consistency, context and symmetry in alberto ginastera’s

Consistency, Context and Symmetry in Alberto Ginastera’s String Quartets Nos. 1 (1948) and 2 (1958, First Version)

By

David L. Sommerville

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Dave Headlam

Department of Music Theory Eastman School of Music

University of Rochester Rochester, New York

2009

Curriculum Vitae

The author was born in Philadelphia, Pennsylvania on February 15, 1969. He

attended Emory University from 1987 to 1991 and graduated with a Bachelor of Arts

degree in International Studies, focusing on Spanish and the history and politics of Latin

America. After several years of pursuing music performance, he began post-

baccalaureate studies in music at Georgia State University, eventually earning a Master

of Music degree in Music Theory from that institution in 1999 and was inducted into the

Pi Kappa Lambda music honor society. He enrolled in the Doctor of Philosophy program

in Music Theory at the Eastman School of Music, University of Rochester in the Fall of

1999, won the outstanding TA prize in 2001, and earned the Master of Arts degree in

Music Theory in 2002. In addition to teaching at Eastman, he also lectured in Music

Theory at Nazareth College. He pursued his research on the music of Alberto Ginastera

under the direction of Professor Dave Headlam.

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Acknowledgements

I wish to acknowledge several people whose influence and support of me during

my doctoral studies at Eastman has been absolutely vital, not only to the completion of

the program, but to the betterment of my life. First, I thank my parents David and Diane

Sommerville for working hard and sacrificing to afford me the numerous opportunities I

have received at every stage in my journey. Second, I thank my wife Maria, who has

delayed her dreams to allow me to pursue mine. Third, I wish to thank the teachers I have

had, from Ken Adkins at Albuquerque Academy so long ago, through Susan Tepping,

John Nelson and Ron Squibbs at Georgia State who introduced me to music theory, to

Dave Headlam (especially), Matthew Brown (especially), John Covach and Steve Laitz at

Eastman who saw me through the happy and the difficult times during my course of

study. Fourth, I wish to thank my other committee members Jeannie Guerrero and

Malena Kuss for their excellent help in the preparation of this document. Fifth, I thank

my friends Jocelyn Kovaleski (née Swigger), Josh Mailman, Sam Ng, Jeff Tucker and

Alfred Vitale for their friendship and support. Finally, I thank my precious daughter

Rory, who is and always will be the greatest part of my life.

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Abstract

Despite its richness, presence in the recital and concert hall, amenity to

established analytic avenues, and its potential to support new directions in music-

theoretic research, the music of Alberto Ginastera has received relatively little attention

among the music theory community. This dissertation attempts to remedy this by

selecting passages from the First and Second String Quartets (1948 and 1958) and

developing appropriate analytic tools rooted in the recent confluence of Perle-based and

transformation theories. The study of these important works in Ginastera’s career reveals

that some of the salient aspects of the musical surface, such as its emphasis on the

ensemble’s open strings, its cyclic pitch structure, and its frequent embrace of repetition

and patterns, are indeed revelatory of deeper structures in the music. In its approach, the

dissertation develops a simple specialized vocabulary to identify and contextualize

significant compositional constructions and relate them in various ways to a few basic

concepts: as a reflection of 1) a cyclic or transformational property of the OS set (open

string) or a significant conjunct subset called a shading or a dyad-space, 2) an interaction

of interval cycles called coloring, or 3) a type of generative motivic pattern called a

source trichord and its subsequent unfolding of larger cyclic pc collections called U-cells

and U-chains. Ultimately, the study uses subtle differences in the above constructions to

engage issues of stylistic development during the decade separating the quartets’

composition.

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After an introductory chapter that provides both a basic context for the analysis of

subsequent chapters and a literature review, the second chapter details the methodological

aspects of the dissertation, discussing not only the established music-theoretic and

analytic approaches the dissertation utilizes, but also the original contribution to

scholarship the dissertation offers. The third and fourth chapters contain analyses of

movements from the First and Second String Quartets, with the former focusing on U-

transformations and dyad-spaces in the first movements of both quartets and the latter

focusing on various internal movements of the Second Quartet. Finally, the fifth chapter

concludes the dissertation by summarizing the previous chapters and offering several

possible avenues of further research.

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Table of Contents

Chapter One Introduction, Review of the Literature, 1 Broad Remarks and Chapter Overviews I. Introduction 1 II. Review of the Literature 5 III. Broad Remarks 8 IV. Chapter Overviews 12 Chapter Two Methodology 14 I. Pitches, Pitch-classes and Pitch-class Collections 15 II. Contextual Transformations K(nm) and U(nm) 18 III. Tiles 28 IV. Crosscuts 29 V. Overview of Main Rhythmic Profiles 31 Chapter Three Dyad-Spaces, U-transformations and Cycles in the 33 First Movements of Quartets One and Two I. Dyad-Spaces and Large-Scale Harmony 38 II. U-transformations and the Melodic/Motivic Aspect 45 Chapter Four Interval Cycles and Cyclic Collections in 60 Internal Movements I. (Mostly) Twelve-Tone Movements Two and Three 61 Of the Second String Quartet Chapter Five Conclusions and Directions for Further Research 86 I. Conclusions 86

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II. Directions for Further Research 89

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List of Examples (Chapters 1-5)

Example Title Page Example 2.1a The OS set 16 Example 2.1b Trichordal subset shadings of the OS set 16 Example 2.1c Musical example of shading Appendix 2 Example 2.2a Musical example of coloring Appendix 2 Example 2.2b Coloring of the 7-cycle dyad G-D Appendix 2 Example 2.3a K-transformations 19 Example 2.3b U-transformations 19 Example 2.4 Quartet No. I/I, mm. 1-2 Appendix 2 Example 2.5a Transformations of source trichord <265> Appendix 2 Example 2.5b Illustration of terminology 23 Example 2.5c Demonstration of labels 23 Example 2.6 Generation of octatonic and hexatonic collections 27 Example 2.7a Generation of pentatonic/diatonic collections 27 Example 2.7b Generation of extended tertian sonorities 27 Example 2.8a Depiction of Tile 1 in Quartet 2/III, mm. 16-19 Appendix 2 Example 2.8b Depiction of Tile 2 in Quartet 2/III, mm. 20-23 Appendix 2 Example 2.8c Depiction of Tile 7 in Quartet 2/III, mm. 44-47 Appendix 2 Example 2.9 Depiction of Crosscut in Quartet 2/I, mm. 32-37 Appendix 2 Example 3.1a Musical Example of Quartet 1/I, mm. 1-7 Appendix 2 Example 3.1b Chords in mm. 2 and 5 Appendix 2

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Example 3.1c Voice-leading between chords in mm. 2 and 5 Appendix 2 Example 3.2a Musical Example of First Theme of Quartet 2/I, Appendix 2 mm. 5-9 Example 3.2b Reduction of First Theme of Quartet 2/I Appendix 2 Example 3.3a Network of OS shadings and dyad-spaces in 42 Quartet 1/I Example 3.3b Network reinterpretation of outer voices of 43 Example 3.3a Example 3.4a Musical Example of Second Theme of Quartet 1/I Appendix 2 Example 3.4b Musical Example of Canon in Second Theme of Appendix 2 Quartet 1/I Example 3.4c Dyad-spaces in Recapitulation of Second Theme Appendix 2 Of Quartet 1/I Example 3.5a Musical Example of Quartet 1/I, mm. 16-24 Appendix 2 Example 3.5b Nested U-cells in First Theme of Quartet 1/I 48 Example 3.6 Five accompanimental chords in First Theme of Appendix 2 Quartet 1/I Example 3.7 Network of voice-leading in accompanimental chords 51 Example 3.8a Musical Example of Transition Theme One in Appendix 2 Quartet 1/I, mm. 60-61 Example 3.8b Musical Example of Transition Theme Two in Appendix 2 Quartet 1/I, mm. 72-73 Example 3.8c Musical Example of Transition Theme Two in Appendix 2 Quartet 1/I, mm. 76-77 Example 3.9a Musical Example of Quartet 2/I, mm. 1-4 Appendix 2 Example 3.9b U-cells and crosscuts in First Theme of Quartet 2/I Appendix 2 Example 3.9c Musical Example of First Theme of Quartet 2/I, Appendix 2 mm. 20-23

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Example 3.10a Musical Example of Second Theme of Quartet 2/I, Appendix 2 mm. 98-104 Example 3.10b Musical Example of Quartet 2/I, mm. 118-121 Appendix 2 Example 4.1a Rows 3-R1 and 3-R2 and their properties in 65 Quartet 2/III Example 4.1b Row 2-R1 and its properties in Quartet 2/III 65 Example 4.2 Row forms in Quartet 2/III 68 Example 4.3a Network of ops in Quartet 2/III 68 Example 4.3b Network of row forms in Quartet 2/III 68 Example 4.3c Network Q2/M3 68 Example 4.4a Musical Example of canon in Quartet 2/III, Appendix 2 mm. 165-167 Example 4.4b Musical Example of 3-cycle dyads in Quartet 2/III, Appendix 2 mm. 165-167 Example 4.5 Row Forms in B section of Quartet 2/III Appendix 2 Example 4.6 Musical Example of canon and interpretive network Appendix 2 In Quartet 2/II Example 4.7a Whole-tone collections in Quartet 2/II, mm. 23-24 Appendix 2 Example 4.7b Whole-tone collections in Quartet 2/II, mm. 27-28 Appendix 2 Example 4.8a Musical Example of hexatonic collections in 80 Quartet 2/III Example 4.8b Musical Example of Tile C1 in Quartet 2/III, Appendix 2 mm. 195-196 Example 4.8c Network interpreting Example 4.8b 80 Example 4.9a Musical Example of Tile C12 in Quartet 2/III, Appendix 2 mm. 202-204 Example 4.9b Network interpretation of Example 4.9a 82

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Example 4.10 B section of Quartet 2/II Appendix 2 Example 4.11 2-/4-cycle collections in Quartet 2/II Appendix 2 Example 5.1a Musical Example of Quartet 2/IV, mm. 1-4 Appendix 2 Example 5.1b Musical Example of Quartet 2/IV, m. 6 Appendix 2 Example 5.1c Musical Example of Quartet 2/IV, mm. 13-14 Appendix 2 Example 5.2 Musical Example of Quartet 3/III, mm. 1-7 Appendix 2 Example 5.3a Reduction of Sonata For Guitar Appendix 2 Example 5.3b K-nets in Sonata for Guitar Appendix 2 Example 5.3c Cycles in Sonata for Guitar Appendix 2 Example 5.3d “Hyper cycles” in Sonata for Guitar Appendix 2 Example 5.4 U-cells in Berg, Op. 4, No. 2 Appendix 2 Example 5.5 U-cells in Berg, Op. 2/III Appendix 2

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List of Tables Table Title Page Table 3.1 Form Chart of Quartet 1/I 34 Table 3.2 Form Chart of Quartet 2/I 35 Table 4.1 Form Chart of Quartet 2/II 62 Table 4.2 Form Chart of Quartet 2/III 63

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List of Examples (Appendix Two) Example Title Page Example 2.1c Musical Example of shading 1 Example 2.2a Musical Example of coloring 2 Example 2.2b Coloring of the 7-cycle dyad G-D 2 Example 2.4 Quartet No. I/I, mm. 1-2 3 Example 2.5a Transformations of source trichord <265> 3 Example 2.8a Depiction of Tile 1 in Quartet 2/iii, mm. 16-19 4 Example 2.8b Depiction of Tile 1 in Quartet 2/iii, mm. 20-23 4 Example 2.8c Depiction of Tile 7 in Quartet 2/iii, mm. 44-47 5 Example 2.9 Depiction of Crosscut in Quartet 2/I, mm. 32-37 6 Example 3.1a Musical Example of Quartet 1/I, mm. 1-7 7 Example 3.1b Chords in Quartet 1/I, mm. 2 and 5 7 Example 3.1c Voice-leading between chords in mm. 2 and 5 7 Example 3.2a First Theme of Quartet 2/I, mm. 5-9 8 Example 3.2b First Theme of Quartet 2/I throughout movement 8 Example 3.4a Second Theme of Quartet 1/I, mm. 85-88 9 Example 3.4b Canon in Second Theme of Quartet 1/I 9 Example 3.4c Dyad-spaces in Recapitulation in Quartet 2/I 10 Example 3.5a Musical Example of Quartet 1/I, mm. 16-24 11 Example 3.6 Five accompanimental chords in First Theme 12 Example 3.8a Transition Theme One in Quartet 1/I, mm. 60-61 12 Example 3.8b Transition Theme Two in Quartet 1/I, mm. 72-73 12

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Example 3.8a Transition Theme Two in Quartet 1/I, mm. 76-77 13 Example 3.9a Opening of Quartet 2/I, mm. 1-5 13 Example 3.9b U-cells in Crosscuts of First Theme in Quartet 2/I 14 Example 3.9c Closing of First Theme in Quartet 2/1, mm. 20-23 14 Example 3.10a Second Theme of Quartet 2/I, mm. 98-104 15 Example 3.10b Tile 15 in Quartet 2/I, mm. 118-121 15 Example 4.4a Canon in Tile A14 of Quartet 2/iii, mm. 165-167 16 Example 4.4b 3-cycle dyads in A14 canon 16 Example 4.5 Row Forms in B section of Quartet 2/iii 17 Example 4.6 Canon and interpretive network in Quartet 2/ii 17 Example 4.7a Whole-tone collections in Quartet 2/ii, mm. 23-24 18 Example 4.7b Whole-tone collections in Quartet 2/ii, mm. 27-28 18 Example 4.8b Tile C1 in Quartet 2/iii, mm. 195-196 19 Example 4.9a Tile C12 in Quartet 2/iii, mm. 202-204 19 Example 4.10 B section of Quartet 2/ii 20 Example 4.11 2-/4-cycle collections in Quartet 2/ii, mm. 13-14 21 Example 5.1a Opening of Quartet 2/iv, mm. 1-4 21 Example 5.1b Second thematic statement in Quartet 2/iv, m. 6 21 Example 5.1c End of Theme in Quartet 2/iv, mm. 13-14 22 Example 5.2 Opening of Quartet 3/iii, mm. 1-7 22 Example 5.3a Opening of Sonata for Guitar 23 Example 5.3b K-nets in Sonata for Guitar 23 Example 5.3c Cycles in Sonata for Guitar 24

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Example 5.3d “Hyper cycles” in Sonata for Guitar 24 Example 5.4 U-cells in Berg, Op. 4, No. 2 24 Example 5.5 U-cells in Berg, Op. 2, No. 3 25

1

Chapter One

Introduction, Review of the Literature, Broad Remarks, and Chapter Overviews

I. Introduction

Since the end of the 19th Century, composers working within the tradition of

Western art music have taken remarkably diverse approaches to composition. In works of

varying style and expression, among other developments, the language of tonality

becomes one choice among many, and formal as well as rhythmic musical elements

associated with regional traditions come to the forefront. In most cases, however, the

composers and specific works that tend to be the objects of music-theoretic research, as it

has developed in the latter half of the 20th century, continue to stem from the Western

European heritage. Moreover, much of the research done by theorists in particular often

disregards the panoply of issues that arises from the diverse national and cultural

influences associated with the language of new musical forms and expression; as

theorists, our approach to research tends to radiate from a Eurocentric nucleus, given that

many of our analytic tools were developed in response to the Western canon.1 Thus,

despite this variety of culturally encoded compositions, most music theorists have

focused their analytic and theory-building efforts on a very select group of works and

composers, relying upon a limited but highly developed set of analytic tools specifically

1 The Western canon is, of course, itself a concept that has been subjected to critical reassessment in recent years. See Patrick McCreless, “Rethinking Contemporary Music Theory,” in Keeping Score: Music, Disciplinarity, Culture, eds. David Schwartz, Anahid Kassabian and Lawrence Siegel. Charlottesville: University Press of Virginia, 1997, 13-52. For broader views on decolonializing discourse on music, see Kofi Agawu, Representing African Music: Postcolonial Notes, Queries, Positions. New York and London: Routledge Press, 2003; and Malena Kuss, “Prologue” to Music in Latin America and the Caribbean, ed. Malena Kuss. Austin: University of Texas Press, 2004, ix-xxvi.

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designed to illustrate technical aspects of specific repertories. Regrettably, this focus has

resulted in the undue neglect of many worthy composers, as in the case of Argentine

composer Alberto Ginastera (1916-1983). Ginastera’s great admiration for and careful

study of the composers who tend to be the objects of study, such as Stravinsky, Bartók,

Schoenberg, Berg, Debussy, and Webern is well documented, not only by others but by

the composer himself.2 Yet only a few writers, principally among them Malena Kuss,

have availed themselves of the potential available for applying new directions in music-

theoretic research to his original approach to pitch, rhythm, and timbric organization.3

As Ginastera weaves into his music a rich web of associations, it is indeed

possible approach his works from many different perspectives. Accordingly, writers have

viewed Ginastera through a variety of lenses. One such focus views Ginastera in the

context of nationalist tendencies. Nationalist approaches tend to concentrate exclusively

on the presence or absence of folk references in the scores. For instance, Deborah

Schwartz-Kates relies on the three stylistic periods Ginastera himself formulated in

2 See Chase, Gilbert. “Alberto Ginastera: Argentine Composer,” The Musical Quarterly XLIII/4 (1957): 439-460; Wallace, David. “Alberto Ginastera: An Analysis of His Style and Techniques of Composition.” Ph.D. diss., Northwestern University, 1964; Suárez-Urtubey, Pola. Alberto Ginastera. Buenos Aires: Ediciones Culturales Argentinas, 1967, and Alberto Ginastera en cinco movimientos. Buenos Aires: Editorial Victor Lerú, 1972, Terrapon, Luc. “Gespräch mit Alberto Ginastera,” in Alberto Ginastera, ed. Friedrich Spangemacher, Bonn: Boosey & Hawkes, 1984 (original French, 1982); Tan, Lillian. “An Interview with Alberto Ginastera,” American Music Teacher XXXIII/3 (1984): 6-8; and Kuss, Malena, “Ginastera, Alberto” in Die Musik in Geschichte und Gegenwart: Allgemeine Enzyclopädie der Musik., 2nd revised edition, ed. Ludwig Finscher. Kassel: Bärenreiter Verlag; Stuttgart: Metzler Verlag, 2002, Personenteil, vol. 7, cols. 974-982. 3 An exception to this assertion of analytical neglect is to be found in the writings of Kuss, who has published numerous articles on Ginastera’s music, as will be discussed below.

