

other words, in terms of the perception of pc set genera, pc set 3-11 (triad) cannot b

tonic chord in this song, even though it has a longer note value than the preceding 3-9

(“appoggiatura” trichord) and is located at the end of each closing subsection. Finally,

Genus 11 (dia), which has the high

in

and the coda. These three subsections, then, seem to project the most unified harmo

fe

110

4.1.4 The Climax

Example 4.4: The segmentation of the thirteenth sub-section, “kein Gedanke,” mm. 17 – 18

t genera helps to identify whether the climactic

ratic harmonic profile. As the pitch reduction in

determine at first glance which “progenitor” is the

actic subsection, since the segmentation identifies

of the possible sets of cardinality three. From this

ichordal pc set dominates the climax’s harmonic

le 4.4 reveals that the segment of the voice is

no. Although the collection of segmentations in the

piano up until the voice enters (the left part from a dotted discretio

lly, l g e

ubsectio as a niq e o dio nc

xample 4.4 shows, it is diffic to

ost inf l cho d i the clim at

ast six ord pc ets one alf

f vi , de rm in h tr

ems crucially impo nt.

n Ex mp

harm ic ep rate fro th pia

n line in Example 4.4)

Fina ana ysis usin pc s

s n h u u r i sy

E ult

m luentia tri r n

le trich al s , h

point o ew te in g w ich

structure se rta

The segm nta oe ti in a

on ally s a d m e

111

looks li

y

usly

ludes

s

ther it

s of the thirteenth subsection, mm. 17 – 18

a) The inclusion of the first half segmentation

5-24

ke a condensed harmonic gesture, harmonies in the last half in the climactic

subsection (the right part from a dotted discretion line in Example 4.4) are more worth

of focus for several reasons. First, pc sets 3-1 and 3-9 have contrasting characteristic

sounds – chromatic and diatonic. In the actual music, these pc sets appear simultaneo

and each phrase is played as a contrapuntal figure, rather than as a component of a

vertical harmony. In this context, we need to determine which trichordal pc set is more

effective or influential as defined by pc set genera. Second, pc set 7-19 which inc

not only pc set 3-5 presented as a vertical trichord, but also a potential 3-2, which i

included in 5-29, is somewhat problematic.94 My interest is whether pc set 7-19

compromises the harmonic features of the respective possible “progenitors,” or whe

is influenced by a specific “progenitor,” i.e., the core harmony in the last half of this

subsection.

Figure 4.8: The inclusion diagram

5-3

4-2 4-Z29

3-1 3-2 3-3 3-5 3-6 3-9

94 Pitch-class set 5-29 {68B12} shown in Example 4.4 includes 3-2 {B12}, but I do not make a segment for that 3-2 for musical contextual perception. While it is possible to regard a single D (pc 2) in the bass clef and a Db (pc 1) that is located just above the D as a connective dyad, the D does not directly connect to Cb (pc B) contained in 3-9.

112

b)

rmal

at

could be considered most influential harmonically.

In contrast, the inclusion relations in the last half of this subsection are

nequivocal in showing that pc set 7-19 is the most strongly associated (see Figure

.8[b]). Thus, we can more easily predict the nexus set here than in the previous diagram.

ne benefit of this diagram is that it illustrates that pc set 7-19 comprises all

The inclusion of the last half segmentation

7-19

5-6 5-29

3-1 3-2 3-3 3-5 3-9

Before analyzing the pitch-class set genera for this subsection, let us first look at

each inclusion diagram subdivided further into that for the first half and that for the last

half of the segmented subsection. Figure 4.8(a) illustrates the inclusion relation for the

first half of the subsection. Trichordal pc sets in bold type represent those that actually

occur in each part (e.g., pc sets 3-2, 3-3, and 3-6 occur in the first half of the subsection,

while pc sets 3-1, 3-5, and 3-9 do not. [Compare Example 4.4 and Figure 4.8]). The

diagram also illustrates the strong association among the three “progenitors” by no

lines, as well as another inclusive relation indicated by dotted lines. This somewhat

hierarchical drawing suggests that a collection of quasi tone-cluster segments in the

piano’s right-hand articulates the principal harmony(ies); specifically, we can assume th

pc set 5-3 may function as the nexus set in the set complex. However, it is difficult to

speculate as to which trichordal pc set

u

4

O

113

“progenitors” at the climax. At the same time, it predicts less well which trichord is the

rongest harmonic set at the climax than in the first half of this subdivision. One might

asonably assume from Figure 4.8(a) that pc set 3-2 is the most significant “progenitor,”

nce it is the only trichordal set that is related to both of the source sets 5-3 and 5-24.