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interviews published by Pola Suárez Urtubey in 1967.4 These follow a rough trajectory,

from direct references to Argentinian folk materials through a sublimation of these

elements into a mature style and an engagement with contemporary techniques. As with

other composers, however, this stylistic division and the general application of

“nationalist” arguments is problematic in its application to Ginastera’s music. Writers

such as Kuss, for instance, have cautioned against employing the term “nationalism” to

Ginastera’s music, as indeed the composer himself has.5 Although the issues surrounding

these positions and the controversies found therein are of great interest and relevance in

understanding Ginastera’s music and its place in relation to other composers, such

concerns are not the focus of this dissertation. Instead, taking a more common music-

theoretic approach, the present work limits its scope to the application of recent advances

in the study of pitch organization, in particular those stemming from the writings of

George Perle, to Ginastera’s String Quartets nos. 1 (1948) and 2 (1958, first version).6

Although the opus numbers of Ginastera’s body of works identify a total of 54

beginning with the ballet Panambí, Op. 1 (1934-37) and ending with the one-movement

4 Deborah Schwartz-Kates. “Ginastera, Alberto,” in Grove Music Online. Oxford Music Online, http://www.oxfordmusiconline.com/subscriber/article/grove/music/11159 (accessed March 29, 2009). 5 In Kuss 2002, specific reference is made to Ginastera’s rejection of the application of the “musical nationalist” label to his music. 6 See Perle, George, The Listening Composer. Berkeley: University of California Press, 1990; Serial Composition and Atonality. 6th ed., rev. Berkeley: University of California Press, 1991; and Twelve-Tone Tonality. 2nd ed. Berkeley: University of California Press, 1996.

http://www.oxfordmusiconline.com/subscriber/article/grove/music/11159

4

Sonata No. 3 for piano, Op. 54 (1982), 53 exist as authorized works.7 Among these

works, several emerge as significant. Foremost of these are his three operas Don Rodrigo,

Op. 31 (1964), Bomarzo, Op. 34 (1967), and Beatrix Cenci, Op. 38 (1971); the Piano

Concerto No. 1, Op. 28 (1961) and Piano Concerto No. 2, Op. 39 (1972); the Sonata for

Piano, Op. 22 (1952) and Sonata for Piano No. 2, Op. 53 (1981); the String Quartet No.

1, Op. 20 (1948), String Quartet No. 2, Op. 26 (1958, revised in 1967) and String Quartet

No. 3, Op. 40 (1973); the symphonic fresco Popol Vuh, Op. 44 (1975-1982), and the

Cantata para América mágica, Op. 27 (1960). The significance of these works stems

from their position in Ginastera’s development as a composer, and, in the case of the

String Quartet No.2 and the opera Don Rodrigo, from critically acclaimed performances

in the United States that helped shape Ginastera’s international recognition. Among these

works, the first and second string quartets suit the present purpose of exploring

Ginastera’s musical language in the context of theoretical methods developed from

studies of European composers. In response to the musical surfaces in selected

movements of the first and second string quartets, this dissertation will develop an idea

rooted in transformation theory that models a process by which a trichordal motive

unfolds systematically into larger symmetrical pitch-class constructions. My analysis of

the quartets, partially based on this process, reveals striking relationships between the

quartets, providing one type of evidence supporting Kuss’ view of Ginastera’s work as a

unified body across which multiple manifestations of musical ideas create a “network of

7 Kuss 2002 (MGG entry) provides the most comprehensive, detailed, and accurate overview of Ginastera’s life and works. Also see Kuss, Malena, Alberto Ginastera: Musikmanuskripte. Winterthur, Schweiz: Amadeus Verlag, 1990, 32 pages (Inventare der Paul Sacher Stiftung, No.8) for additional discussion the issue of authorized and unauthorized works.

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intratextual relationships” (Netz intratextueller Verbindungen) which problematizes the

categorization of the composer’s work into “style periods” as several authors have

attempted to do.8

II. Review of the Literature

In 1942 Aaron Copland brought attention to a young Ginastera in his well-known

article published in Modern Music.9 Gilbert Chase, writing in 1957, established a context

for the composers works written up to the Harp Concerto of 1956, surveying relevant

aspects of Argentina’s history and geography as well as the composer’s family,

education, and works up to 1956, with examples of his music.10 Pola Suárez Urtubey

authored two Spanish-language books (1967 and 1972) containing interviews with the

composer, published critical reactions to specific works, and brief analytical references

with musical examples.11 Malena Kuss’ numerous publications over a 40-year span begin

with an article in 1970 detailing the composer’s approach to teaching.12 Her analytic

writings on Ginastera’s works begin with her 1976 dissertation on Argentine operas at the

8 Based on the comprehensive sketch study of Ginastera’s works, Kuss 2002 recognizes within them Schoenbergs’s concept of the Entelechie relevant, a single energy or impulse whose various musical manifestations recur in new forms to create this intratextual network and identifies numerous examples of reappearing musical ideas within the composer’s oeuvre; see also Kuss 1990. 9 Copland, Aaron. “The Composers of South America.” Modern Music 19/2 (1942): 75-82. 10 Chase, Gilbert. “Alberto Ginastera: Argentine Composer,” The Musical Quarterly XLIII/4 (1957): 439-460. 11 Suárez Urtubey, Pola. Alberto Ginastera. Buenos Aires: Ediciones Culturales Argentinas, 1967 and Alberto Ginastera en cinco movimientos. Buenos Aires: Editorial Víctor Lerú, 1972. 12 Kuss, Malena. “Alberto Ginastera and the Early Training of the Composer.” Heterofonía 3/1 (Enero-Febrero 1970, 13-17 (abstracted in Melos [September 1970], 350).

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Teatro Colón (1908-1972), a work that establishes one of the major themes in her work,

namely musical structure and dramaturgy in the composer’s three operas.13 Articles on

this specific subject of opera analysis include studies of the encoding of a native idiom in

Don Rodrigo’s (1964) basic twelve-tone row and its variety of structural roles in 1980,

dramaturgic and musical structure in Bomarzo (1967) in 1984 and 2002, and a study of

the idiosyncratic assimilation of nativistic strands in (among others) Don Rodrigo in 1990

and 1992.14 Outside of the operas, Kuss has also discussed structural elements of many of

the composer’s other works, including the Second String Quartet (1958) in 1989 and

2002, the Piano Sonata No. 1 (1952) in 2002, organized the inventory of the composer’s

collected works for the Paul Sacher Stiftung in 1990, authored an extensive entry in Die

13 Ibid, Nativistic Strains in Argentine Operas Premiered at the Teatro Colón (1908-1972). Ph.D. dissertation, Historical Musicology, University of California at Los Angeles, 1976. Ann Arbor, Michigan: University Microfilms International, 76-28570, 533 pages. 14 Ibid, “Type, Derivation, and Use of Folk Idioms in Ginastera’s Don Rodrigo (1964),” Latin American Music Review 1/2 (Fall-Winter 1980), 176-196; “Symbol und Phantasie in Ginasteras Bomarzo (1967)” in Alberto Ginastera, edited by Friedrich Spangemacher. Bonn: Boosey & Hawkes, 1984, 88-102 (Series Musik der Zeit: Dokumentationen und Studien, No. 4); “Identity and change: Nativism in operas from Argentina, Brazil, and Mexico” in Musical Repercussions of 1492: Encounters in Text and Performance, edited by Carol E. Robertson. Washington, D.C.: The Smithsonian Institution Press, 1992, 299-335 (Proceedings of the symposium on Musical Repercussions of 1492, Quincentennary Program, Smithsonian Institution, 1988); “The structural role of folk elements in 20th-century art music” in Proceedings of the XIVth Congress, International Musicological Society: Transmission and Reception of Musical Culture, edited by Lorenzo Bianconi, F. Alberto Gallo, Angelo Pompilio, and Donatella Restanti. Torino: EDT/Musica, 1990, vol. III, 99-120; and “’Si quieres saber de mí, te lo dirán unas piedras’: Alberto Ginastera, autor de Bomarzo” in Ópera en España e Hispanoamérica, 2 vols., edited by Emilio Casares Rodicio and Álvaro Torrente. Madrid: ICCMU (Instituto Complutense de Ciencias Musicales), 2002, vol. II, 393-411.

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Musik in Geschichte und Gegenwart in 2002.15 The composer’s death in 1983 inspired a

spate of articles and interviews appearing in Alberto Ginastera (1984) and the Latin

American Music Review 6/1 (1985), the latter edited by Kuss. The former includes

German translations of essays by Ginastera, a definitive analysis of the Cantata para

América mágica (1960) by Hans-Werner Heister, and an interview with the composer by

Luc Terrapon, while the latter contains articles by Chase, Stuart St. Pope, Carleton

Sprague Smith, and Robert Stevenson.16 In addition, dissertations by Wallace in 1964,

Richards in 1985, and Campbell in 1991 provide analyses ranging from broad treatments

of the composer’s early work (Wallace), pitch-structure in Don Rodrigo (Richards), and

the three piano sonatas (Campbell).17 The most recent research on the composer is a book

by Antonieta Sottile in 2007.18

15 Ibid, “Alberto Ginastera,” Mitteilungen der Paul Sacher Stiftung (Basel, Switzerland), No. 2 (Januar 1989), 17-18; “Ginastera, Alberto” in Die Musik in Geschichte und Gegenwart: Allgemeine Enzyclopädie der Musik., 2nd revised edition, ed. Ludwig Finscher. Kassel: Bärenreiter Verlag; Stuttgart: Metzler Verlag, 2002, Personenteil vol. 7, cols. 974-982; “La poética referencial de Astor Piazzolla,” in Estudios sobre la obra de Astor Piazzolla, edited by Omar García Brunelli. Buenos Aires: Gourmet Musical Ediciones, 2007, 57-76 and Revista del Instituto Superior de Música (Santa Fe, Argentina, Universidad Nacional del Litoral), No. 9 (2002), 11-29; “La certidumbre de la utopía: Estrategias interpretativas para una historia musical americana,” Música (La Habana, Boletín de Casa de las Américas), Nueva Época, No. 4 (2000), 4-24; and Alberto Ginastera: Musikmanuskripte. Winterthur, Schweiz: Amadeus Verlag, 1990, 32 pages (Inventare der Paul Sacher Stiftung, No.8). 16 See Alberto Ginastera, edited by Friedrich Spangemacher. Bonn: Boosey & Hawkes, 1984, 88-102 (Series Musik der Zeit: Dokumentationen und Studien, No. 4). 17 Wallace, David. “Alberto Ginastera: An Analysis of His Style and Techniques of Composition.” Ph.D. diss., Northwestern University, 1964; Richards, James Edward, Jr. “Pitch Structure in the Opera Don Rodrigo.” Ph.D. diss., University of Rochester, 1985; and Campbell, Grace M. “Evolution, Symmetrization, and Synthesis: The Piano Sonatas of Alberto Ginastera.” D.M.A. diss., University of North Texas, 1991 (advised by Kuss). 18 Sottile, Antonieta. Alberto Ginastera: Le(s) style(s) d’un compositeur argentin. Paris: L’Harmattan, 2007.

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III. Broad Remarks

As mentioned previously, this dissertation is narrowly focused to analyses of

Ginastera’s first and second string quartets and their engagement with the European

tradition.19 Despite the pivotal roles the first and second string quartets play in

understanding the evolution of Ginastera’s compositional career, little analytic work has

been done on these quartets outside of Kuss 1989 and 2002. In published works, only five

authors have engaged the quartets from an analytic perspective, but in most cases, the

analyses are used in broad discussions of large issues such as musical style, nationalism

and 12-tone techniques.20 Of these authors and works, only Wallace 1964 addresses

specific issues of form, pitch and rhythm that arise in the quartets as autonomous works.

Although the First String Quartet (1948) and the Second String Quartet (1958)

are ten years apart, the works share many striking similarities. Both quartets reflect

neoclassic approaches to form, texture, phrasing, and tonal centricity. Formally, the

quartets contain four and five movements respectively, employing thematic sonata

forms21 (first movement of both quartets), three-part Rondo forms (ABA in movements

two and three of the first quartet and movements two and four of the second), five-part

19 Two versions of the second quartet exist. The earlier version (1958) is the one this dissertation will address. The revised version (1967) omits both a reference to a folk rhythm in its third movement and a quotation from “Triste” in the fourth movement of the composer’s Cinco canciones populares argentines, Op. 10 (1943). These changes were first introduced in the Concerto per corde, Op. 33 (1965), an arrangement of the last four movements of the quartet for string orchestra (see Kuss 2002). 20 See Chase 1957, pp. 441, 450-451; Wallace 1964, pp. 144-152, 219-245; Suárez Urtubey 1967, pp. 27-29 and 1972, pp. 44-46, 58-60, pp. 875 and 877; and Kuss 2002, cols. 976, 980. Parts of the second and third movements of the second quartet are 12-tone. Chapter four of this dissertation analyzes these movements in detail. 21 In the present case, “thematic” sonata form refers to the use of two contrasting themes within a three-part, exposition-development-recapitulation rhetorical plan. Unlike traditional sonata forms, however, the first movements do not feature the transposition of exposition themes in the recapitulation.

9

Rondo forms (ABABA in movement four of the first quartet and ABACA in movement

three of the second quartet), and Theme and Variations (movement four of the second

quartet).22 Texturally, the pieces most often employ melody and accompaniment,

imitative counterpoint, and pointillistic approaches.

In addition to broad ideas of form and texture, issues of pitch are of paramount

importance in the quartets. Significant pitches generally appear with salience on the

musical surface as extended pedal points, registral boundaries, points of imitation or

initiation of themes and other significant constructions; a study of the various analytic

contexts surrounding the salient pitch (and more generally pitch-class) appearance is the

central aim of this dissertation.23

One such analytic context is found in the frequent appearance of salient pcs as

subsets of familiar referential collections.24 A quintessential example of this type of

context appears in the first two measures of the first movement of the String Quartet No.

1, in which the pc D, the central pc of the movement, initiates a six eighth-note octatonic

22 Kuss 2002 in particular notes the connection between the second quartet and Bartók’s Fourth String Quartet (1928) and Berg’s Lyric Suite (1926). The connection to the Lyric Suite is also made explicit in Irving Lowens’ article “Current Chronicle,” MQ 44 (1958), p. 378. Also see Ginastera, “Hommage à Bartók” and Terrapon 1982/4 for references to other composers Ginastera admired. 23 A list of ten criteria for salience in atonal music appear in Lerdahl, Fred. “Atonal Prolongational Structure,” Contemporary Music Review 4 (1989): 64-87. Lerdahl’s study determines that an event is salient if it is: a) attacked within a particular time span, b) in a relatively strong metrical position, c) relatively loud, d) relatively prominent timbrally, e) in an extreme registral position, f) relatively dense, g) relatively long in duration, h) relatively important motivically, i) next to a relatively large grouping boundary, and j) parallel to a choice made elsewhere in the analysis. This dissertation does not rigorously reconcile the musical surfaces studied with these exact criteria, but its claims regarding the salience of the events comport comfortably with them. These criteria are tested and refined in Dibben, Nicola. “The Perception of Structural Stability in Atonal Music: The Influence of Salience, Stability, Horizontal Motion, Pitch Commonality, and Dissonance,” Music Perception 16/3 (1999): 265-294. 24 These include pentatonic, diatonic, hexatonic, octatonic, and whole-tone collections.

10

gesture that unfolds from what might be termed a three-note Grundgestalt <D-F#-F>.25

Several authors above refer to various pc collections, establishing a precedent for delving

into the issue of octatonicism in the composer’s music.26 Beginning with Wallace 1964,

some of these authors refer to the multiple salient appearances of different combinations

of major and minor triads in a larger sample of pieces as “polytonality.”27 Since the mid-

1980s, but formulating it in print in 2002, Kuss has described Ginastera’s pre-12-tone

pitch language as “octatonic with modal interaction.”28 The present dissertation develops

the issue of the interaction between motive and collection in terms of interval cycles

(cyclic pc collections) that often result from a systematic transformation of motives.29

25 This example illustrates the multifaceted role D plays in the movement, as D 1) establishes itself as a focal pc, 2) initiates the preeminent motive of the movement, and 3) locates that motive within a familiar pc collection. 26 See Wallace 1964, Kuss 1980 and 2002; Heister, “Trauer eines Halbkontinents und Vergegenwärtigung von Geschichte: Ginasteras ‘Cantata para América mágica, Op. 27,” in Alberto Ginastera, edited by Friedrich Spangemacher. Bonn: Boosey & Hawkes, 1984, 45-75 (Series Musik der Zeit: Dokumentationen und Studien, No. 4); Campbell, Grace M. “Evolution, Symmetrization, and Synthesis: The Piano Sonatas of Alberto Ginastera,” D.M.A. diss., University of North Texas, 1991; and Tabor, Michelle, “Alberto Ginastera’s Late Instrumental Style,” Latin American Music Review 15/1 (1994): 1-31. 27 See Berger, Arthur. “Problems of Pitch Organization in Stravinsky,” Perspectives of New Music 2/1 (1963): 11-42; and Lendvai, Ernö. Béla Bartók: an analysis of his music. London: Kahn & Averill, 1973 for seminal music-theoretic work on the interactions of diatonic and octatonic collections in Bartók’s music. 28 See “oktatonisch mit modaler Interaktion,” col. 979. Although Kuss’ use of the term in this case is technically limited to the Piano Sonata (1952), the term is generally applicable to a noteworthy amount of Ginastera’s music. The original concept of octatonic-diatonic interaction was first introduced in such a form by Pieter C.van den Toorn in The Music of Igor Stravinsky. New Haven, Connecticut: Yale University Press, 1983. 29 Composer and musicologist George Perle was the first to develop a theory based on interval cycles (see Perle 1962, 1977, and 1990 for a comprehensive exposition of interval cycles). Also see Antokoletz, Elliott. The Music of Béla Bartók. Berkeley: University of California Press, 1984, and Headlam, Dave. The Music of Alban Berg. New Haven: Yale University Press, 1996.

11

Salient pitches, pcs, collections and motives are often concrete manifestations of

the more general idea of patterning in Ginastera’s music. The use of patterns of notes

and/or rhythms is a hallmark of the composer’s style and can be found readily in many of

the composer’s works. In the present quartets, the aforementioned salient musical

features are in many cases manifestations of patterns, and an understanding of the

mechanism behind the patterns helps to provide a more complete view of a major source

of consistency between the quartets. In its analysis of the quartets, this dissertation offers

one possible transformational model of the above unfolding process, the pattern’s source,

which links salient pc and motive to referential collection.

The use of cyclic, hence symmetrical, pitch and pc collections invites inquiry into

potential ways various types of symmetry manifest themselves on the musical surface and

structure. However, the music-theoretic literature on symmetry within the context of

music analysis is far too vast to discuss presently. In literature on Ginastera’s work,

Chase 1957 discusses formal symmetry, as does Suárez Urtubey 1967, Kuss 1980, 1984,

2000 and 2002, and Heister 1984; Kuss 1980, 1984, 1989, and 2002, and Campbell 1991

discuss symmetry in pitch. The present dissertation discusses various formal and

pitch/pc-based symmetries in the quartets. The latter receives special attention due to the

ubiquity of interval cycles in the music.

Much like the presence of interval cycles in music invites symmetry-based

analytic approaches, the presence of formal symmetry in both the scholarly literature and

the music on which that literature focuses invites a more detailed investigation into any

role symmetry may play in the music’s rhetoric and form. In its approach of this subject,

the present dissertation develops a simple concept that incorporates the above notion of

12

mechanism into a small-scale formal apparatus that comports with aspects of pitch/pc

language and hypermeter. Specifically, the device provides a facile vocabulary for

discussing the issue of seemingly discontinuous rhetorical elements and their relationship

to the pitch/pc element of the music.

Despite various attempts over the years to divide Ginastera’s compositional

output into labeled style periods, the picture that emerges from the present study of the

first two quartets is one of tremendous continuity that transcends the traditional labels

often applied to the composer’s music.

IV. Chapter Overviews

To accomplish the goals stated above, the dissertation offers five chapters. The

present chapter provides a cursory overview of the composer’s works and summarizes

some of the scholarly community’s reception of them. In this initial Chapter One, we

have learned about Ginastera scholarship, significant broad aspects of the composer’s

style, the pivotal roles the first two string quartets play in stylistic assessments of the

composer’s works, and the present author’s basic intuitions about the musical contents of

the quartets. Chapter Two discusses the basic methodology behind the present approach,

reviewing applicable existing analytic approaches and providing several original

contributions stemming from the present author’s experience with the quartets. Chapter

Three presents detailed analytic commentary on the first movements of the quartets based

on the new methodology discussed in Chapter Two, while Chapter Four engages in a

more traditional cyclic analysis of two internal movements of the latter quartet. Finally,

13

Chapter Five offers conclusions based upon the previous chapters and proposes several

avenues for further research.

14

Chapter Two

Methodology

As discussed in the preceding chapter, Ginastera’s First and Second String

Quartets play significant roles in his oeuvre in a number of contexts. This chapter

presents a methodology for the detailed analyses to follow in chapters 3 and 4. The main

features of the music to be examined are its melodic and rhythmic patterns, symmetry,

and overt repetition of focal pitches. In addition, the relationship of motivic elements to

larger octatonic and other symmetrical collections will be articulated. Building upon

well-established ways to relate individual pitch-classes (pcs) to small cyclic and non-

cyclic pitch-class sets (pcsets as articulated in pcset theory, transformation theory, and

cyclic theory), the methodology introduces seven analytic/interpretive devices which

highlight the roles of the above features in specific ways, allowing the idiosyncrasies of

the quartets to participate meaningfully in this interpretation of their musical processes.30

These seven devices fall into three broad categories. The first is based on pc- and

pc-collections and includes definitions of “dyad-space,” as well as “coloring” and

“shading” basic collections. The second stems from transformational and interpretive

devices which demonstrate order and invariant relationships, given below as K(nm), U(nm),

and J00. Finally, the third emerges from formal/rhetorical devices described as “tiles” and

“crosscuts.” The seven devices in these categories combine to present a more

comprehensive picture of the musical processes in the first two quartets.

30 Large-scale representatives of these approaches are contained in Forte 1973, Lewin 1987, and Perle 1962, 1977 and 1990.

15

I. Pitches, Pitch-classes and Pitch-Class Collections

[Examples 2.1a, b and c]

Each of the nine movements of the first two quartets significantly incorporates the

ensemble’s open strings, either as a referential collection or specified subset.31 Example

2.1a illustrates the basic 7-cycle pentatonic collection {CDEGA} of the string quartet’s

component instruments; each instrument contributes a conjunct subset of the overall

collection. Contextual emphasis on a specific conjunct subset to the overall collection

will be referred to as a “shading” of the basic collection. For example, the leftmost arrow

in Example 2.1a indicates a cello/viola “shading” of the basic collection, emphasizing the

open string tetrachord {CGDA} within the overall pentachord. The rightmost depicts a

violin shading {GDAE}. Example 2.1b shows two different shadings of the overall

collection focused on conjunct trichords {CDG} and {DAE}. Example 2.1c provides an

example of how such shadings could be represented on a musical surface, showing a

passage from the first theme from the Second String Quartet in which the sonority on the

downbeats of mm. 16, 18 and 20 is considered a {CGD} trichordal shading of the OS

set.32 Owing to the ubiquity of such collections in the music, the notion of shading proves

useful in discussing connections between formal and tonal components of the musical

surface.

[Examples 2.2a and b]

Within collections associated with shading, pairs of pcs may emerge as structural

units; these pairs also extend beyond the shading note groups. Any pair of pcs interpreted

31 This is reminiscent of Berg’s Violin Concerto, which is discussed at length in Headlam 1996. 32 The shaded subset can be merely a pc (and not necessarily pitch) subset. In addition, shading does not have to be associated with a specific instrument.

16

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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as either a transposition or an inversion could be considered to establish or participate in

an established “dyad-space.” In general, a dyad-space involving pcs q and r is formally

defined as Tn/In(q) = r (and expressed by the argument q/r-(Tn/In)), meaning pcs q and r

are Tn- or In-related. Thus, a dyad-space is considered to be a transformational space

where the path from one pc to the other is an active transformation, rather than viewing

the dyad as a harmonic entity. An example of a dyad-space is given in Examples 2.2a and

b; Example 2.2a provides the score of the violin I melody from mm. 5-16 of the first

movement of the Second Quartet, while Example 2.2b depicts a reduction of the part.33

Pitch-classes G and D, the boundary pcs of the passage, are familiar from the previous

figure as members of the open string pc collection {CDEGA}. The two networks below

Example 2.2b provide two transformational interpretations of the passage. The leftmost

network interprets the reduction with two transpositions and two inversions, while the

rightmost interprets it as four transpositions.34 In the present context, salient pcs G and D

form two dyad-spaces, G/D-(I9) and G/D-(T7) depending upon the preferred

interpretation of the passage. Herein lies the advantage of the concept of the dyad-space:

various musical situations could highlight these two pcs in different ways. Interpreting

them as dyad-spaces with specific components (pcs and a transformation) facilitates the

many comparisons of a limited number of privileged pcs and/or transformations the

music frequently features.

Examples 2.2a and b also provide an illustration of the “coloring” of the

fundamental 7-cycle with a 6-cycle. As a member of set-class [0167], pc set <G-Ab-Db-

33 This passage is associated with the First Theme of the movement’s sonata form. 34 These are indeed well-formed K-net and L-net interpretations, but this fact is not germane to the present discussion since the networks are not included in a larger network demonstrating hyper-transformations.

18

D> exhibits both 5-/7-cycle and 6-cycle properties. Since G and D recur throughout the

quartets as privileged members of the fundamental open string collection while Ab and

Db do not, it becomes advantageous to refer to the pc set as projecting a primary 7-cycle

<G-D> colored by secondary 6-cycles, since one cycle is central to the quartets’ tonal

language and the other is not, thereby implying a relative hierarchic superiority of the

dyad <G-D>. The quartets very commonly feature passages in which multiple cycles and

cyclic processes occur simultaneously, and the ability to characterize these passages

accurately and easily is of value.

II. Contextual Transformations K(nm) and U(nm)


The contextual transformations K(nm) and U(nm) are I and RI operators performed

on ordered trichords. Let both n and m be one of three order-positions f(0), s(1), or t(2); f,

s, and t stand for the first, second, and third terms in an ordered trichord <fst> called a

“source trichord.” For example, let pcset <265> be the source trichord for a particular

passage with f=2, s=6, and t=5. K(ff) performed on <265> inverts it via the inversion

operator that maps the first pc, in this case 2, into itself. This inversional index (in this

case I4) is then applied to the entire trichord, yielding <2te> (sum 4). Thus,

K(ff)(<265>)=<2te>.35 The K transformation can be applied to ff, fs, ft, ss, st and tt for a

35 As defined, K(ff) is limited to ordered trichords. The label J00 will be used for the same essential operation (inversion mapping the first element of an ordered pc set onto itself) performed on larger sets. For example, the tone-row <05e2819t7436> (the first row in the third movement of the second quartet) is J00-related to <071t4e325896>. Thus, in the present case, J00 is I0.

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20

total of six possible transformations. The six K-transformations for the trichord <265>

are given in Example 2.3a.

Since K(nm) is an inversion operation, it is an involution; successive applications

of the identical transformation will toggle back and forth between the domain and its

range. The last column (I index) gives the familiar operation that would produce the

identical range for each K-transform. Again, since K/U-transforms are based on order-

positions in a source trichord, the In index would change given a different source trichord.

U(nm) works similarly, except it generates a retrograde-inversion of the given

source trichord. For example, U(st) performed on source trichord <265> inverts it via the

inversion operation that maps the second term, in this case 6, into the third term, in this

case 5. Thus, U(st) in this case performs RIe on <265>, yielding <659>.36 Since U(nm)

generates a retrograde-inversion of the source trichord, multiple applications may not

result in mere back-and-forth toggling between two pc sets. Example 2.3b provides the

same information for U-transforms as Example 2.3a gave for K-transforms, but it

incorporates the necessary concept of a p-value, which reflects the number of subsequent

identical transformations (the “order” of the transformation) required to return to the

original trichord.37 For instance, U(ff) on <265> requires four transformations to return

<265> and thus has an order of 4: <265> to <et2> (inversional sum 4 with a retrograde),

<et2> to <80e> (inversional sum t with a retrograde), <80e> to <548> (inversional sum 4

with a retrograde), and finally <548> to <265> (inversional sum t with a retrograde). The

considerable difference in the amount of information contained in both tables reflects the

36 Herein lies an intimate connection with Lewin’s RICH transformation. See Generalized Musical Intervals and Transformations, pp.180-181. 37 The notational convention in the leftmost column is intended to parallel mathematical exponents; p stands for “power.”

21

appreciable effect of retrogression in U-transforms. Despite these differences, however,

one similarity persists: like all K-transforms, U(ft) and U(ss) are involutions (order 2)

regardless of the pc membership of the source trichord. However, the remaining four

transformations are not involutions, and their order is dependent upon the source

trichord’s pc membership. For example, given source trichord <265>, U(ff) and U(tt) return

to the original source trichord <265> after four transformations, or are of order 4 ((U(ff))4

and/or (U(tt))4 of <265> is <265>) while U(ft) and U(st) are of order 8.38

[Example 2.4]

Looking at the RIn indices in Example 2.3b, we notice the cyclic nature of the “n”

values: RI4/t in a 6-cycle, T/I8/5/2/e in a 9- or 3-cycle. We also notice a symmetry, where

U(st) has the retrograde of the “n” values of U(fs), and U(tt) has the retrograde U(ff). Aside

from their symmetry, the “n” values are in interval cycles, and thus examples 2.3a and b

readily underscore the great relevance of interval cycles to a passage of music modeled

by K/U-transformations, especially U-transforms with p-values greater than 1. Example

2.4 provides such a passage, taken from the first two measures of the first quartet. Here

the initial forceful gesture is <D-F#-F-A-Ab> in violin I and cello, while violin II and

viola punctuate the 3-cycle <D-F-Ab>. The addition of violin I’s C5 on the downbeat of

m.2 finishes the gesture and extends the leftover dyad <F#-A> into another 3-cycle <F#-

A-C>. The held chord in m.2 is comprised entirely of the notes from m.1 (plus the violin

C), symmetrically registered <D2-A2-F#3-Ab3-F4-C5> around G3. In addition to its

cyclic interpretation as two partial 3-cycle collections {D,F,Ab} and {F#,A,C}, the chord

could also be considered as interlocking D major and F minor triads, an interpretation

38 A more detailed account of each step of the six U-transformations appears after in Appendix One, which appears after Chapter 5.

22

bolstered by the conspicuous placement of the D major triad (with the P5 D2-A2) in the

lower voices and F minor triad (with the P5 F4-C5) in the upper. Despite these tonal

readings, however, the current methodology sheds light on the generative process lurking

beneath the passage; indeed, the theme of surface convention and underlying generation

will recur throughout this dissertation. Figures 2.3a and b illustrate each step of the

process.

[Examples 2.5a, b, and c]

Example 2.5a depicts the initial three-note gesture <D-F#-F> as a pc motive

<265> that is transformed first into <659>, and then <598> and <980> by RIe, RI2, and

RI5, respectively, providing the individual “frames” in a larger “film” showing <265>

transforming into <980> via RI2. This succession of imbricated trichords extends one

pitch-class at a time. Ultimately, the film created by the chaining of individual frames is,

in a general sense, well-formed; each successive frame is generated by the same

homomorphism.

The passage is modeled in a more organic sense in Example 2.5b, which

interprets the same passage as three individual U(st) transformations ((U(st))3).39 The

source trichord is indicated on the bottom left of the figure, the dotted box highlights the

distinct four-note U-cell, and the dashed box highlights the six-note U-chain. Below the

figure appears a chart that provides the p-value (the order, or the number of times the

transformation is applied), the nomenclature for the pc set that successive transformations

39 The use of the term organic is intended to convey that the operations that create U-cells and U-chains from a source trichord derive from the trichord itself; a property of the source trichord determines the transformational path of the chain. In the example, U(st) is chosen because it is the only one of the twelve contextual transformations that replicates the int<4e4e4> in the chain.

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24

generate, followed by the respective pc sets, Tn-types, and Tn/TnI-types. Example 2.5c

depicts how the objects appear in subsequent examples or prose. For example, the whole

incipit can now be meaningfully discussed as the U-chain <265>-<980>, a pc

construction made by unfolding the first term, source trichord <265>, into the second

term, final trichord <980> by means of U(st)3.

The network of Example 2.5b demonstrates two of the basic advantages of

contextual K-/U-transformations. First, they locate the origin of an ordered trichord’s

transformational path within the trichord itself. Thus, inversional sums become less

abstract and thereby enhance their role in a piece of music that unequivocally

demonstrates a clear context for inversional symmetry.40 Second, by utilizing only one

transformation, they underscore the potential for maximal transformational efficiency.

As the above discussion demonstrates, K-/U-transformations systematically

unfold IS pcsets from a single ordered trichord, and each unfolded set is a member of the

same set-class as the source trichord. However, this dissertation will distinguish between

I-related Tn-types, thus the source trichord and its first K-/U-transform are considered as

members of Tn-types [034] ({256}) and [014] ({569}). The union of a source trichord

and its first K/U-transform constitute a “K-cell” or “U-cell” (p=1, see Table 2.1). In the

above example, the tetrachord <2659>, the first four notes of the gesture, is a U-cell

<256>-<9>.41 The cardinality of K/U-cells is defined by numbers of distinct notes, and

ranges from three (all common tones between the source trichord and its K/U-transform)

40 The analyses in Chapters 3 and 4 attest to the context for inversional symmetry in Ginastera’s first and second quartets. 41 Technically, they form a “U(st)-cell,” but the informal name suffices for prose. Initially, “U” stood for “union.” Cells of this particular type are especially germane to Ginastera’s quartets, thus the remainder of the discussion will focus on this type exclusively.