onversely, one might conclude that pc set 3-5 does not figure in an overall harmonic

ructure of the climax (Figure 4.8[b]). In fact, pc set 3-5 is not located within the

ontrapuntal confrontation between the voice and piano. Because pc set 3-5 does not

articipate in structuring the climax, it is unlikely that pc set 3-5 here functions musically

s the primary generic harmony.

igure 4.9: The tables of genera and the Squo indices for two subdivisions in the thirteenth subsection

a) First subdivision b) Last subdivision

Set Name: G1 G5 G7 G8 G9 G11 3-1 √

st

re

si

C

st

c

p

a

F

Set Name: G1 G5 G6 G11 3-1 √

3-2 √ √ 3-3 √ √ 3-5 √ 3-9 √ 5-6 √ √

5-29 √ √ √ 7-19 √ √ √ Counts: 8 4 2 3 2 2 2

3-2 √ √ 3-3 √

3-5 √ 3-9 √ 4-2 √ √ 4-Z29 √ 5-3 √ √ 5-24 √ √

Counts: 9 3 4 4 2 G1 G5 G6 G11 Total no. sets: 63 29 45 29

Status Quotient (Squo) Indices

G5

G1 0.053 0.153 (chroma)

G11 0.077 G6 0.099

G1 G5 G7 G8 G9 G1

Status Quotient (Squo) Indices

G1 0.079 G5

1 Total no. sets: 63 29 45 41 41 29

0.086 (chromG7 0.083 G8 0.061 G9 0.061

G11

a)

0.086 (dia)

114

The tables of genera and Squo indices above in Figure 4.9 verify the speculative

analytic conclusions argued in the two previous paragraphs. In Figure 4.9(a), the

-3

2 which

of

as the nexus set. In

ocal

e

hypothetical nexus, pc set 5-3, in the first subdivision is marked in genera 5 and 6, which

are the first and second highest Squo values; moreover, pc set 4-2 which is a subset of 5

marks exactly the same genera as 5-3. Inevitably, our attention is led to pc set 3-

also marks the identical genera to those of 5-3 and 4-2. These three pc sets are the

primary members of Genus 5 (chroma) which dominates the first subdivision, in addition

to a potential 3-1. Thus, we conclude that pc set 3-2 is the most significant “progenitor.”

The table for the last subdivision in Figure 4.9(b) is intriguing. No checkmark

pc set 7-19 in the table belongs to two the primary genera, 5 (chroma) and G11 (dia).

This suggests that the harmonic set 7-19 is too vague or undefined to determine the

subsection’s particular harmonic profile, even if it is designated

contrast, the relationship between trichordal pc sets and the primary genera clearly

reaffirms the subtle harmonic conflict derived from a contrapuntal parallel of trichordal

polarity (3-1 and 3-9). In fact, these two trichordal sets play an important role in

determining the highest value of the Squo indices, which confirms the ambiguous

harmonic feature of the climax (see Figure 4.9[b]). The mingled sonority of harmonic

polarization by which pc sets 3-1 and 3-9 sound simultaneously corresponds to the

musical and textual context of the climax. The harmonic set 3-1, marked by f in the v

line, emphasizes the text’s last negation, “kein Gedanke (no thought),” while th

harmonic set 3-9 in the piano’s right hand flows with “espressivo,” as we discussed in

Chapter 3.

115

4.1.5 Genera Succession of “Nur ein Wort”

The first section: G11 – G6 – G5 – G7 – G5 – G11

Figure 4.10: The Genera succession of “Nur ein Wort”

Based upon the structure of the sub-sections, the progression of genera in “Nur

ein Wort” unfolds an interesting formal design. As Figure 4.10 indicates, the progression

of genera in the first section is to some extent stable. Genus 11 which structures the

introduction and the codetta suggests harmonically a conventional closed form. In

addition, it is obvious that Genus 5 dominates the middle of the section. In contrast, the

genera in the second section are more varied and seemingly lack cohesion. Interestingly,

a genera succession in the opening of the piece G11 – G6 – G5 reappears at the end in

retrograde G5 – G6 – G11 (see Figure 4.10). And the concluding Genus 11 of the coda is

also a kind of resolution of the primary series. Although I have divided “Nur ein Wort”

into two large sections, they do not suggest any conventional binary form; rather, the

form seems best described as through-composed. Nonetheless, according to the analysis

of pitch-class set genera, Genus 11 (dia) seems to be “tonicized” as the center of generic

harmony. In fact, Genus 11 remotely connects the song’s introduction, codetta, and coda.

Above all, the tonicization of Genus 11 at the coda is the most effective means of

represe osure

to the s e

The second section: G3 – G11 – G7 or G3 – G1 – G5 – G6 – G11

nting how Krenek skillfully dramatizes the song’s last few measures. As a cl

ong, the relationship between the musical effect and the text is exquisite: th

116

isolated

iano remains after the voice disappears, suggesting that by the final measure the speaker

has fallen into a deep spiritual sleep.

Eb “Schlaf (sleep)” in m. 20 is sung when the coda is played. Furthermore, the

p

117

Conclusion

everal analytic approaches have been applied to “Nur ein Wort, nur eine Bitte”

of Fünf Lieder. In Chapter 1, particular pitch-class sets derived from partitioning the

primary twelve-tone series have revealed how respective pc sets function as harmonic

aterial for structuring “Nur ein Wort.” Above all, the discovery of significant trichords

otivic [pc set 3-2], “(cantabile) espressivo” [pc set 3-9], triadic [pc set 3-11] harmonies)

nd one pentachordal set (pc set 5-29) that potentially contains 3-2 and 3-9 have pointed

Krenek’s preferred sets for providing harmonic structure.

In Chapter 2, we have focused mainly on examining how contour structures the

ocal line. This approach has contributed considerably both to revealing the structural

onfigurations of the vocal line, and to suggesting several long-range structural

harmonies. More importantly, one of the gre pplying pitch-contour

theory to “Nur ein Wort” is that it reveals to some extent how the music and Kafka’s text

are exquisitely related. From a larger perspec ve, for instance, we have confirmed that

emphasized pitches – i.e., the highest and lowest pitches in each phrase – are closely

lated to particular words in the text. Also, the formation of dyads by these pitches

veals a transformational scheme that structures not only the overall shape of the vocal

line, but also the projection of a key word in the text, “Bitte.”

Chapter 3 derived from Kre f tension-degrees extends his

classification from three to twelve tension-degrees, and organizes them hierarchically.

This classification allows us to apply tension-degrees to particular harmonies in several

important subsections, regardless of whether the harmony is presented horizontally or

S

m

(m

a

to

v

c

atest benefits of a

ti

re

re

nek’s own concept o

118

vertically. Although inconsistency occurs between my rough mathematical model and

Krenek’s tension-degrees, the ordering of tw ve tension-degrees by this eclectic means

bears fruit in exploring how “voice-leading” works in “Nur ein Wort.” Moreover, I

believe that even though my classification of tension-degrees relies partly on a

mathematical justification, it should not distract from Krenek’s original concept, since

frequency ratios and LCMs are merely used to determine twelve orderings to

accommodate the total number of trichordal s sets. That is, unlike Hindemith’s

concept with which Krenek disagrees, the purpose of employing frequency ratios is to

classify trichords in twelve kinds, not to deal with real overtones or imagined undertones.

However, it may be necessary to consider tension-degrees in terms of actual pitch register

(a feature that Krenek scarcely mentions), since different registers should project

different psycho-acoustic effects that must be directly related to the degree of tension.

Finally, Chapter 4 applies the theory of pitch-class set genera to reveal particular

harmonic species of several selected subsecti ns of “Nur ein Wort” and to derive more

concrete representations. Since most harmonic objects presented in the analysis of “Nur

ein Wort” during the previous three chapters re trichordal pitch-class sets, labeling

generic characteristics according to Forte’s theory better accommodates the song’s

specific harmonies. alize the harmonic

network or cohesion in respective subsections. In particular, it determines the

predominating genus (genera), where the values of two or more status quotients are the

same or nearly the same.

The constant change of time signatures in “Nur ein Wort” suggests that Krenek

has a unique way of structuring rhythm. Particularly in musical works with texts, Krenek

el

pitch-clas

o

a

In addition, Parks’ inclusion diagram helps to visu

119

is likely to be extraordinarily sensitive to the metrical and rhythmic proportions. In any

event, Krenek’s notion of meter or rhythm in music is to provide material to create a

musical space, but not to measure the exact time points. Under the title of “Dealing with

the Time Factor” in Music Here and Now, Krenek describes how he thinks meter works

in an atonal context:

Every freely articulated single phrase in atonality must first produce its own meter. The meter is not founded on a filling-in of a rhythmic scheme which would continue to pulsate in silence underneath the music; it is based on a free distribution of weights in a continuity that is indefinite in its direction. No abstractly progressing 4/4 or 6/8 time signature is thus presupposed; instead, the metric figure is created simultaneously with the musical idea.95

Krenek uses the term “direction” in the context of musical meter, which may remind us

of a parameter in total serialism. Regarding meter, Krenek proposes an intriguing idea

that may prove essential for future methods of manipulating time (point) and note value

in a musical work. In fact, Krenek’s description of how atonal time differs from tonal

time is quite suggestive:

Time is divested of the character which tonality has agreed on, and becomes, instead, a spacelike continuity. When there is no longer a steadily pulsating meter to suggest the familiar “one-way-street” of time, the time measurement to be filled with music seemingly assumes the quality of space through which we can move in many directions, even in opposite directions. Thus, musical retrogression could well create the impression of time moving back. It appears as though the time quantity consumed by the forward-moving musical unit would be “repaid” by its retrograde statement.96

Although Krenek here does not directly suggest the idea of a rhythmic series, the “time

quantity” in “retrograde statement” suggests aspects of the methods later used by Milton

Babbitt and Pierre Boulez to structure rhythm through serial procedures.