25

to five (one common tone) depending on the particularities of the source trichord and its

first transform. In the above example, U(st) mapped <265> into <659>, but the common

tones F# and F were not articulated a second time.42 A second transformation results in

an “extended U-cell” (<265>-<98>, p=2), and a third (see Figure 2.1b) yields a “U-

chain” (<265>-<980>, p=3) in which each member of the source trichord has been

mapped once (9=(6+5)-2, 8=(5+9)-6, and 0=(9+8)-5). Further extensions of the chain (p

= or >4) are “extended

U-chains.”

The cyclic nature of U-cells, extended U-cells, U-chains and extended U-chains

causes them to produce familiar symmetrical pc collections. For example, the chromatic

scale (1-cycle) is an extended U-chain on a source trichord <pc, pc+1, pc+2>; both

whole-tone scales (2-cycles) are U-chains of source trichord <pc, pc+2, pc+4>, the three

3-cycle collections are U-cells of source trichord <pc, pc+3, pc+6>, etc.43 The string

quartet’s open string collection itself is an extended U-chain of source trichord <CGD>.

In the abstract, such “purely cyclic” collections are best considered as T-cycles, although

it is possible to consider such collections in a piece of music as U-transformations given

sufficient context through the overt presence of IS in p-space, a cyclic motive, or some

connection or pattern in the (R)In indices implied by significant pairs of pcs. However,

42 Strictly speaking, the “real” transformation in this case is, given <fst>, let the next note be (s+t)-f mod12. Each of the twelve K(nm)/U(nm) transforms can be defined similarly. 43 The initial ideas on cycles are explicitly discussed by Perle, but the present discussion has more direct roots in Headlam 1996, pp.14-17. Collections comprised of aligned interval cycles, such as the octatonic and hexatonic, are formed by interpolating another pc from outside the basic cycle, as seen with the trichord <265> in Example 2.3b.

26

non-symmetrical source trichords produce other familiar IS collections in surprising

ways.44

[Example 2.6]

Contextual K-/U-transformations can also be intimately linked with familiar

cyclic pc collections such as the octatonic [0134679t] and hexatonic [014589]. Example

2.6 depicts two such cases in which a subtle re-ordering of a source trichord’s elements

generates different pc collections. In the top example, source trichord <265> generates

the C/D octatonic collection as an extended U-chain made by five U(st) transformations

(imbricated trichords <265>-<659>-<598>-<980>-<80e>-<0e3>); the bottom example

re-orders the source trichord from <265> to <256> and generates the C#/D hexatonic

collection as a U-chain via three U(st) transformations (imbricated trichords <256>-

<569>-<69t>-<9t1>).45 Thus, a link between cycles and U-transforms is established: the

interval from f to t in a source trichord determines the fundamental cyclic bent of the pc

collection its (extended) U-chain generates. In the example, the equations “t-f” appearing

below the identification of the common collections indicate that the intervals from the

first to the third elements in the source trichords are 3 and 4 respectively. These intervals

in turn give rise to the 3-cycle octatonic and 4-cycle hexatonic collections.


The chromatic, whole-tone, octatonic and hexatonic are not the only familiar

collections that unfold via U-transforms. Extended U-chains from source trichords of Tn-

44 For example, the pentatonic collection {CDEGA} can be generated as an extended U-cell (U(st)2) of source trichords <CGD> ([027]) or <EGA> ([025]); the former yields the ordered collection <C-G-D-A-E>, while the latter yields <E-G-A-C-D>. 45 K-/U-transformations cannot link the two source trichords, as the latter is r2R0 of the former.

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28

types [025] and [035] eventually unfold the pc aggregate, but the initial extensions of the

given source trichords can unfold various pentatonic 5-cycle chords, given that the

trichord’s ic5 appears in f and t. Example 2.7a illustrates the unfolding of source trichord

<247> (Tn [025]) via U-transform into U-cell <247>-<9> ([0257]), and extended U-cell

<247>-<90> (pentatonic [02479]). At this point in the chain, the next transformation

yields trichord <902>, which returns pc 2 before continuing on to trichord <025> and

completing the diatonic hexachord [024579]. Thus, the unfolding has created a multi-set

U-chain <247>-<902> (still [02479]) on its way to a multi-set extended U-chain (and

diatonic hexachord [024579]) <247>-<9025>. Example 2.7b depicts the unfolding

extended U-cell, with some tonal labels, from source trichords <269> (D major) and

<259> (D minor) and interprets these unfoldings in staff notation and provides common

nomenclature for the extended tertian sonorities they create (up to seven members).

III. Tiles

In general, much of the quartets’ musical material is composite in nature, often

resembling a type of musical mosaic in which (roughly) 4-measure blocks of one type of

gesture dominates the musical surface. These concatenated blocks are hereafter referred

to as “tiles,” in deference to the overall aesthetic associated with visual and material art

bearing the moniker “mosaic.”46 Generally, a tile is roughly analogous to a musical

phrase due to its size and focus on one particular musical aspect, but the inherent

46 The meaning of the term “mosaic” here is not to be confused with that associated with Martino (1961), Morris and Alegant (1998) and Mead (1998). In general, the term connotes a series of discrete events in the present context, and while I will be discussing 12-tone rows here, I will not utilize the aggregate-partitioning methodology and terminology of the above authors.

29

implications of the term “phrase” complicate its broad application to many of the musical

situations that arise in the quartets. Ultimately, the definition of a tile is intentionally

flexible.


Examples 2.8a, b and c each present four-measure tiles from the third movement

of the second quartet. Tile 1 (Example 2.8a) presents the first complete iteration (mm. 16-

20) of the movement’s theme, which is in essence two statements of the movement’s first

row (3-R1) as a stream of staccato eighth notes orchestrating row A’s discrete trichords

in each instrument. The same description applies to the musical surfaces of Tiles 2

(Example 2.8b) and 7 (Example 2.8c); only the row-forms change. The uniformity in the

presentation of each four-measure unit challenges the efficacy of the term “phrase” for

each, yet each maintains an individual identity within the overall formal section. Indeed,

it is possible to discuss these groups of measures without labeling them as tiles. However,

the concept will greatly streamline such discussions.

IV. Crosscuts

[Example 2.9]

In addition to tiles, the quartets contain an intriguing musical analogue to a film

editing technique, which also has its roots in Stravinsky’s music.47 Crosscutting is a

technique in film editing which describes the “[alternation] of [movie] shots from one

47 See Cone, Edward T. Musical Form and Musical Performance, Perspectives on Schoenberg and Stravinsky, ed. Benjamin Boretz and Edward T. Cone. Princeton: Princeton University Press, 1968, for a relevant discussion of stratification, interlock, and synthesis.

30

line of action in one place with shots of other events in other places.”48 In film studies

and filmmaking, it is a narrative device designed to “tie together…different lines of

action,” allowing spatially distinct events to exist together within one brief time span.49

The quartets occasionally yet significantly feature a musical analogue to the technique.50

Example 2.9 presents a six-measure passage from the Second quartet in which two

distinct gestures are temporally linked like the different lines of action discussed above.

The first gesture, appearing in mm.32-33 and 35-36, is referred to beneath the example as

“downbeat chords” in each of the instruments. The second appears in the rising canonic

figure in all voices in m. 34 and m. 37.

The networks appearing below Example 2.9 show two distinct processes at work

in the passage. The upper network interprets the downbeat chords as four members of

[0347] articulated as two pairs of T6-related tetrachords in dashed boxes occurring on

beats one and two of the aforementioned measures; the T7 arrow interprets the

relationship between the pairs.51 The lower network models the two canonic crosscuts,

again in two pairs, as four U-chains of source trichords <69e>, <t13> (twice), and <257>,

members of Tn [035] unfolding diatonic hexachords [024579]. Each pair of U-chains,

again appearing in dashed boxes, is T4-related, while the pairs themselves are T8-related.

Thus, the two distinct gestures feature their own transformational process; the downbeat

48 Bordwell, David and Thompson, Kristin. Film Art: An Introduction, 5th ed. McGraw-Hill, 1997, pp. 297-298. Significant examples of early uses of the technique occur in D.W. Griffith’s The Battle at Elderbrush Gulch (1913) and Sergei Eisenstein’s Battleship Potenmkin (1925). 49 Ibid. 50 This temporal juxtaposition of material from seemingly disparate sources is reminiscent of the discussion of “event time” versus “gestural time” in Kramer, Jonathan. “Questions of Time in the Music of Beethoven,” Perspectives of New Music XI/2 (1973): [pages]. 51 The two pairs of [0347]s result in C/C# and C#/D octatonic collections.

31

chords of [0347] not only incorporate interval 7s in each, but also reflect a 7-cycle in

their transpositional index, while the U-chains are based on a 4-cycle. The initiating pc of

each U-chain, F#, Bb (twice) and D, extends the pronounced influence of these three pcs

established in the immediately preceding section and on a larger level, reflect the coloring

of a fundamental 7-cycle with a 4-cycle.

Ultimately, crosscuts are local phenomena and do not significantly impact large-

scale formal design, nor do they appear in every movement this dissertation discusses.

Rather, they are a technique the composer uses to temporarily interrupt a musical flow

with either related or contrasting material. Crosscuts enrich the musical passage in which

they appear in diverse ways, often through the development of a predominant motive, the

introduction of a new motive or compositional technique (such as an inverted canon), or

the invigoration of a static ostinato with elements of surprise.

V. Overview of Main Rhythmic Profiles

In general, surface rhythms engage the notated meters in the nine movements of

the quartets, demonstrating a high degree of consistency through their frequent

employment of motives and ostinati. In a very broad sense, three basic rhythmic profiles

emerge: 1) uniform, in which the predominant rhythmic figures reinforce the notated

meter, 2) free contrapuntal, in which the figures are in tension with the meter, and 3)

rhapsodic, where the figures negate or suspend the meter. Uniform rhythmic profiles,

which dominate movements 1, 2 and 4 of the First Quartet and movements 1, 3 and 5 of

the Second Quartet, can be characterized as textures in which rhythmic and melodic

events unfold with temporal regularity. In such textures, the predominant rhythmic and

32

melodic events such as accompanimental or motivic ostinati and canonic thematic

entrances generally confirm the notated meter. Free contrapuntal textures, which appear

in movement 3 of the First Quartet and movement 2 of the Second Quartet, feature

significant thematic and motivic material that neither contradicts nor overtly supports the

notated meter. Rhapsodic rhythmic profiles, which occur significantly in movements 2, 3

and 4 of the Second Quartet, are the freest of the three and feature solo passages, either

alone or accompanied, that temporarily suspend the notated meter. The passage in

Example 2.9 depicts a uniform rhythmic profile, employing pronounced chords on both

downbeats of the 6/8 meter in mm. 32-33 and 35-36; the crosscuts in mm. 34 and 37,

while representing a clear interruption of the prevailing chordal punctuations, maintain

the overall rhythmic flow of the excerpt.

33

Chapter Three: Dyad-Spaces, U-transformations and Cycles in the First Movements of Quartets One and Two

Despite the divergent opinions of scholars regarding the respective roles the first

and second quartets play in Ginastera’s compositional career, the pieces themselves

present remarkably similar musical surfaces and structures. The respective first

movements in particular appear more alike than any other pair of movements. Both are,

in essence, thematic sonata forms that feature traditionally aggressive First Themes and

placid Second Themes. Both quartets present salient, central pitch-classes (pcs) as

touchstones that structure the movements through their pairing with other pcs in “dyad-

spaces.” In addition to their role in these spaces, the salient pcs also engage thematic

elements as initiation points of U-cells, U-chains, wedges and other symmetric

formations. These salient pcs also play fundamental roles in the harmonic dimension of

the movements, figuring prominently as members of interval cycles and familiar cyclic

pc collections (diatonic/pentatonic, octatonic, hexatonic and whole-tone) that appear with

regularity on the musical surface. The present chapter analyzes the first movements of

both quartets within the context of these essential constructs, juxtaposing specific

passages from both movements to investigate the various how pcs interact with

formal/rhetoric-based elements in an effort to understand and answer the many intriguing

questions the movements engage.

[Tables 3.1, 3.2]

34



35



36

In a broad sense, the thematic sonata form of both first movements is articulated

in part by salient presentations of each movement’s central pcs, D (first quartet) and G

(second quartet). Tables 3.1 and 3.2 provide form charts for both movements

respectively, detailing specific measure numbers for both the large overarching formal

features of exposition, development, recapitulation, etc., and the smaller formal features

of individual tiles within the larger sections.52 Some of the larger sections have salient

pcs in parentheses appearing after their formal designation, indicating particular emphasis

on these pcs in conjunction with the overall central pcs. A casual observance of the

parenthetic pcs on the tables reveals that as a whole, the pcs associated with the open

strings of the string quartet as a musical ensemble are of particular importance in the

music. In general, the central pcs G and D, along with the other open string pcs, become a

referential collection for both movements; the influence of this referential collection is

not limited exclusively to the pitch and pc arenas in the pieces, as it has ramifications for

their form as well.

As a referential pc set, the open string pcset (hereafter “OS set”) {C-G-D-A-E}

contains the potential for tonal and cyclic (7-cycle) associations, and its influence

52 Tiles are defined in Chapter 2; briefly, a tile is flexibly defined as a small formal unit, roughly the size of a phrase, which expresses one basic idea. The charts also include parenthetic descriptions of the general contents of each constituent tile. For example, tile 3 in the FTA of the first quartet (Table 3.1) contains the ascent of the first main theme, tile 4 contains the descent of the theme, tile 5 contains material that reworks and intensifies previous material, etc. On occasion, the qualifying parentheses contain numbers referring to previous tiles. For example, tile 15, which appears in the recapitulation of the STA in the same quartet, returns material previously heard in tile 9.

37

is unmistakable in both quartets’ first movements.53 The first movement of the first

quartet provides many instances of salient OS pcs: the establishment of fundamental pc D

in the Introduction, the FTA in both Exposition and Recapitulation, and the incorporation

of pcs A and E in the STA in the same formal sections are just a few noteworthy

examples. The second quartet also prominently features its fundamental pc G along with

other OS set pcs C and D in its FTA and pcs D, A and E in its STA, broadly projecting

{C-G-D} and {D-A-E} shadings as local referential collections. Despite both

movements’ significant reliance upon subsets of the OS set, however, the first quartet’s

movement employs the set and its subset shadings more consistently throughout the

movement than does the second quartet’s movement.

The remainder of the present chapter provides some analytic details about the first

movements of both quartets. It divides into two basic sections. The first provides a

general discussion of the large-scale harmonic (vertical) dimension of the quartets,

identifying significant dyad-spaces and demonstrating their structuring of the music.

After the broad framework established in the first section, the second section focuses on

U-transformations and their articulations of dyad-spaces in the thematic and motivic

dimensions of the pieces. The second section also incorporates significant harmonic

constructions local to the melody and motive under discussion.

53 Whether considered as a pentatonic collection, a partial diatonic collection, a 5/7-cycle collection, or a stack of conjunct T7 transformations, sonorities of this type are very common occurrences in all of Ginastera’s work and understanding the various contexts surrounding their use is essential to gaining insight into large amounts of his work. Ginastera also frequently employs the ordered pitch collection <E2-A2-D3-G3-B3-E4>, also known as the “guitar chord” (the standard-tuned guitar’s OS set), in pieces for other instruments and ensembles (see Chase 1957, Wallace 1964, Suárez Urtubey 1967 and 1972, and Kuss 1980 and 2002). The view of the close associations between Ginastera’s use of the chord and Argentine nationalism is often discussed by the above authors and the others discussed in Chapter One.

38

I. Dyad-spaces and large-scale harmony

While a particular shading of the 7-cycle OS set can suggest various relationships

among pcs on a musical surface, the interpretive aspect of a dyad-space demonstrates one

way OS subsets create harmonic structures beneath the musical surface. The introduction

of the first quartet’s movement provides a prime example of how a structural dyad-space

can emerge from an OS set shading.


Example 3.1a, from mm. 1-7 (tile 1 and the beginning of tile 2), features three rising

gestures <D-F#-F-A-Ab-C> in mm. 1, 4 and 7 punctuated by two vertical hexachords

(low to high) <D-A-F#-Ab-F-C> and <G-D-Bb-E-G#-B-(C)> in mm. 2 and 5 (C5 is

considered a non-chord tone).54 These gestures and the punctuating chords that follow

establish many of the movement’s essential features: the salience of D as the gestures’

initiation points and as the lowest cello note of the chord of m.2, the movement’s main

motive (source trichord) <D-F#-F> and its contextual transformation, the movement’s

texture and tessitura, and transformational relationships that form an essential part of the

movement’s structure. Significantly, the first chord is a verticalization of the rising

gestures orchestrated as a D major triad beneath an F minor triad, while the second chord

is a G minor triad beneath an E major triad; both strongly affirm the chord’s 7-cycle

54 Chapter Two discussed this passage in the context of U-transformations.

39

element.55 Example 3.1b isolates both chords and indicates their membership in the

octatonic subset [013479]. The latter example also highlights the lowest voices in the

cello and their pc multiset c.o.=<D-A-G-D> (see the dashed box), a tetrachord that recurs

throughout the movement as an agent of harmonic balance.56 For instance, this tetrachord

accompanies the closure of the FTA in the exposition (tile 6), recapitulation (tile 20) and

the end of the Coda (tile 25). Although the two chords are not immediately adjacent,

Example 3.1c provides two models for the voice-leading between these chords; the

leftmost diagram treats all motions from the first to the second chord as T2, while the

rightmost treats all as I4. The smoothness of the I4 voice-leading in the rightmost model

(compare the arrow lines connecting the two chords in Ex. 3.1c) makes it the more

attractive of the two interpretations. Thus, given the musical context for the “balance”

tetrachord <D-A-G-D>, the two Ds are I4-related, as are the notes A and the G. Such an

interpretation suggests that within each of the two chords, the intervals from D2 to A2 in

the first chord and G2 to D3 in the second are T7, reinforcing the “conventional wisdom”

that the open strings of the cello are T7-related and affording the balance tetrachord the

common tonal suggestiveness associated with 7-cycle sonorities. The inversional sum 4

pairing of D with itself is further supported by the music in tile 2, the first measure of

55 The use of superimposed triads here is reminiscent of Berg’s Op.2/IV, which prominently features members of [014589] expressed as concomitant E minor and Ab major triads in m. 22 and A minor and Db major triads in m. 23. 56 The abbreviation “c.o.” stands for “canonic ordering,” which, given an inversionally symmetrical pc set, locates the sum pairs in the first/fourth and second/third positions within the angle brackets customarily used for ordered sets. In the present example, c.o.=<D-A-G-D> locates the sum 4 pc pair D/D in the first and fourth slots and the other sum 4 pair A/G in the second and third. The c.o. can be generalized as follows: given pcs <wxyz>, w+z=x+y. The c.o. translates easily into a 2T/2I K-net with the intervals from w to z and x to y labeled with I-arrows and the intervals from either w to x or x to w and y to z or z to y labeled with T-arrows.

40

which (m.7) appears at the end of Example 3.1a. By featuring the salient pc D as the

origin of the pitch wedge made by inversionally-related instances of the basic gesture, the

continuation of the first several measures fortifies D’s role as the initiator of the dyad-

space D/D-(I4), itself a structural entity which defines the transformational relationships

involving the {G-D-A} shading of the OS collection.


Like the first quartet’s opening movement, the second quartet’s movement also

features a “balance tetrachord” based upon a trichordal shading of the OS set. The left of

Example 3.2a depicts the beginning of the movement’s first theme (tile 2, mm. 5-7),

while Example 3.2b presents a reduction of the entire theme (tile 2, mm. 5-19).57 The

theme itself begins as a focal OS pc G and unfolds a wedge through violin I pcs Ab

(m.7), Db (m.9) to OS pc D (m.16) and viola pcs F# (m.7), C# (m.9) to OS pc C (m.16).

The culmination of the theme occurs simultaneously with the arrival of the balance

tetrachord c.o.=<G-D-C-G>, whose shading {C-G-D} sounds on the downbeat of m.

16.58 In analogy to the pairing of D with itself at sum 4 in the previous example, the

present pairs G with itself in the dyad-space G/G-(I2), grouping the other OS pcs C and D

as a sum 2 pair within the multiset tetrachord. As is the case in the first quartet, the

activation of harmonic dyad-spaces in first theme resonates throughout the movement,

thereby extending the dyad-space’s structuring capacity throughout the entire movement.

The use of structuring dyad-spaces to create and interpret referential sonorities in both

quartets illustrates a point of consistency between the quartets and supports a recent trend

57 This theme recurs at pitch in mm. 55-97 and 302-319. 58 The details of this unfolding are discussed as resulting from U-transformations later in the present chapter.

41

in Ginastera scholarship to challenge the stringent period-based understanding of his

compositional career.59

Of the two movements in question, the earlier quartet’s movement places greater

emphasis on harmonic dyad-spaces than the later quartet. As established above, the

Introduction, important elements of the FTA, and the Coda significantly employ the

balance tetrachord <D-A-G-D>, a 7-cycle pc set structured by a single dyad-space D/D-

(I4). The movement’s remaining formal sections are largely structured by three additional

harmonic dyad-spaces: A/E-(I1) in the STA (tiles 9-11), C/G-(I7) in the recapitulation of

the FTA (tile 19), and D/Ab-(It) (tile 8) in the transition. Of the three remaining structural

dyad-spaces, two, A/E-(I1) and C/G-(I7), incorporate pairs of OS pcs, while the third,

D/Ab-(It), does not. The following paragraphs present additional analytic details

regarding the musical context in which these harmonic dyad-spaces function.


In addition to possessing distinct themes, the earlier quartet’s first movement

generally features a distinct OS set shading and dyad-space. Example 3.3a provides a

broad overview of the OS shadings, dyad-spaces, and some small transformation

networks that relate salient pcs in each of the above components. The selection and

presentation of elements in the figure highlights several important features of the music

and implies some inherent methodological assumptions. First, as can be seen in the

horizontal space identifying the movement’s formal sections, the movement features a

59 See Tabor 1994.

42



43



reverse recapitulation, returning the STA before the FTA in the form.60 Second, as

implied by the list of OS shadings, motion between shadings does not result from pc

transformations, but from a change in perspective toward the a priori OS set. Third,

multiple dyad-spaces can be projected within one section, but the pcs involved in that

projection do not owe exclusive allegiance to one particular space. For example, the STA

in both exposition and recapitulation projects two dyad-spaces, sometimes

simultaneously. In addition, both spaces within this formal section involve pc A, placing

it in two distinct relationships: as the sum 1 partner of E and the sum e partner of D.

Finally, the transformation network atop the figure suggests that the movement’s focal pc

D is an active agent throughout the movement. As D “moves” throughout the piece via

T0, it appears in a number of distinct transformational relationships with other pcs. The

60 Additional discussion of this phenomenon appears later in the present chapter.

44

following analysis discusses each of these features and assumptions, demonstrating their

significance in their respective movements.

Despite myriad local relationships entered into by the fundamental pc D, one

particular set of relationships unifies the various dyad-spaces, OS shadings, and the

transformation network depicted in Figure 3.3, linking them all to the formal structure of

the movement. The transformation network in Figure 3.3a interprets the various

relationships between the fundamental D, pictured along the bottom horizontal as

resulting from four T0 transformations, and the four basic dyad-spaces that help to define

structure in the movement. As the movement progresses chronologically (reading left-to-

right on the Figure), the fundamental D enters four basic relationships. The first

relationship pairs D with D via I4 in the balance tetrachord at the beginning and end of

the movement. The transition section between the two main themes pairs D with Ab via

It; this change from D/D to D/Ab is interpreted in the figure as the upper left D moving to

Ab via T6 while the fundamental D moves via T0. The STA precipitates another

inversional pairing of D with B via I1 within the dyad-space A/E-(I1). In the figure, the

motion from Ab to B is interpreted as T3. The recapitulation of the FTA begins with a

reharmonization (c.o.=<C-D-F-G>) of the balance tetrachord, articulating the dyad-space

C/G-(I7). Within this dyad-space, D appears paired with F at I7, while the motion from the

STA D/B pair to the FTA D/F pair is interpreted as B moving to F via T6. The final

motion from D/F back to the balance tetrachord D/D at the end of the movement is

interpreted as F moving to D via T9. The network of Example 3.3b is a reinterpretation of

the topmost and bottommost horizontals in Example 3.2a. Example 3.3b depicts the

fundamental D as the lower of the two Ds linked by the vertical arrow labeled I4. The

45

upper D is a subset of the 3-cycle {D-F-Ab-B} that appears in the square L-net atop

Example 3.3.61 This upper square is a reinterpretation of the top line of Example 3.3a that

highlights the cyclic underpinnings of the set of fundamental D partners. Thus, the

unifying device connecting the various pc constructions to musical structure results from

the inversional pairing of the fundamental pc D with each member of the complete 3-

cycle collection {D-F-Ab-B}.

II. U-transformations and the melodic/motivic aspect

In addition to dyad-spaces and their connection to the large-scale harmonic

dimension of the pieces, U-transformations and their connection to the melodic and

motivic aspects of the pieces play a central role as well. As opposed to the earlier

movement’s emphasis on vertical harmonies, the later movement’s emphasis on motivic

connections and processes affords greater opportunity for the interaction of motivic and

harmonic dimensions. Indeed, much of both movements’ richness results from the

blending of the music’s harmonic and melodic flavors. As the following analysis reveals,

the gentle receding of the OS set from the first quartet to the second diminishes, but does

not destroy, the pronounced influence of the 7-cycle associated with the first quartet.

Although significant uses of the OS set still occur in the latter movement, most 7-cycle

elements that appear are often colored by their blending with 2-, 3-, 4- and 6-cycle

elements that result from numerous melodic constructions, which are modeled by U-

transformations. The promotion of the motive and the ability of U-transformations to

61 L-nets are “all T” networks discussed in O’Donnell 1998.

46

embed competing interval cycles again appear to distinguish between the earlier and

latter quartets. Ironically, however, it is the persistent presence of pitch constructions

easily modeled by U-transformations that ultimately challenges the efficacy of the

concept of distinct style periods in Ginastera’s music.

In the earlier quartet’s first movement, the specific cyclic coloring of the

structuring 7-cycle OS set shadings resonates within the movement’s formal layout. For

example, the FTA and Coda feature the 3-cycle coloring of the structural OS set shading

{G-D-A} by U-transformations of the main motive <D-F#-F> discussed above. OS set

shadings recede in the two main parts of the exposition’s transition, as the D whole-tone

collection and dyad-space D/Ab-(It) dominates tile 7 and the dyad-space F/F-(It)

dominates tile 8.62 However, the STA’s OS set shading {D-A-E} remains essentially

uncolored by a competing cycle, except for a brief 4-cycle U-chain of the motive <Bb-A-

F#> in tile 17, resulting in the C#/D hexatonic collection. The following paragraphs

provide analytic detail surrounding each of these sections.


The STA in both exposition and recapitulation offers the clearest example of the

uncolored 7-cycle OS set and its constituent dyad-spaces. As discussed above, the STA is

based on the {DAE} shading of the OS set; both dyad-spaces A/E-(I1) and D/A-(Ie) fulfill

structural roles in the section. Example 3.4a depicts the beginning of tile 9, the first

measures of the Second Theme, which significantly features a 7-cycle pentachordal

descent in reference to Amerindian musical practice.63 In its affect, the theme counteracts

62 The OS shading {G-D-A} must recede, as the D whole-tone collection does not contain pcs A and G. 63 I thank Malena Kuss for informing me of this connection.

47

the aggression of the First Theme, exploring the metric fluidity of juxtaposed rhythmic

figures at home both in the notated 3/4 meter (m. 85) and in 6/8 (m. 86). The theme

begins with an initial definition of the dyad-space A/E-(I1), establishing sum 1 as

referential. The figuration associated with the theme reveals a pitch axis of C/C#

operating throughout the entire section; in tile 9, the axis lies at C4/C#4 when the theme

appears in viola and cello and migrates up an octave to C5/C#5 in tiles 10 and 11 as the

theme moves into the violins. The second OS set dyad-space D/A-(Ie) structures the

presentation of the theme in the lower voices in tile 11, which occur in canon with the

upper voice A/E-(I1) dyad-space, as shown in Example 3.4b. The recapitulation of the

STA in tile 15 recalls the canon of tile 11, featuring a different pairing of voices (dyad-

space D/A-(Ie) in violin I and viola and A/E-(I1) in violin II and cello) and a different

time interval (one measure in the exposition, one eighth note in the recapitulation), as

shown in Example 3.4c. In this final presentation, the canons are transposed up one

semitone, allowing a new high point Bb5 to emerge from the texture and initiate the

aforementioned 4-cycle-based ({D-F#-Bb}) U-chain from source trichord <Bb-A-F#> to

close the section. Although the movement’s fundamental D temporarily recedes from the

salient prominence it enjoyed in the FTA, it nonetheless maintains a vital role in the STA

through its I1 association with B.


As discussed briefly in Chapters One and Two, Kuss’ description of

Ginastera’s tonal language as “octatonic with modal interaction” readily applies to the

First Theme, though the present dissertation would describe this tonal language as a

48



7-cycle with 3-cycle coloring. In general, the First Theme, the score of which appears in

Example 3.5a, is a large arch, ascending in violin I from the focal D4 to C5 in tile 3, and

descending back to its origin in tile 4, followed by an motivic intensification in tile 5 and

closing gesture in tile 6. The large, overarching slur in Example 3.5b summarizes the

broad ascent from the first theme’s beginning on D4 in m. 16 to its apex on C5 in m. 24,

outlining the first theme’s broad dyad-space D/C-(I2) in a type of “composing out” of the

opening gesture in m. 1. On a broad scale, a new trichord <D-F-A> is transformed into

U-cell <D-F-A>-<C> via one U(st) transformation, as highlighted by the large slur

connecting the dyad-space’s boundaries.64 However, this broad motion is actually

comprised of two smaller, nested U-cells within the larger motion, as highlighted by the

64 The source trichord <D-F-A> incorporates significant notes from the theme’s first four notes, U-cell <D-F-F#>-<A>.

49

two smaller slurs in Example 3.5 connecting D4 to A5 in mm. 16 and 19 and F4 to A5

beginning in m. 20. The first nested U-cell, shown under the first small slur, begins with a

reordering of the opening gesture’s source trichord <D-F-F#> (r2RT0 of <D-F#-F>) and

transforms it into U-cell <D-F-F#>-<A> via U(st) culminating on the local melodic apex

A4 in m. 19, spanning the OS dyad-space D/A-(Ie). The second nested U-cell, shown

under the second small slur, re-interprets the new second and third terms of the previous

source trichord <F-F#> as the first and second terms of the new source trichord <F-F#-

G#>. The new source is in turn transformed into U-cell <F-F#-G#>-<A>, re-engaging the

original local melodic apex A4. The emphasis on the musical surface of pcs D, F and A

articulates the new, overarching source trichord <D-F-A> for the entire ascent of the first

theme’s melodic arch, as discussed above. The U-chain melody throughout this section is

comprised exclusively of the C/D octatonic collection, the only version that contains the

vital OS set dyad D/A as a subset.

The D/A-(Ie) dyad-space that structures the opening gesture’s U-cell <D-F#-F>-

<A> also defines the space in which aspects of this cell are developed as a Grundgestalt.

The subtle re-ordering of the opening gesture’s ordering of the U-cell’s four elements

<D-F#-F>-<A> as <D-F-F#>-<A> establishes a dialogue between the two sets. The

former occurs twice throughout the piece in the Introduction (tiles 1 and 2) and the Coda

(tile 24), while the latter recurs with all instances of the first theme (tiles 3 and 19). The

tension between these orderings is reflected in subtly distinct versions of the “cadence

motive” appearing as section-closing motivic gestures (the Grundgestalt) in mm. 33-34,

mm. 46-47 (tile 5), 51-52 (tile 6) in the exposition and 183-184 (tile 20) and 200-201 (tile

21) in the recapitulation. The first, fourth and fifth of these are essentially identical and

50

appear in Example 3.3a. Thus, after the gesture’s original statement in mm. 33-34, in

which it closes the theme but not the section, it returns to close not only the theme but the

main body of the sonata form.65 The second and third versions (Example 3.3b) challenge

the ordering of the initial statement, affording its formal return at the end of the

recapitulation a sense of closure by returning a conclusive version.

[Examples 3.6 and 3.7]

While the theme in this section highlights the C/D octatonic collection, the

accompanimental chords in the lowest three voices maintain a general 7-cycle harmonic

profile. Within the context of cyclic shadings and colorings, the accompaniment,

expressed on the musical surface as eighth note block chords, provides the 7-cycle

material that the 3-cycle melody colors.66 Example 3.6 depicts the five exclusive chords

which accompany the complete melodic arch in tiles 3 and 4, indicating the measure of

the chord’s first appearance and the set-class membership of each chord to illustrate the

considerable degree to which 5-/7-cycles permeate this musical figure. The first two and

last chords of this group are purely 7-cycles, while the third and fourth are multi-cyclic as

shown by the p-space transformations modeled in Example 3.7. In the figure, the first

chord, a multiset pentachord {2449e}, is segmented into an upper trichord {249} ([027])

and a lower dyad {4e} (Tn[07]) based on register.67 The upper rank of [027] trichords

65 This is analogous to Warren Darcy and James Hepokoski’s “Essential Sonata Closure (ESC).” See Darcy and Hepokoski 1997. This is especially true because of the reverse recapitulation, which leaves the first theme for last. 66 The texture of the accompaniment is essentially identical to the famous accompanimental ostinato in The Augurs of Spring from Stravinsky’s Rite of Spring. 67 The essential difference between the exposition and recapitulation of the FTA is that the latter features a reharmonization of the first several accompanimental chords. Most notably, the initial multiset {2449e} reappears as {0257} (c.o.=<CDFG>), expressed

51



follows a p-space transformational path of <T(p)+2, T(p)-4, T(p)+6/+1, T(p)-6/-1> throughout the

section; the lower rank essentially mirrors the upper, exchanging + for – and vice-versa

within the individual transformations. The first three chords contain five pitches, thus the

motions between them are easily represented as single transformations. As the passage

moves into the fourth and fifth (final) chords, the texture initially thickens to a hexachord

in m. 24 containing the greatest registral span (C2 to F#4) before collapsing into the final,

most registrally limited chord in m. 31. From a transformational perspective, the motion

to and from these final chords involves multiple paths. The fourth chord is a hexachord in

which the top tetrachord {1368} is considered as two overlapping [027] trichords {138}

and {168}, necessitating the interpretation of two different T(p) transformations (+6/+1

into the chord and -6/-1 out of it) in the upper [027]-based rank. The first four maintain a

sum 6 dyad in the outer voices (D and E in the first and second chords, C and F# in the

third and fourth), suggesting an accompanimental OS set dyad-space D/E-(I6) throughout

this section. The common thread between melody and accompaniment in the FTA is the

within the OS set dyad-space C/G-(I7) (see Example 3.2a). However, both versions of the theme terminate with the balance tetrachord <DAGD>.

52

emphasis on the fundamental pc D and the contexts in which it appears: as the initiation

point of the melodic/thematic 3-cycle U-chain and as a member of dyad-space D/E-(I6)

that governs the 7-cycle accompanimental chords.