95 Krenek, Music Here and Now, 211-12. 96 Itlicized by Krenek, ibid.

120

It is unclear whether Krenek applied the idea of flipping or retrograding rhythmic

durations when he composed “Nur ein Wort,

rther. Clearly, Krenek’s manipulation of the meter and rhythm in “Nur ein Wort”

ontributes significantly to its aesthetic appeal. Using Christopher Hasty’s terminology,

ere are in fact contiguous rhythmic repetitions or patterns in “Nur ein Wort.” 97 Hasty’s

ethodology for measuring the meter and time point(s) may well be useful in exploring

rther how Krenek structures meter both in “Nur ein Wort” and his later twelve-tone

orks.

” but this feature needs to be explored

fu

c

th

m

fu

w

97 Christopher Hasty, Meter as Rhythm (New York: Oxford University Press, 1997).

121

APPEND

IX I

MUSICAL SCORE

122

“Nur ein Wort, nur eine Bitte” from Fünf Lieder

123

124

APPENDIX II

TABLES FOR TENSION-DEGREES

125

Tension-degrees in LCM

Pc set nprime for

ree withi eachoup

Position among all trichords

order

ame/ m

LCM Deg n gr

3-11 [037] 60 cons. 1 1 3-9 [027] 72 m. 1 2 3-4 [015] 240 sh. 1 3 3-12 [048] 400 cons. 2 4 3-2 [013] 720 sh. 2 5 3-10 [036] 900 cons. 3 6 3-7 [025] 1080 m. 2 7 3-8 [026] 1440 m. 4 8 3-3 [ s014] 2400 h. 3 9 3-5 [ 0 sh. 1016] 288 4 0 3-6 [0 ] 4 m. 124 518 3 1 3-1 [0 ] 0 sh. 112 5760 1 2

126

The table for the f quency rat CONS

3-11 [C-Eb-G] C = 5 Eb-G = 4:5 → 2 LCM = 60 (1st inversion) E = 4 G-C = 3:4 → 12 15:20 60

(2nd inversion) G = 3:4 → 0:24 120 3-12 [C-E-G#] C = 4:5 → 16:20:25 400 3-10 [C-Eb-Gb] C 5 Eb-Gb = 5:6 → 2 6 900 (1st inversion) Gb-C = 32:45 115:138:180 4140 (2nd inversion) C-Eb = 5:6 54 4320

M 3-6 [C-D-E C-D = 8: - → 64:72:81 LCM = 5184

D-E = 8:9 E-C = 5:8 → 40 45:72 360 C-D = 8:9 → 5:8:9 360 3-7 [C-D-F] C-D = 8:9 D-F = 5:6 → 40 1080 D 5:6 F-C = 2:3 → 5:6:9 90 F 2:3 C-D = 8:9 → 16:24:27 432

3-8 [C-D-F#] C-D = 8:9 D-F# = 4:5 → 32:36:45 1440 D-F# = 4:5 F#-C = 32:45→ 128:160:225 28800 F#-C = 32:45 C-D = 8:9 → 256:360:405 103680

D-G = 3:4 G-C = 3:4 → 9:12:16 144

G-C = 3:4 C-D = 8:9 → 6:8:9 72

3-1 [C-C#-D] C-C# = 15:16, C#-D = 15:16 → 225:240:256 LCM = 57600 C#-D = 15:16, D-C = 5:9 → 75:80:144 3600 D-C = 5:9, C-C# = 15:16 → 80:135:144 2160

→ 15:16:18 720 :5 → 8:9:15 360

Eb-C = 3:5, C-Db = 15:16 → 9:15:16 720 3-3 # # :1 # 6 7 :96 2 C#-E = 5:6, E-C = :8 :48 120

= 8, C-C# 15: 7 20:1 96

4 Db C-Db = 15:16 Db-F 4:5 :20 24 -F :5, 8:10:15 120 F-C = , C-Db 15:1 → 10:15:16 240 5 C# # 5:1 45:48:64 2880 C#-F# = 4:3, → 24:32:45 1440 C 2:45 C-C# = 15:16 → 2:45: 1440

re ios and LCM of trichordal pc sets

-Eb :6 10:1 :15b-G :5 :-C C-Eb = 5:6 15:2

-E E-Ab = 4:5

-Eb = :6 5:30:3Eb-Gb = 5:6 →Gb-C =32:45 → 32:45:

] 9 D E = 8:9 :

E-C = 5:8

:45:54 -F = -C =

3-9 [C-D-G] C-D = 8:9 D-G = 3:4 → 8:9:12 72 SH

3-2 [C-Db-Eb] C-Db = 15:16, Db-Eb = 8:9 Db-Eb = 8:9, Eb-C = 3

[C-C -E] C-C = 15 6, C -E = 5: → 5:80 400 5 → 25:30 0

E-C 5: = 16 → 5:1 28 00

3- [C- -F] , = → 15:16 0 Db = 4 F-C = 2:3 →

2:3 = 6

3- [C- -F#] C-C = 1 6, C#-F# = 3:4 → F#-C = 45:32 F#- = 3 , 3 48

127

The table for the frequency ratios and LCM of trichordal pc sets in another means CONS

3-11 [C-Eb-G] = 1: (6/5): (3/2) = 10:12:15 LCM = 60 [C-E-A](1st inversion) = 1: (5/4): (5/3) = 12:15:20 60 [C-F-Ab](2nd inversion) = 1: (4/3): (8/5) = 15:20:24 120

800 3-10 : ) = 1 14400 -Eb-A] inve ion 1: 5) /3) 15: :25 450 -F#-A] inv sion 1: /32): (5/3) = 96:135:160 4320

M 3-6 [ -D-E] 5/ = 8:9:10 LCM = 360

[ -D-Bb] = 1: (6/5): (9/5 5:72 360 [ -Ab-Bb = 1: (8/5): (9/ = 5:8:9 360 3-7 [ -D-F] 4/3 = 24:27: 2 864 [ -Eb-Bb] = 1: (6/5): (9/5 = 5:6:9 90 [ -G-A] 5/ = :1 90 3-8 [ 4 ) 2 1440 [ = 1: (5/4): (9 900 [ = 1 5 ): ) 6 5 57600 3-9 [ (3/2 72 [C-D = 1: (9/8): (3/2 = 8 :12 72 [C-F- = 1: (4/3): (9/5 = 15:20: 7 540 SH

3-1 [C-C#-D] = 1 16/ ): (9/8) = 120:128:135 LCM = 17280 [C-C#-B] = 1 16/ ): (15/8) = 120:128:225 28800

[C-Bb-B] = 1: (9/5): (15/8) = 40:72:75 1800

[C-A-Bb] = 1: (5/3): (9/5) = 15:25:27 675 3-3 [C-C#-E] = 1: (16/15): (5/4) = 60:64:75 4800 [C-Eb-B] = 1: (6/5): (15/8) = 40:48:75 1200 [C-Ab-A] = 1: (8/5): (5/3) = 15:24:25 600

3-4 [C-Db-F] = 1: (16/15): (4/3) = 15:16:20 240 [C-E-B] = 1: (5/4): (15/8) = 8:10:15 120 [C-G-Ab] = 1: (3/2): (8/5) = 10:15:16 240

3-5 [C-C#-F#] = 1: (16/15): (45/32) = 480:512:675 3456 0 [C-F-B] = 1: (4/3): (15/8) = 24:32:45 1440

[C-F#-G] = 1: (45/32): (3/2) = 32:45:48 1440

3-12 [C-E-G#] = 1: (5/4): (8/5) = 20:25:32

[C-Eb-Gb] = 1 ( /56 : 45( /3 ) 2 1 0:6 92:225 [C [C

(1st (2nd

rser

) )

= =

(6/(45

: (5 = 18

C = 1: (9/8): ( 4) C ) = 40:4 C ] 5)

C = 1: (9/8): ( ) 3 C ) C = 1: (3/2): ( 3) 6:9 0

C-D-F#] = 1: (9/8): ( 5/32/ =

= 3 2

:36:45

5

C-E-Bb]

-Gb-Ab] 5) ( /5

20: = 1

:360:22

C : (4 /32 8 :2 65

C-D- = 1: (4/3)G] : ) = 6:8:9 -G] ) :9

Bb] ) 2

: ( 15

: ( 15 3-2 [C-Db-Eb] = 1: (16/15): (6/5) = 15:16:18 720 [C-D-B] = 1: (9/8): (15/8) = 8:9:15 360

0

128

APPENDIX III

TABLES, DIAGRAMS, AND PITCH REDUC IONS FOR PC SET GENERA

T

129

K- & Kh-relation of the second subsection, mm. 1 – 3

4-2

5-1

7-265-26

7-9Z 6

3-3 3-4 5-9 6-

3

4-2 Kh K Kh K Kh Kh 5-1 Kh K Kh K

7-26/5-26 Kh Kh K 7-9/5-9 Kh Kh Kh

6-Z36 Kh Kh Kh K

6-Z36: #Kh = 3, #K = 1 #Kh = 35-9/7-9:

K- & Kh-relations in Codetta, m. 9

3-2

3-5

3-9

3-11

4-13

4-14

4-22

4-27

7-29 7-Z36 5-Z36

6-Z47

4-13 Kh Kh K K Kh Kh Kh 4-14 kh K Kh Kh Kh K Kh 4-22 K K Kh Kh K Kh Kh 4-27 K K K Kh Kh K Kh

7-29 Kh Kh Kh Kh Kh Kh K Kh Kh 5-Z36 7-Z36

Kh Kh Kh Kh Kh K Kh K K

6-Z47 Kh Kh Kh Kh Kh Kh Kh Kh Kh K

- & Kh-relation of the thirteenth subsection, “kein Gedanke,” mm. 17 – 18

5-3: #Kh = 5, #K = 3

7-19: #Kh = 4, #K = 2 5-24: #Kh = 5, #K = 3 5-29: #Kh = 3, #K = 5

3- 3-2 3-3 3-5 3-6 3-9 4-2 4-Z29 5-3 5-6 5-24

5-29

K

7-19 1

4-2 Kh Kh h Kh K Kh K K K K K 4-Z29 K K Kh K Kh K K K K h Kh K

5-3 Kh h Kh Kh K Kh K K K 5-6 Kh K Kh Kh K K K K

7-19 K K h Kh Kh Kh K 5-24 K Kh K Kh Kh Kh K h K 5-29 K Kh K Kh K Kh K K

130

The table of genera and the Squo indices for the overall thirteenth subsection, mm. 17 – 18

Set Name: G1 G5 G6 G7 G11 3-1 √ 3-2 √ √ √ 3-3 √ 3-5 √

√ √ √

√ 19) √ √ [7-19] 4 √ 9 √ √

6 4 4 5 G1 G5 G6 G7

o. se : 63 29 45 45

uo nt uo dic G1 79 G

3-9 √ 4-2 √ √ 4-Z2

5 3 9

√

-

5-6(5-

5-2 √ √ 5-2 √Counts:

12

3G11

Total n ts 29 Status Q tie (Sq

0.0) In

es

5 0.115 (chroma) G 4 G 3

Segmentation of the third subsection, “nur eine Bewegen der Luft, nur ein Beweis,” mm. 3 – 6

6 0.0

0.097

7

G11 0.086

131

K- & Kh-relation of the third subsection, mm 3 – 6

3-1 3-2 3-4 4-1 4-2 4-21 4-Z29 5-2 5-5 5-24

5-35

6-Z40

.

9-1 9-2 8-2

4-1 Kh Kh h h K K K K K K8-2/4-2 Kh Kh Kh K Kh K K

4-21 K K K K K K Kh4-Z29 K K Kh Kh K K Kh Kh

5-2 Kh Kh Kh Kh Kh K K 5-5 Kh Kh Kh Kh K K K

5-24 K h Kh Kh K K Kh K 5-35 K K K K

6-Z40 Kh Kh Kh K Kh K K K

5-2: #Kh = 5, #K = 2 6-Z40: #Kh = 4, #K = 4

The table of genera and the Squo indices for the third subsection, mm 3 – 6

Set Name: G2 G5 G6 G7 G8 G10 G11 3-1 √ [9-1]

3-2 √ √ √ [9-2] 3-4 √ √ 4-1 √ 4-2 √ √ [8-2] 4-21 √ 4-Z29 √ 5-2 √ √ √ √ 5-5 √ √ √ 5- √ √ √ √ 6-Z40 √ √ √ √ √ √ √

Tot no. sets: 65 29 45 45 41 41 29 Sta Quotie quo) dices

G2 0.0 G5

24 5-35

Counts: 12 5 7 4 4 3 3 3 G2 G5 G6 G7 G8 G10 G11

al

tus nt (S In64

0.201 (chroma) G6 0.0 G7 0.0 G8 0.0

0 0.0 086

74 74 61

G1 61 G11 0.

132

The segmentation of the fourth subsection, “dass du noch lebst und wartest,” mm. 6 – 8

K- & Kh-relations of the fourth subsection, mm. 6 – 8

9-2 3-2

4-13

4-21

7-9

5-28

5-29

6-Z11

4-13 Kh K K Kh Kh 4-21 K Kh K K

7-9 Kh K Kh 5-28 Kh K K 5-29 Kh Kh K

6-Z11 Kh Kh

6-Z11: #Kh = 2 7-9: #Kh = 2, #K = 1

he inclusion diagram of the fourth subsection, mm. 6 – 8

9-2

7-9

6-Z11

5-28 5-29

4-21 4-13

3-2

T

133

The table of genera and the Squo indices for the fourth subsection, mm. 6 – 8

Set Name: G2 G3 G5 G6 G7 3-2 √ √ √ 4-13 √ √ 4-21 √ 5-9 √ √ √ √ 5-28 √ √ √

Counts: 7 5 4 3 4 5 G2 G3 G5 G6 G7 Total no. sets: 65 45 29 45 45 Status Quotient (Squo) Indices