Both the exposition and the recapitulation feature transition sections between the

first and second themes. Of the two, the first transition section, tiles 7 and 8, significantly

explores the cyclic coloring of the OS set.68 The division of the section into two tiles

reflects the basic binary division of the section according to stylistic, motivic and tonal

associations. Stylistically, the section bridges the gap between the aggressive first theme

and placid second theme through two intermediary steps. The first step (tile 7) maintains

the former’s rhythmic intensity while softening its impact by thinning the ensemble

texture; the second step (tile 8) initiates the long, sweeping phrases that will characterize

the STA. A similar transformation occurs motivically as initially forceful rhythmic

motives also give way to smoother figuration, gradually relaxing the first theme’s tension

as the music moves into the serene second theme.


The area of greatest significance to the large-scale processes of the movement,

however, encompasses tonal issues. Broadly, each of the transition’s two main sections

projects two different cyclic collections. The first is based on the D whole-tone

collection, while the second is based on 5-/7-cycle pc sets {C-F-Bb} and {F#-B-E},

which simultaneously recall familiar OS harmonies and depart from OS set shadings,

which intensifies the return of the OS shading {D-A-E} at the arrival of the second

theme. Example 3.8a presents the first two measures of tile 7, depicting both D whole-

68 The second transition section in tile 18 contains only developmental rhythmic figuration of the OS set.

53

tone-based passages (<Ab-F#> in violin 1 and <D-E> in vcl in mm. 60-61) and gapped 5-

cycle collections {E-F#-A} (m. 61). The figure in violin I, viola and cello in m. 60 is the

first of three similar statements appearing in tile 7. The outer-voice dyad initiates the

structural dyad-space D/Ab-(It), establishing sum t as the primary transformational

relationship across tile 7. The second and third statements (not pictured) allow the sum t

dyad Bb-C to emerge at the beginning of tile 8, where this dyad is incorporated into the

vertical trichord {F-Bb-C} in m. 73, as shown in Example 3.8b.69 This trichord is the first

of two salient sum t 5-cycle sonorities; the trichord {E-F#-B} (Example 3.8c) emerges in

m. 77, recalling the sum t dyad {E-F#} from m. 61 but placing it in a new, non-gapped 5-

cycle context {E-F#-B}. Thus, the significant sum t orientation of the transition is

initiated by the replacing of the D/D-(I4) and D/A-(Ie) dyad-spaces of the first theme with

D/Ab-(It), a dyad within the D whole-tone collection. Example 3.3a interprets this change

as a motion via T6 from the D atop the balance tetrachord in the FTA to the Ab paired

with D in the transition.

As discussed above, the later quartet’s first movement shares many similarities

with its earlier counterpart. Indeed, both first movements possess common attributes in

five general areas. First, both are thematic sonata forms featuring aggressive first themes

and tranquil second themes. Second, the OS set figures prominently as a referential set in

both movements’ thematic organization, with specific shadings of the OS set aiding in the

distinction between formal sections. Third, dyad-spaces, both inclusive and non-inclusive

of OS pcs, structure much of the musical surface. Fourth, the various dyad-spaces that

structure elements within sections are colored by cyclic pc constructions. Finally, U-

69 The C appears on beat two in the viola.

54

transformations effectively model many of both movements’ melodic and motivic

constructions, often resonating in significant ways with the above-enumerated

similarities.

Although both movements employ many of the same devices, however, the later

movement distinguishes itself from the earlier movement in two basic areas. The first

distinction results from the expansion in the number and role of motivic pc sets recurring

as fundamental thematic and developmental sets throughout the movement. For example,

as the above analysis demonstrates, members of Tn-type [034] play a pivotal role in the

movement’s introduction and first theme; in the later movement, members of Tn/TnI-

types [014] and [016] feature prominently in virtually all formal sections, inhabiting both

melodic and harmonic dimensions. The second distinction results from the deeper

coloring of the basic OS set by 3-cycle and 4-cycle collections, with the latter often

appearing as a foil to the former. For example, the earlier movement’s first theme

features the coloring of its fundamental 7-cycle OS shading {G-D-A} by the 3-cycle

collection {D-F-Ab} as a result of multiple U-transformations of the motivic source

trichord <D-F#-F>; the resulting C/D octatonicism permeated important sections of the

movement. As the analysis below demonstrates, octatonic sections resulting from the 3-

cycle coloring of 7-cycle sets also include significant, salient 4-cycle coloring as well,

enriching melodic and harmonic dimensions. Thus, the expanded number of motives and

their relation to larger structural devices result not only in an expanded cyclic harmonic

palette, which also is reflected in broader compositional areas. The following analytic

detail develops these various issues in an attempt to answer the fundamental questions

posed in the present dissertation’s early chapters.

55

Like the first quartet’s first movement, OS set shadings and the dyad-spaces they

imply play an important role in establishing local referential collections through their

association with the movement’s major formal components. As discussed above, the

second quartet’s first movement is based on a {C-G-D} shading of the OS set and is

articulated by the balance tetrachord c.o=<G-D-C-G>, which opens and closes the

movement (tiles 2 and 37). The exposition’s transition (tiles 4-9) is generally based upon

a {G-D-A} shading of the OS set, while the exposition’s second theme (tiles 12-17 in the

exposition and 31-34 in the recapitulation) is generally based upon a {D-A-E} shading.

However, the later quartet’s movement only features clear OS set shadings in the

exposition; the recapitulations of the first and second themes depart from the tight

harmonic organization of the exposition versions.

[Example 3.9a]

Each of these OS set shadings combines with motives to suggest several locally

structural dyad-spaces. In essence, most of the significant motivic elements of the

movement derive from the two salient trichordal motives appearing in the movement’s

first four measures (tile 1). Example 3.9a provides the first four measures of the

movement. This dramatic opening statement introduces two salient trichordal motives

<D-Bb-C#> ([014]) and <Bb-F-E> ([016]) which appear in boxes in the example.

Members of these Tn/TnI classes constitute an integral part of all major themes in the

movement, and in each case significantly involve pcs and dyad-spaces associated with

the trichordal shadings of the OS set.

[Example 3.9b]

56

Both the movement’s first and second themes significantly feature members of

both of the boxed motives’ Tn-types [016] and [014]. The diagram accompanying

Example 3.9b shows two different views of the first theme’s wedge construction (also

depicted in Example 3.2b) on two bass staves. The top staff illustrates the vn 1 and viola

wedge emerging from the focal OS pc G in m. 5. The upper and lower voice link the

focal G to D and C via single U(st) transformations of source trichords <G-Ab-Db>-<D>

and <G-F#-C#>-<C> in m. 16, where they sound together as the familiar OS set shading

{C-G-D} and participate in the movement’s structure as the balance tetrachord.70 The

lower staff presents a different layer of the thematic wedge, bounded by the same OS set

shading. In this layer, the source trichords are members of [014], referencing the first

three notes of the piece. The notes involving the source trichords’ distinctive ic 3, the

upper-voice A# and B and lower-voice E and D#, appear in a distinctive rhythmic pattern

on the musical surface, establishing a model for subsequent motivic interpolations

discussed in Chapter 2 as “crosscuts.” The union of both top and bottom staves, and

thereby the union of [014] and [016] motives within the dyad-spaces G/D-(I9) and G/C-

(I7), reveals that the upper and lower “ramps” of the wedge radiating outward from the

central G are indeed statements of octatonic collections <G-Ab-A#-B-Db-D> (C#/D

octatonic) and <G-F#-E-D#-C#-C> (C/C# octatonic). The endpoints of these collections

reaffirm the controlling OS shading {C-G-D} and balance tetrachord c.o.=<G-D-C-G>.

Thus, the main body of the first theme illustrates the startling interconnectedness of

motive, U-cell, dyad-space, and OS shading.

[Example 3.9c]

70 Octave doublings of the focal G are included in the multiset balance tetrachord.

57

While the first theme’s wedge engages the aforementioned first stylistic

distinction, the final measures of the first theme, a passage immediately following the

wedge, clearly illustrates the second distinction. The cyclic interaction associated

with the wedge transforms from a 7-/3-cycle interaction into a 7-/4-cycle interaction as

depicted in Example 3.9c. In the example, the central OS pc G initiates a rising figure

<G-A-Bb-B-C#-D> in violins I and II and the cello; this figure is then transposed via a T8

cycle from its initial G in m. 20 to Eb in mm. 21 and 22, to B in m. 23, conjuring a

contrasting 4-cycle {G-Eb-B} that replaces the embedded 3-cycle {G-A#-Db-E}

contained in upper and lower wedge ramps.71 Thus, the first theme casts G in three

distinct cyclic layers: as a member of the 7-cycle OS shading {C-G-D}, the coloring 3-

cycle {G-A#-Db-E} and the contrasting 4-cycle {G-Eb-B}.

[Example 3.10a]

Like the first theme, the second theme also significantly features members of

[014] and [016] within its OS shading {D-A-E}. Broadly, the second theme divides into

two large sections, tiles 12-14 and 15-18, which respectively present main thematic

material and develop main thematic material. Example 3.10a shows gestures 1 and 2 as

they appear in tile 12, the first presentation of the second theme. The excerpt depicts a 7-

measure melodic arch formed by rising gesture 1 and “cascading” gesture 2. Gesture 1 is

formed by a concatenation of members of [014] unfolding the dyad-space D/A-(Ie) in a

U-cell <D-Ab-Eb>-<A> as points of imitation; the C/D octatonic pc collection results

from the union of these points of imitation and the held notes F (vcl), B(vla), F# (vn2)

and C (vn1) appearing at the tail end of the three-note [014] gestures. Another rising

71 This cycle appears at the beginning of each transposition, which occurs on the second eighth note of each measure.

58

figure in vn 1 beginning in m. 99 further opens the tessitura until the attainment of salient

OS pc E on the downbeat of m. 101. This arrival on E fully articulates the second theme’s

{DAE} shading of the OS set and initiates a downward cascade featuring two [016]-

based U-cells <E-B-Bb>-<F> and <Gb-Db-C>-<G>; unlike the first ascent, however, the

union of these U-cells is not octatonic. The return of the second theme in the

recapitulation (tile 31) features pcs B, A, G and Eb as points of imitation, disavowing the

initial U-cell construction and embracing the 4-cycle collection {G-B-Eb} from the end

of the first theme as a structural component. Indeed, the later quartet’s main themes boast

similar types of constructions as the earlier quartet’s themes: a balance tetrachord and

melodic U-cells which articulate the sections’ structuring dyad-spaces and OS set

shadings. However, the significant incorporation of U-cells comprised of two source

trichords of differing set-class membership and the incorporation of a structural 4-cycle

“foil” represent a substantive development from earlier practice.

[Example 3.10b]

After the thematic presentation appearing in tile 12, the tile 15 developmental

subsection of the second theme further underscores the expanded role of the 4-cycle.

Example 3.10b depicts the first two (of four) developmental subphrases in tile 15. Each

subphrase is in essence an amalgam of two tile 12 elements. The first, beginning with the

cello Db3 in m. 118 and the violin II F4 in m. 119, is an extension of the three-note [014]

motive featured in Example 3.10a; the second, appearing after the extended motives,

recalls the longer rhythms appearing in the [016] cascade. The four developmental

subphrases present 4-cycle pcs <Db-F-A-F> in points of imitation, as opposed to the

unfolded U-cell points appearing in tile 12. The symmetrical properties of these new 4-

59

cycle points of imitation also figure prominently on the musical surface, as the third

subphrase (beginning on A) replicates the first It note-for-note (beginning on Db), and the

fourth (beginning on F) replicates the second is It (also beginning on F).72 Thus, the

expanding role of cyclic pc sets not only results in a more variegated tonal palette, but it

also facilitates the deeper exploration of inversional symmetry associated with the

stylistic change between first and second quartets.

72 The inversional symmetry of this passage is reflected in p- as well as pc-spaces.

60

Chapter Four: Interval Cycles and Cyclic Collections in Internal Movements

In Chapter 3, we employed the various analytic devices described in Chapter 2 to

elucidate connections among form, tonal structure, and motive in first movements of

Ginastera’s String Quartets Nos. 1 and 2; the present chapter analyzes two internal

movements from the second quartet with the same fundamental goal. Of the four

movements discussed in the present chapter, two (movements two and three of the second

quartet) exhibit significant interaction between 12-tone and free atonal material, calling to

mind Berg’s early twelve-tone works, the Chamber Concerto and Lyric Suite.

Accordingly, the present chapter divides into two basic parts, broadly reflecting the use

of serial and non-serial techniques in the above movements. However, as the chapter

demonstrates, the structuring of musical material based on the interactions of interval

cycles transcends modes of surface pc presentation, further strengthening the connections

between the present movements and Berg’s music as discussed by Headlam in The Music

of Alban Berg.

Like most 20th-Century composers who utilized twelve-tone rows in their

compositions, Ginastera employed the technique idiosyncratically.73 In discussing his

73 See Wallace 1964, Kuss 1976, 1980, 2000, 2002 (Bomarzo), and 2002 (MGG), and Schwartz-Kates 2001. In general, Ginastera’s approach is most reminiscent of Webern’s linear unfolding of a row’s pc content as demonstrated in works beginning with Op. 21. His approach also bears similarities with that of Berg in its exploitation of connections between interval cycles and order positions (hereafter referred to as ops and indicated with underlines such as 012 for the trichord in a row’s first, second and third ops) and its use of multiple rows associated with operatic characters (see Perle 1985, Headlam 1985, 1990, 1996), Kuss 1976, 1980, 2000, 2002 (Bomarzo), and 2002 (MGG), and Richards 1986).

61

approach to twelve-tone composition, the composer himself observed in 1959 that his

technique “is adapted to the expressive necessities of the music,” revealing a conscious

flexibility to depart from an inviolable ordering of pitch material to accommodate an

aesthetic impetus.74 The following analyses of the second and third movements of the

second quartet confirm the above assertion, demonstrating the frequent coexistence of

three basic approaches to musical material within a single movement: 1) twelve-tone

serialism, 2) unordered pitch-class (pc) aggregates, and 3) unordered, non-aggregate-

based constructions. Within the first of the above three approaches, we may superimpose

three distinct musical contexts: 1) concatenated twelve-tone rows, 2) canonic row

presentations, and 3) segmented/partitioned rows (rows divided into adjacent and non-

adjacent collections) and ordered sub-collections.

I. (Mostly) Twelve-Tone Movements Two and Three of the Second String Quartet

[Tables 4.1 and 4.2]

Although the rondo-like forms of the third (A-B-A1-C-A2) and second (ABA’)

movements of the latter quartet are reflected in their rhetorical and textural dimensions,

the presence and/or absence of tone rows also plays a vital role in the delineation of

formal sections.75 Neither of these movements is entirely twelve-tone. The third

movement largely reflects the aforementioned first basic approach (twelve-

74 Ginastera, Alberto, Boletín Interamericano de Música, No. 14, Nov., 1959, pp. 3-4. 75 I will discuss the third movement before the second due to its stricter adherence to traditional twelve-tone orthodoxy. See also Kuss 2002 (MGG).

62



63



64

tone serialism) by significantly employing two source rows labeled 3-R1 and 3-R2 in its

constituent A, A1 and A2 (hereafter referred to as “A-based”) and B sections (see Table

4.1). The movement’s C section reflects the aforementioned second basic approach,

unordered pc aggregates. The second movement employs a single tone row labeled 2-R1

(see Table 4.2) in its outer A-based sections, interspersing manifestations of the third

basic approach (unordered, non-aggregate-based constructions) with row statements,

while the B section employs material derived from the second and third basic approach

material.

In addition to indicating the large formal sections of both movements, Tables 4.1

and 4.2 delineate smaller formal units as well. Table 4.1 depicts the presence of a main

theme in each of the A-based sections in the third movement, whose outer sections (A

and A2) feature only thematic and codetta sections and whose inner section (A1)

alternates between main thematic and contrasting sections. The B section divides into

four smaller subsections, each featuring a rhapsodic instrumental solo by the four

members of the string quartet. The A-based sections and the C section divide comfortably

into tiles that predominantly reflect 4-bar hypermetric units; in the A-based sections,

these units often are comprised of a single row statement. Table 4.2 illustrates the second

movement’s three overarching sections and subdivides them into subsections labeled

according to criteria other than tiles. The analytic paragraphs in the following section

discuss each section in greater detail.


65



66

Each of the three row forms appearing in the latter quartet embeds interval

cycles in particular ways; the composer often exploits these encoded cycles through

choices of row form and row presentation on the musical surface. Example 4.1a provides

a basic harmonic profile of rows 3-R1 and 3-R2 (appearing in the third movement), while

Example 4.1b does the same for row 2-R1 (second movement). The discrete hexachords

of rows 3-R1 and 2-R1 are members of different set-classes ([012369]/[013467] for row

3-R1 and [013478]/[012569] for row 2-R1), while the discrete hexachords of row 3-R2

are members of the same set-class (both [012346]).76 As illustrated by the interval-class

vectors in the figure, the row 3-R1 hexachords favor ic3, and the row 2-R1 hexachords

favors ic4; the row 3-R2 hexachords are nearly chromatic and favor both ics 1 and 2.