G2 0.110 G3 0.127 G5 0.148 (chroma) G6 0.127 G7

5-29 √ √ √ 6-Z11 √ √ √ √ √

0.159 (chroma-dia)

The segmentations of the seventh subsection, “Nein, keine Bitte,” mm. 10 – 11

134

K- & Kh-rel tions, th seven s n e k B tte,” mm 10 – 11

7-6 Z12 5-19

5-25

5-32

6-Z50

a e th ubsectio , “N in, eine i .

3-11 4-13 4-14 4-Z29

5-

4-13 K Kh K K Kh K h Kh 4-14 K K K K K K h K

4-Z29 K K K K K h K h Kh 7-6 K K K K 5-Z12 K Kh K K

5-19 Kh Kh K Kh K 5-25 Kh Kh K Kh K 5-32 Kh K K K K

6-Z50 Kh Kh K K K K K

5-19: #Kh = 3, #K = 2 5-25: #Kh = 3, #K = 2

6-Z50: #Kh = 2, #K = 5

The table of genera and the Squo indices for the seventh subsection, mm. 10 – 11

Set Name: G1 G3 G7 G9 G10 G12 3-11 √ √ √ 4-13 √ √ √ 4-14 √ 4-Z29 √ (5-6) √ [7-6] 5-Z12 √ √ √ 5-19 √ √ √ √ 5-25 √ √ √ √ 5-32 √ √ √ √ 6-Z50 √ √ √ √ √ Counts: 10 8 6 5 4 2 4

Total . sets: 63 45 45 41 41 45

tat uo t ( es 7 ( )

G1 G3 G7 G9 G10 G12 no

S us Q tien Squo) Indic

G1 0.12 atonal G3 0.133 ( ini )

1 8 9 9

dim shedG7 0.11 G9 0.09

G10 0.04 G12 0.08

135

The se n n e ghth ub on; e t ” m . 1 – 12gme tatio of th ei s secti “nur in A men, m 1

K- & Kh-relations of the eighth subsection, “nur ein Atmen,” mm. 11 – 12

3-2

3-6

4-8

4-138-18

4-22

5-29

6-Z47 4-8 K K K K

4-13 Kh K Kh Kh 8-18 K K K K

4-22 K Kh K Kh 5-29 Kh K K Kh K K Kh

6-Z47 Kh Kh K Kh K Kh Kh

6-Z47: #Kh = 5, #K = 2

he inclusion diagram of the eighth subsection, mm. 11 – 12

T

6-Z47

5-29

4-8 4-13 4-22

3-2 3-6

136

The table of enera e q in ce . 11 – 12 Set Na 3 7 11 3-2

g and th S uo di s for the eighth subsection, mm

me: G1 G G G √

4-8 -13

(4-1 [8-18] 4-22

5-29 √ √ √ √ √ √ √

Counts: 7 4 3 1 3 7 11 Total no. sets: 3 5

Q t quo) Indices G1 0.113 G3 0.127 G7 0.127 (chroma-dia)

G11

√ 4 √ √ √

8) √ √ √

6-Z47 √5 4

G G G G 6 4 45 29

Status uotien (S

0.148 (dia)

The segmentations of the ninth and tenth subsections, “kein Atmen, nur ein Bereitsein,” mm. 13 – 14

137

K- & Kh-relations of the ninth and tenth subsections, mm. 13 – 14

3-8 4-1 4-13 4-21 4-Z29 5-4 5-25

6-33

6-Z47

9-6

4-1 K K K Kh Kh Kh 4-13 K K Kh Kh Kh Kh 4-21 Kh Kh K K Kh

4-Z29 K Kh K Kh Kh K 5-4 K Kh Kh Kh K K

5-25 K Kh Kh K Kh Kh K 6-33 Kh Kh Kh Kh Kh Kh

6-Z47 Kh Kh Kh K K

6-33: #Kh = 6

he inclusion diagram for the ninth and tenth subsections, mm. 13 – 14

T

8-13

6-Z47 6-33

5-255-4

4-1 4-13 4-21 4-Z29

3-6 3-8

9-6

138

The table of genera and the Squo indices for the ninth and tenth subsections, mm. 13 – 14

Set Name: G2 G3 G5 G7 G11 3-8 √ 4-1 √ 4-13 √ √

5-25 √ √ √ √

5 3 G2 G3 G5 G7 G11

G3

4-21 √ 4-Z29 √

5-4 √ √ √ √ 6-33 √ √ √ √ 6-Z47 √ √ √ √ Counts: 9 7 5 2

Total no. sets: 65 45 29 45 29

Status Quotient (Squo) Indices G2 0.120 (whole-tone)