However, further analysis reveals members of 3-cycle [0369] and [016] appearing in

segmental 1234 and partitional 01e are of paramount importance in the A-based sections

of the third movement; the same is true of members of 4-cycle [048] in 023 and 69t (the

two combine to create a member of 2-cycle [02468t]) in the A-based sections of row 2-

R1 of the second movement. The discrete trichords of row A are members of [016] and

[013], with the pair of [016]s RI1-related and the [013]s I1-related; of these discrete

trichords, the first figures significantly in the music. The discrete trichords of row 3-R2

are all members of different set-classes [016], [012], [015], and [014], and as subsequent

analysis demonstrates, the manner in which these discrete trichords are presented is a

vital component in the music. The discrete trichords of row 2-R1 feature three members

of [037] at Te, Ie, and I0 (the remaining trichord is a member of [016]). Again, as is the

76 The hexachords for row 2-R1, members of [013478] and [012569], are the famous Z-related pair often used by Schoenberg, as the latter contains the six pcs in the composer’s last name. Allen Forte labels these as 6-z19 and 6-z44 respectively.

67

case with the discrete trichords of rows 3-R1 and 3-R2, the cyclic properties of these

discrete trichords are of great relevance to the musical surface of the A-based sections of

the second movement.

As introduced earlier, one of the main assertions about the later quartet’s third

movement is that while mostly twelve-tone, its fundamental structuring aspects are cyclic

in origin. While ideas and procedures typically associated with twelve-tone music, such

as the segmentation of the musical surface into ops and row-forms are useful and are

employed in subsequent analytic observations, these devices are interpreted here in their

role as demonstrating the underlying influence of cyclic thinking in the movement. The

fundamental aim of the following analysis is to reveal the types of connections between

cyclic composition and twelve-tone expression in the third movement. To this end, the

analysis begins by identifying one basic motive and two cyclic pc sets on the musical

surface and illustrating how twelve-tone compositional procedures participate in their

development into some of the movement’s essential structural features. Of paramount

importance to the movement is the establishment of a “signature” opening trichord <C-F-

B> ([016]) which emerges as a fundamental structural component.

[Example 4.2]

The fundamental cyclic element of the A-based sections derives from the pcs and

symmetric properties of one cyclic pc set (and its subsets), the 6-cycle tetrachord

68



69

{B-C-F-F#} (a member of [0167]). As indicated above in Table 4.1, the outer A sections

feature only two main forms of row A, P/I0(3-R1). Each of these row forms embeds the

tritone {C-F#}({06}) at 0e due to the dyad’s inversional symmetry at I0. The

incorporation of this dyad with the row’s “signature” trichord <C-F-B> (<05e>) in 012

completes the above 6-cycle tetrachord, which appears in row partition 012e. The

inversional symmetry of this fundamental tetrachord facilitates its role as an overarching

element of unity among all of the movement’s A-based sections; its isomorphic

partitioning in two of the row forms appears in the inner A1 section.77 As indicated in

Table 4.1, the inner A section features four row forms in two J00-related pairs, P/Ie(3-R1)

and P/I5(3-R1). Two of these row forms, Ie(3-R1) and I5(3-R1), contain different

orderings of this tetrachord in 012e. Thus, this tetrachord’s appearance in the above row

partition provides the common element in all A-based sections. Example 4.2 provides a

chart detailing each row form in the movement. The leftmost column indicates the formal

sections in which the middle column rows appear, while the rightmost column identifies

the abovementioned musical contexts in which the row forms appear. The middle column

provides the actual rows used, highlighting in bold type the members of the above 6-

cycle tetrachord. Again, their conspicuous placement at the beginning (or end) of the row

forms underscores the importance of this tetrachord or its subsets on the musical

surface.78


77 For definitions and discussions of isomorphic partitions, see Haimo, Ethan and Paul Johnson. “Isomorphic Partitioning and Schoenberg’s Fourth String Quartet.” Journal of Music Theory 28 (1984): 47-72. 78 Members of this tetrachord ([0167]) feature prominently in Bartók’s String Quartet No. 4 and Berg’s Lulu (see Perle 1955, 1985, Antokoletz 1984 and Headlam 1996.)

70

The first discrete trichord of the first row-form P0(3-R1), pcset <C-F-B>, referred

to above as the “signature trichord,” establishes the main transformational relationships

explored in the A-sections of the piece. The pcs of this trichord occupy not only 012 of

the basic row-from, but they occupy 0 in each of the aforementioned J00-related pairs that

exclusively comprise the movement. Thus, the trichord in 012 of P0(3-R1) is “telescoped”

across the entire movement. Example 4.3a provides a transformation network interpreting

the various relationships between both the pc set <C-F-B> as 012 of P0(3-R1) and the

relationships between the six main row-forms of the movement, each of which contain C,

F or B at 0. The network begins with C and interprets its motion through F and B as T5

and T6 respectively, as represented in the leftmost “northeastern” arrows. While these

motions describe the transformations within 012 of P0(3-R1), they also describe the

transformations among the pcs at 0 of P0(3-R1), P5(3-R1) and Pe(3-R1). Each of these pcs

is also paired with itself via inversion, as interpreted on the figure in the three horizontal

arrows bearing I operators (bottom to top) I0, It and It. While these inversions are only

implicit in the pcs of 012 in P0(3-R1), they are explicit within the context of 0 in row-

forms I0(3-R1), I5(3-R1) and Ie(3-R1), as shown by the row-form pairs appearing to the

right of the figure. Example 4.3b depicts this explicit connection by showing a

transformation network isomorphic to that in Example 4.3a in which row-forms are

substituted for the initial pcs.79 The isomorphic networks of Examples 4.3a and b imply

the transformation graph appearing in Example 4.3c, which models the transformational

relationships among all basic row forms in the movements’ A-based sections.

[Example 4.4a]

79 The J00 appearing below the I operators are intended only to illustrate the connection and are not intended to operate along with the T operators.

71

While the emphasis of 0e in the concatenated, individually presented J00-related

row-pairs highlights the 6-cycle subsets of the outer A-sections’ main pc set {B-C-F-F#},

similar partitionings of canonic row pairs highlights the interval 5 (and potential 5-cycle)

subsets {C-F} and {B-F#} in the inner A1 section. Example 4.4a depicts the beginning of

Tile A14, which models the canonic passage within the movement’s middle A1 section.

The canonic passages of Tiles A14 and A15 present concatenated row-forms in seamless

running eighth notes in the upper three instruments. The canon arises from the

simultaneous statements of row-forms in staggered entrances five eighth notes apart

beginning with viola F4 (RPe(3-R1)) on the downbeat of m. 165, followed by violin II B4

(RP5(3-R1)) on the sixth eighth note of the same measure and the violin I F5 (RP5(3-R1))

on the fifth eighth note of the following measure.80 The pcs F and B, which return

conspicuously throughout the A1 section, are stated at e in each pc’s row-form, appearing

temporally first as a result of the retrograded row-forms. Each two-measure row

statement roughly unfolds a “pseudo-wedge” with e occupying a registral midpoint

between the highest pc at 5 seven semitones higher and the lowest pc at 0 six semitones

lower. For example, the viola’s F4 at e of RPe(3-R1) is a registral midpoint between the

viola’s highest pc C5 at 5 and B4 at 0.81 In the case of the viola (the canonic dux), the

placement of the registral high point on the downbeat of m. 166 further adds to the

connection between the initial F4 and registral apex C5, linking the two pcs in an interval

5 dyad {C-F}; the comes in violin II connects pcs {F#-B} in an identical fashion. Thus,

the fundamental pcset {B-C-F-F#} exists in Tile A14 as the union of the above interval 5

dyads that occur in 5e of the two main row-forms comprising the canonic row pair.

80 This is reminiscent of mm. 46-69 of Berg’s Lyric Suite, movement III. 81 These are indicated by small boxes on the example.

72

[Example 4.4b]

Whereas all A sections are unified through their use of the isomorphic partitioning

of {B-C-F-F#} at 012e of critical row forms, the inner A section stands out due to its

additional development and salient projection of the 3-cycle pcset {D-F-G#-B}. In

addition to projecting 6- and 5-cycles, the canon of Tile A14 also explores the nascent 3-

cycle tetrachord {D-F-G#-B} initially appearing in the row segment at 1234 of the

original row-form P0(3-R1).82 In the row-class of row (3-R1), the three members of

[0369] appear as the row segment 1234 and row partitions 06te and 5789; in the two row-

forms in Tile A14, the latter segments contain {D-F-G#-B}. Example 4.4b shows the

same passage as appeared in the preceding example, giving the first three measures of

Tile A14. However, in the present example, the boxed notes highlight salient members of

{D-F-G#-B} from 06te in both row-forms. Both pcsets {B-C-F-F#} and {D-F-G#-B}

share the dyad {F-B} at 0e, constituting another salient appearance of these particular

pcs. The remaining 6-cycle dyad {D-G#} appears conspicuously highlighted in two basic

ways on the musical surface, the first melodic and the second harmonic. First, since pcs D

and G# appear at t of row-forms Pe(3-R1) and P5(3-R1), the first sounding dyads in each

instrument’s canonic entrance are <F-D> and <B-G#> (see viola and violins II and I),

emphasizing the “3-cycleness” within the staggered entrances. Second, the metric

alignments of the canonic entrances highlight three dyadic subsets of the basic 3-cycle

collection: 1) {G#-B} between viola and violin II on the sixth eighth note of m. 165

(boxed), 2) {D-F} between violins I and II on the fifth eighth note of m. 166, and 3){B-

D} between violin I and viola on the sixth eighth note of m. 166. The subsequent musical

82 The relevant passage is depicted in the second box in Example 4.1.

73

passage (Tile A15, mm. 169-172, not pictured) essentially repeats the canons with J00-

related row-forms RIe(3-R1) and RI5(3-R1), which exchanges the members of {D-F-G#-

B} appearing in the above horizontal and vertical dyads: the horizontal dyads <F-D> and

<B-G#> become <F-G#> and <B-D> and the vertical dyads {Ab-B}, {D-F} and {F-G#}

become {B-D}, {F-G#} and {G#-B} since the contextual inversion J00 is isomorphic to

the non-contextual transformation It in the present case.

[Example 4.5]

The trio B section contrasts the frenetic and strongly metric A-based sections by

presenting four rhapsodic instrumental solos (see Table 4.2) harmonized by long, held

notes in the other voices.83 The solos essentially present concatenated versions of the J00-

related row-form pairing of P/I4(3-R2), while the long accompanimental chords state

discrete trichords from the solo instrument’s row. Example 4.5 summarizes the first

(cello) solo beginning in m. 67, indicating the ops of the main row P4(3-R2) in both the

solo and the high accompanimental chords. As evident from the passage beginning in m.

72 of the example, the solo features row segments in addition to the complete form. The

(unordered) first discrete trichord {E-F-Bb} in row B is identical to the (unordered) first

discrete trichord in row-form P5(3-R1) appearing in the canons in Tiles A14 and A15 in

section A1, connecting the movement’s two main row-classes. Like the row-forms of the

A-based sections, the B section row-forms are represented by the J00 pair P4(3-R2) and

I4(3-R2). As opposed to the tritone at 0e of row 3-R1, row 3-R2 features the interval 4

dyad <E-C> at 0e. The J00-transform (and its non-contextual analogue I8) preserves the

first pc of the dyad and maps the second onto pc 8, completing the 4-cycle collection {C-

83 This is reminiscent of Lyric Suite, movement 5.

74

E-Ab}. The exchanging pcs C and Ab are foreshadowed in the introduction of the

movement, where they form salient registral boundaries of the slow, methodical assembly

of row A’s first discrete pentachord <C-F-B-D-Ab> that characterizes the section.

The second quartet’s second movement presents a lyrical, contrapuntal adagio in

which twelve-tone and free atonal compositional strata interact freely. As indicated in

Table 4.1, the movement is an ABA’ form in which the A-based sections contain all

twelve-tone material (along with non-twelve-tone material), and the B section is freely

atonal.84 The basic compositional scheme in the A-based sections involves two

introductory individual row statements followed by five canonic pairings of

transpositionally-related row forms at ever-decreasing time intervals. Table 4.1 indicates

that the two thematic row statements occur in mm. 1 and 9, followed by three canonic

pairs labeled as Pairs 1-5 in mm. 13-20, 21-24, 25-26, 27 and 28 (labeled as “All

Voices”). The first two thematic statements of the A section present complete row

statements in one of the string quartet’s voices (viola in m. 1 and cello in m. 9), while the

following five stretto pairings stack the canonic entrances at a distance of two measures

(violins I and II in mm. 13-20 and viola and cello in mm. 21-24), one measure (viola in

mm. 25-26), one half note (violins I and II in m. 27), and one quarter note (all four voices

in m. 28); the final A’ section features one individual statement (m. 48) and a final

canonic pairing at a two bar interval (mm. 54-56). The compositional material

accompanying each of these thematic row presentations is freely atonal in style.

However, as the following analyses demonstrate, both twelve-tone and free atonal

accompanimental material are united in their use of interval cycles.

84 See Lyric Suite, movement 3.

75

[Example 4.6]

The fundamental cyclic component of the thematic row forms in this movement is

based on whole-tone 2-cycle and 4-cycle collections. Example 4.6 presents the first

canonic pairing beginning in m. 13, providing the clearest example of the complete row,

its thematic expression, and its ramifications when presented in canon with other row

forms. Row form P9(2-R1) unfolds in violin I over the four measures of the excerpt

beginning in m. 13, while P7(2-R1) enters two bars later in violin II, providing a

counterpoint to the second hexachord of violin I’s row. From a cyclic perspective, this

example depicts two main ideas. The first underscores the essential 4-cycle+ component

of the row’s first discrete tetrachord <A-D-F-C#>, in which the pure 4-cycle collection

{C#FA} combines with “outside” pc D in an expression of [0148] from the excerpt’s first

measure to the downbeat of the second. The “flag” network appearing below the music

interprets this tetrachord, attaching the gesture’s lowest point D to its initial pc A via T7

and treating this as distinct from the 4-cycle upper tones. The second main idea

incorporates the gesture’s 4-cycle into the larger 2-cycle C# whole-tone collection in

02369t. Members of this collection appear saliently in the example on the downbeats of

each measure in violin I (<A-C#-B-Eb>) and in the registral extreme point of G6 on beat

three of the example’s final measure. This collection is reinforced on the musical surface

as the second canonic voice enters in m. 15. Its members G and B appear aligned with the

B and Eb on the downbeats of mm. 15 and 16 that result from the particular alignment of

6-e of P9(2-R1) in violin I and 0-5 of P7(2-R1) in violin II. Thus, both the melody and its

canonic pairing at the temporal interval of two measures firmly emphasizes the salience

76

and structural importance of the C# whole-tone collection via its two component 4-cycle

subsets.


The preservation of salient whole-tone sonorities remains throughout the next

passage despite the addition of non-twelve-tone material and the re-ordering of

material within a row form. Example 4.7a presents a reduction of the musical surface of

the beginnings of measures 23 and 24, in which the canonic pairing of P0(2-R1) and Pt(2-

R1) in the viola and cello are accompanied by a second stratum of non-twelve-tone

material in violins I and II. While the previous example features the projection of the C#

whole-tone collection through its alignment of P9(2-R1) and P7(2-R1), the present

example features the projection of the C whole-tone collection through its alignment of

P0(2-R1) and Pt(2-R1), resulting in the D5 and Bb3 appearing at the beginning of m. 23

and Gb5 and D4 appearing at the beginning of m. 24. In the example, the inner ic4s

between the outer registral extremes derive from an aligned triplet figure in vns I and II,

which emphasize dyads D4/F#4 in m. 23 and E4/G#4 and C5/Ab4 in salient locations in

m. 24. Thus, the resulting sonorities in both twelve-tone and non-twelve-tone strata, {Bb-

D-F#} and {D-E-Gb-Ab-C}, reinforce the local dominance of the C whole-tone

collection.

The localized projection of whole-tone collections persists throughout the next

passage in mm. 27 and 28, though it is accomplished through different means. At this

point in the piece, the gradual thickening of the musical texture combined with the

methodical reduction in the time interval between canonic voices dramatically increases

the tumultuousness of the end of the A section as it heightens in intensity going into the

77

movement’s climactic B section. One of the mechanisms by which the intensity increases

is the rapid toggling between the C and C# whole-tone collections, as depicted in

Example 4.7b. As illustrated by the example, the quartet’s upper voices significantly

project pentachordal subsets of the C whole-tone collections {D-E-F#-Ab-Bb} and {D-E-

F#-G#-Bb} in beats one and two of mm. 27 and 28, and the C# whole-tone collection

{Db-D#-F-G-A} in beats three and four of m. 27. While these beats do not exclusively

contain pcs from these collections, constituent members of these collections appear on

metric downbeats of these measures, thereby gaining emphasis. As indicated below the

staff, the instruments contain hexachords of four different row forms, P6(2-R1), P7(2-R1),

P5(2-R1) and P2(2-R1). Three of these row forms, P7(2-R1) and P6(2-R1) and P5(2-R1),

feature retrograded second hexachords, which facilitates the downbeat projection of the

corresponding whole-tone collections. Thus, the uniform maintenance of whole-tone

collections at structurally significant points in the present case necessitates the

manipulation of the order of pcs within row forms, providing a concrete example of the

composer’s adaptation of strict twelve-tone procedures to accommodate the expressive

necessities of the music.

Since the juxtaposition between the two whole-tone collections is one of the

prime agents of tension within the outer A sections of the movement, it follows that the

absence of one of the collections could be interpreted as a relaxation of this tension. For

example, the initial statement of P9(2-R1) as depicted in Example 4.6 and its canonic

pairing with P7(2-R1) reinforced the C# whole-tone collection nascent in the row form.