0.123 G5 0.077 G7 0.123 (chroma-dia)

G11 0.115 The segmentations of the eleventh subsection, “kein Bereitsein, nur ein Gedanke,”

-16 mm. 15

139

K- & Kh-relations of the eleventh subsection, mm. 15 – 16

3-2

3-5

3-8

3-9

4-13

4-Z15

4-Z29

5-9

5-13

7-15 5-20

5-29

6-21

6-22

4-13 Kh Kh K K K K K Kh 4-Z15 K Kh Kh K Kh K K K K Kh Kh 4-Z29 Kh Kh Kh K K Kh K Kh K Kh Kh 5-9 Kh Kh Kh K K Kh K Kh Kh 5-13 Kh Kh Kh K K Kh Kh Kh 7-15 K Kh Kh Kh K K K Kh 5-20 Kh Kh Kh Kh K K Kh 5-29 Kh Kh Kh Kh Kh K K 6-21 Kh Kh Kh Kh Kh Kh Kh 6-22 Kh Kh Kh Kh Kh Kh Kh Kh Kh

6-22: #Kh = 9 The inclusion diagram of the eleventh subsection, mm. 15 – 16

9-5

7-15

6-21 6-22

5-9 5-13 5-205-29

4-13 4-Z15 4-Z29

3-2 3-5 3-8 3-9

140

The table of genera and the Squo indices for the eleventh subsection, mm. 15 – 16

Set Name: G1 G2 G4 G5 G11 3-2 √ 3-5 √ 3-8 √ 3-9 √ 4-13 √ 4-Z15 √ √ 4-Z29 √ √ 5-9 √ √ √ 5-13 √ √ √ √ (5-15) √ √ [7-15] 5-20 √ √ 5-29 √ √ √ 6-21 √ √ √ √ 6-22 √ √ √ √ √ Counts: 14 11 10 3 5 3 G1 G2 G4 G5 G11 Total no. sets: 63 65 21 29 29 Status Quotient (Squo) Indices

G1 0.125 (atonal) G2 0.110 G4 0.102 G5 0.123 (chroma)

G11 0.074 The segmentations of the fourteenth subsection, “nur ruhiger Schlaf,” mm. 19 – 20

141

K- & Kh-relations of the fourteenth subsection, “nur ruhiger Schlaf,” mm. 19 – 20

3-2

4-3

4-Z15

8-Z29 5-3

5-11

5-28

4-3 Kh Kh K 4-Z15 K K Kh 8-Z29 Kh K K Kh

5-3 Kh Kh K 5-11 Kh K K K 5-28 Kh Kh Kh

5-28: #Kh = 3

The inclusion diagram for the fourteenth subsection, mm. 19 – 20

The table of genera and the Squo indices for the fourteenth subsection, mm. 19 – 20

Set Name: G1 G5 G6 3-2 √ √ 4-3 √ 4-Z15 √ 4-Z29 √ 5-3 √ √ 5-11 √ √ 5-28 √ √ Counts: 7 3 3 5 G1 G5 G6 Total no. sets: 63 29 45 Status Quotient (Squo) Indices

G1 0.068 G5 0.148 G6 0.159 (semichroma)

8-Z29

5-28 5-11 5-3

4-Z154-3

3-2

142

BIBLIOGRAPHY

Backus, John. The Acoustical Foundations of Music, 2d ed. New York: W. W. Norton

and Company, 1977. Bernard, Jonathan W. “Pitch/Register in the Music of Edgard Varèse.” Music

Theory Spectrum 3 (Spring 1981): 1-25. Bowles, Garrett H. Ernst Krenek: A Bio-Bibliography. New York: Greenwood

Press, 1989. _______________. “Krenek, Ernst.” In The New Grove Dictionary of Music and

Musicians, 2d ed., ed. Stanley Sadie and John Tyrrell, 895-899. London: Macmillan Publishers Limited, 2001.

Burns, Edward M. and W. Dixon Ward: “Intervals, Scales, and Tuning.” In The

Psychology of Music, ed. Diana Deutsch, 241-269. New York: Academic Press, 1982.

Carlson, Effie B. A Bibliographical Dictionary of Twelve-Tone and Serial Composers.

Metuchen: The Scarecrow Press, 1970. Erickson, Richard. “Krenek’s Later Music.” The Music Review 9 (1948): 29-44. Foley, Gretchen. “Pitch and Interval Structures in George Perle’s Theory of

Twelve-Tone Tonality.” Ph.D. diss., University of Western Ontario, 1998. Forte, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973. _________. “Pitch-Class Set Genera and the Origin of Modern Harmonic Species.”

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