Thus, any odd T-level row form will project the C# whole-tone collection. As illustrated

in Table 4.1, the piece does not begin with the above canonic pairing, but with T6-related

78

P3(2-R1) and P9(2-R1) row forms. However, the odd T-levels tether these initial row

forms to the family of row forms that, given requisite compositional emphasis, project the

C# whole-tone collection. Although the movement contains ten of the twelve available

transpositions of the fundamental row class, the odd transpositions enjoy an elevated

status resulting from their privilege-of-place appearance as the first four presentations in

the A section and final three presentations, which occur as an initial solo statement of

Pe(2-R1) beginning in m. 48 and a final canonic pairing of P7(2-R1) and P9(2-R1) in mm.

54-59, similar to their appearance in mm. 13-17.85 Thus, from a cyclic harmonic

perspective, the C# whole tone collection initially establishes itself as referential, is

challenged through various juxtapositions with the C whole-tone collection, and

ultimately emerges to close the piece and resolve compositional tension.

At the core of twelve-tone composition is the idea that transformations of a

complete, ordered pc aggregate constitute a major structuring principle of a piece.

However, the utilization of a complete, unordered pc aggregate may also be a central

feature of music untethered to a specified ordering of the pc aggregate.86 Such is the case

with parts of the second quartet’s second and third movements. Specifically, the C

section of the third movement and the B section of the second movement significantly

feature localized pc aggregates without 12-tone rows. The following analysis provides a

detailed examination of the variety of ways pc aggregates operate in these sections,

describing, where applicable, issues of simple aggregate partitioning, interval cycles, and

operator cycles.

85 No I-forms appear in the movement. 86 Morris (2001) describes such constructions as “free arrays” (p. 181). The passages in question are brief and do not require most of the “aggregate partition” apparatus discussed by Morris (1987, 2001) and Mead (1988).

79

In the (non-twelve-tone) C section of the third movement, aggregate completion is

fundamentally achieved via the use of the four 4-cycle pcs sets and the 4-cycle T- and I-

cycles often associated with them. The analyses below demonstrate two distinct uses of

4-cycle-based, aggregate-producing processes in the two basic subsections within the

larger C section (Tiles C1-C3 and C4-C8). In general, this emphasis on 4-cycles within

the C section allows it to contrast the 6-, 5- and 3-cycles of the A-based sections while

linking it with the 4-cycle collection {048} that figures prominently in the B section.


In the first subsection (see tile C1 in Table 4.1), pc aggregates are created by

uniting pairs of trichordal members of Tn-types [037] and [047] within complementary

hexatonic collections. Example 4.8a depicts the pc aggregate divided into two members

of the 4-cycle-based C#/D and B/C hexatonic collections, both of which are members of

[014589]. Both of these pc collections are further subdivided into the I-related pairs of

Tn-types [037] and [047] whose union is the complete hexatonic collection. For example,

the C#/D hexatonic collection {C#-D-F-F#-A-Bb} can be “disunited” into the I3-related

pair {D-F-A} ([037]) and {F#-Bb-C#} ([047]), the I7-related pair {Bb-C#-F} and{D-F#-

A}, or the Ie-related pair {F#-A-C#} and {Bb-D-F}, as appears on the left of the diagram.

An analogous situation obtains via the same operators within the B/C hexatonic

collection {Eb-E-G-Ab-B-C}, as shown on the right of the figure. The following analysis

confirms the relevance of these hexatonic collections and their disunification into minor

and major triads within the C section.

The presentation of members of [037] and [047] in Tiles C1-C3 (mm. 195-211)

highlights the transformational connection between the I-related trichords via the metric

80





placement of representative I-related pc pairs. Example 4.8b depicts the first two

measures of Tile C1, while Figure 4.8c interprets them. As evident in Example 4.8b, two

complete pc aggregates, one per measure, are segmented into members of [037] and

[047] appearing as vertical triads in m. 195 and small arpeggios in m. 196. The diagram

81

to the left of the example indicates that the verticalities expressed in the instrumental

pairings of the upper and lower voices on the first and second eighth notes of m. 195 do

not voice hexatonic collections, but rather pairs of I5-related triads which unite into

members of [023679] and [014679]; the I5 transformations between the individual voices

within each pair are depicted on the left column in Example 4.8c. However, the arpeggios

of m. 196 accompany a migration of the trichords within the upper three voices, as

indicated by the solid lines linking violin 1 to the viola, violin 2 to violin 1, viola to violin

2 and cello to itself from mm. 195 to 196. This re-orchestration precipitates a change in

the set-class membership of the upper and lower voice pairs from the above non-

hexatonic collections illustrated to the left of the example to the hexatonic collections on

the right. This migration and the ensuing emphasis on hexatonic collections appears in

the right column in Example 4.8c, which shows the I3- and I7-related pairings familiar

from Example 4.8a moving from outer and inner voice pairs that are displaced in time to

upper and lower voice pairs sounding together. Significantly, the cello’s arpeggio in m.

196 is distinct from the remaining voices in its presentation of a rotated (“second

inversion”) trichord, which aligns the I3 dyad {Gb-A} on the downbeat of the measure,

underscoring the presence of I3 through its metric placement.


The combinatorial potential of [014589] is further explored in Tile C3 (mm. 202-

211) as a result of canons featuring 4-cycle operators. Example 4.9a provides the first

three measures of Tile C12, which maintain the instrumental figuration of m. 196 in Tile

C1 above. In addition to the familiar segmentation of members of [014589] into the I-

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related pairing of [037]/[047], Tile C12 incorporates I-related [014589] subsets



[015]/[045], using them in tandem with the [037]/[047] segments to provide contrast

within the broader context of aggregate completion. The three boxes appearing in the

example each contain the pc aggregate; Example 4.9b illustrates the manner in which the

pc sets within the three boxes create and maintain aggregates throughout the entire

passage. The example begins by defining two pc sets A <Bb-D-F> ([047]) and B <Ab-E-

Eb> ([015]) which act as the excerpt’s most basic building blocks. The leftmost part of

the example depicts the C#/D hexatonic collection resulting from the familiar, metrically

supported pairing of T0(3-R1) and Ie(3-R1) appearing in the violins at the beginning of m.

202. The T8 arrows along the top rank of hexachords show the cyclic transposition of the

I-related pairs of trichord A that appear in beat one of each measure, from the sum e

83

pairing above through pairings of sums 3 (T8(3-R1)/I7(3-R1)) and 7 (T4(3-R1)/I3(3-R1)).

The “southeast” T8 arrow initiates a second cycle of T8 transpositions that anticipate the

upper rank pairings by one beat. A virtually identical situation obtains with the pairing of

T0(3-R2) and I3(3-R2) in the violins on the second beat of m. 202 and their subsequent

transformation via a T8-cycle through the same sums. The resulting texture is one

comprised of two interlocking T8-cycles featuring pc aggregates expressed in three

dimensions. The first exists in the violins within each measure, as beat one contains the

C#/D hexatonic collection in an [037]/[047] pairing and beat two contains the

complementary B/C hexatonic collection in an [015]/[045] pairing on beat two. The

second occurs in canon with the first in the viola and cello, with the [037]/[047] and

[015]/[045] pairings occurring on opposite beats. The third appears in all four voices on

each beat beginning on beat two of m. 202, as the necessary transpositions and inversions

of both trichords A and B align as a result of the canon between upper and lower voices.

In Example 4.9b, any pair of hexachords connected by a dashed line indicates a pc

aggregate. The first dimension appears in the top horizontal, the second in the bottom,

and the third in the vertical slices.

[Example 4.10]

Like the Second String Quartet’s third movement, the second movement also

integrates twelve-tone and free atonal material. The latter movement’s B section, itself

the center section of an ABA’ form, is not based upon a twelve-tone row. However, the

main thematic idea, a series of six hexachordal descents occurring within a very dense

musical texture, dialogues directly with the main twelve-tone theme of the outer A

sections through associations based upon membership in the same set-class. Example

84

4.10 presents a detailed diagram of these descents, depicting them as noteheads on the

staff and indicating the measures in which they occur. Below the staff is another detailed

diagram demonstrating the two transformational paths the initial row’s hexachords

traverse within the B section.87 The leftmost section of the staff, corresponding to mm.

13-16, presents the row form P9(2-R1) (discussed above as the first row of the first

canonic pairing) divided into its constituent hexachords. These hexachords are also

interpreted below the staff divided in two ways: above and below the boxes as members

of [013478] (above) and [012569] (below) and broken into the row’s discrete trichords as

members of Tn-types [037]. The modest transformation network above the boxed figures

depicts the motion of the row’s first hexachord through the upper register hexachords

appearing on the musical surface as quarter-note triplet descents in mm. 36, 38 and 44.

As illustrated by the arrows connecting them, these hexachords are all transpositionally-

related, moving via T5, Tt and T5.88 The bottom network depicts the motion of the row’s

second hexachord through the lower register hexachords appearing on the musical

surface in a variety of rhythmic presentations in mm. 37, 39 and 45 (immediately

following their upper register presentation). As illustrated by their arrow labels, the first

of the B section’s lower hexachords is inversionally-related to its A section analogue,

appearing in its Ie form. However, a second inversion I2 returns it to its original form

before its final motion via T5, underscoring the role of the m. 39 presentation as

restorative.89

[Example 4.11]

87 The diagram uses the standard mod12 integers to represent pcs. 88 On the musical surface, these figures all appear in violin I. 89 The first two of these hexachords appear in the cello, while the last appears in the viola.

85

While the hexachordal aspect of the B section provides the strongest link between

the twelve-tone A sections and the free atonal B section, the trichordal aspect is revealing

as well. The above analysis of the row and its cyclic structure places a premium on the

first hexachord’s whole-tone elements, especially emphasizing the 4-cycle {F-A-C#} as

its fundamental structural component. When viewed from a cyclic perspective, P9(2-

R1)’s first hexachord disunites into two whole-tone trichords {F-A-C#} [048] and {D-E-

G#} [026], from the opposing C# and C whole-tone collections. The final hexachord of

the B section, appearing in the rightmost box in Example 4.10, reveals that this initial

abstract segmentation is articulated concretely on the musical surface in the closing

melodic gesture of the B section. The viola statement <B-G-Eb-Gb-Ab-D>, pictured to

the right of the double bar in Example 4.11, presents members of set-classes [048] and

[026], stating overtly an initially abstract yet fundamental relationship of the row’s

signature gesture.

86

Chapter Five: Conclusions and Implications for Further Research

I. Conclusions

The methodology developed and applied through the analysis in this dissertation

reveals consistency among the diverse musical surfaces presented in Ginastera’s First and

Second String Quartets. In particular, three main avenues of compositional consistency

emerge as relevant in the chosen quartets. The first avenue addresses general issues of

tonality, such as the existence of and compositional contexts for referential collections

and their relationships to the various structural tonal centers and axes of symmetry

appearing in the music. The second avenue engages motivic issues and the consistent

manner in which pc motives emerge from the quartets’ structural cyclic tonal background

to comprise significant thematic and developmental constructions. The third avenue lies

in consistencies in the rhetorical unfolding of material in specific movements and how

that material is linked to the movements’ tonal, motivic, and formal constructions. As the

analytic chapters demonstrate, the fundamental precept supporting each of these avenues

of consistency is the interval cycle.

From a tonal perspective, Kuss’ ascription of the Berger/Van Den Toorn concept

of octatonic-diatonic interaction to Ginastera’s music is indeed borne out in the quartets

and provides a general framework for understanding the various pitch structures that

appear consistently and saliently in the music. As established by the detailed analyses of

recurring pitch structures in Chapters 3 and 4, the basic octatonic-diatonic frame can be

defined in terms of interval cycles and cyclic collections undergirding the diatonic and

octatonic collections. When these 3-cycle and 5-cycle collections combine with hexatonic

87

and whole-tone collections created by aligned 4-cycle collections (which initially appear

in the First Quartet and recur ubiquitously in the Second), a more complete and nuanced

picture of Ginastera’s tonal language emerges. For example, in the first movement of

both quartets, the 5-cycle pentachord {C-G-D-A-E} formed by the open strings of the

violin, viola and cello provides an a priori referential set whose constituent conjunct

trichords, referred to as shadings of the overall collection, play a critical role in defining

local referential collections. Both First Themes in the quartets’ first movements are

significantly octatonic, yet the particular 3-cycle-based collections within the themes are

ultimately controlled by the 5-cycle dyad spaces, resulting in passages in which the 3-

cycle melodic material colors the overarching referential 5-cycle. Ultimately, the analyses

of Chapters 3 and 4 demonstrate this nuanced cyclic approach in both free atonal and

twelve-tone music, strengthening the connection between Ginastera and major

proponents of cyclic organization, Bartòk and (above all) Berg.

In addition to tonal issues, motivic issues also comprise a major component in the

quartets under study. Specifically, the main thematic construction of the quartets’ first

movements can be modeled by a systematic unfolding of a single ordered, motivic

trichord via multiple U(st)-transformations, thereby locating within the motive and its

intervallic properties the potential for significant musical expression. For example, the

Chapter 3 analysis of the First Theme of the First Quartet’s first movement considers the

entire theme as a U-chain formed by three successive transformations of the movement’s

first three notes <D-F#-F>. Three of the four major thematic elements of the four main

themes of both quartets’ first movements feature U-transformations at their very core,

and the terminology developed in Chapter 2 greatly facilitates the discussions of the

88

movements’ themes and their roles within their respective movements. The analyses

based on U-transformations not only value highly the role of the motive in the quartets,

but they also participate significantly in the cyclic aspect of the music, as multiple U(st)-

transformations of a source set result in cyclic pc collections.

The unfolding of connected trichords also links Ginastera to Webern, who

famously works with small sts in such pieces as the Fünf Sätze für Streichquartett Op. 5,

no. 3 and the Concerto for Nine Instruments, Op. 24. It is well known that the former

piece and movement feature significant trichordal material as the basis of its composition;

the movement even features two inversionally related extended U-chains <C#-A-G#-E-

D#-B-Bb> and < C-E-F-A-Bb-D-Eb> in violin I and vcl respectively in m. 7. The row of

the concerto is a derived row, exclusively featuring members of [014] in each of its

discrete trichords, and while no overt connection exists between the contextual

transformations defined in Chapter 2 of this dissertation, it is plausible that similar

mechanisms could be defined for a detailed investigation of the work. The notion of

connected trichords also links Ginastera to Schoenberg, who in his well-known radio

address on the Op. 22 Orchestral Songs describes the “developing variation” style of

unfolding trichords and discusses the concepts of Grundgestalt and Gedanke. These play

critical roles not only in his works but also appear in the works of Berg, Webern, and, as

discussed in Chapter 3 of this dissertation, Ginastera.90

90 As Boss 1992 indicates, “the original typescript of the lecture, in German, is in the archive of the Arnold Schoenberg Institute in Los Angeles. An English translation by Claudio Spies was published as ‘Analysis of the Four Orchestral Songs Op. 22’ in Perspectives of New Music 3, no. 2 (1965) and appears in Perspectives on Schoenberg and Stravinsky 2nd ed., Benjamin Boretz and Edward T. Cone (New York: Norton, 1972.) Currently, the archive resides in Vienna.

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Returning to the Ginastera quartets, each of the movements studied employs a

similar rhetorical structure at significant formal junctures of the pieces. Major formal

sections often begin with a salient motivic and/or harmonic statement that establishes a

context for subsequent compositional reinterpretation. In the First Quartet’s first

movement, the aforementioned motive (and potential Grundgestalt) <D-F#-F>

participates significantly in the movement’s first theme and its tonal center, while in the

Second Quartet’s first movement, the initial trichord <D-Bb-C#> [014] and salient

trichord <Bb-F-E> [016] in tile 1 establish contexts for their significant appearance in the

motivic and harmonic character of the main themes. In the Second Quartet’s mostly serial

second and third movements, initial motivic statements <Eb-Ab-Cb-G> [0148] and <C-F-

B> [016] and their internal cyclic intervallic properties provide not only the source of

subsequent melodic constructions, but the statements’ transformation establishes cyclic

harmonic constructions critical to the piece’s harmonic and formal structures. Ultimately,

such consistency in the movements’ rhetorical unfolding could provide a potent insight

into the composer’s practice and should continue to do so in other of his pieces.

II. Suggestions for Further Research

Despite the considerable amount of highly-focused analytic detail on significant

aspects of the quartets’ tonal language and thematic construction offered in the previous

chapters, it represents a mere fraction of the potential analytic and music-theoretic work

suggested not only in the pieces themselves, but in all of the composer’s work and

beyond. The continuation of the present chapter elucidates several avenues for further

research. In general, the suggestions divide into two main categories, analytic and

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theoretic. The former category begins with the first steps of an analysis of the 14-bar

theme of the Second Quartet’s fourth movement, a theme and variations featuring

rhapsodic solo passages in each of the string quartet’s voices. This initial analytic

category continues in illustrations of brief passages from the composer’s Third String

Quartet Op. 40 (1973) and Sonata for Guitar Op. 47 (1976) that clearly demonstrate the

applicability and relevance of the dissertation’s methodology to a broader sample of the

composer’s work. It closes with a brief analytic demonstration of the general

methodology in a passage from Berg’s Op. 4, No. 2 in an effort to place Ginastera’s

works in dialogue with other music that has received considerable analytic attention. The

discussion of the latter category addresses some theoretic elements introduced in Chapter

2 and speculates on how they may be developed further.


The theme of the Second Quartet’s fourth movement is a sparsely-accompanied

14-measure rhapsodic violin solo whose three basic thematic statements establish 3-, 4-

and 5-cycle contexts for the theme’s focal pc G. The first two statements, depicted in

Examples 5.1a and b respectively, present two versions of the movement’s main idea.

The former clearly illustrates the establishment of G4 as the gesture’s melodic touchstone

in mm. 1-4, while the latter depicts the note’s maintenance within the second statement

beginning in m. 6. Both of these beginnings show the placement of the central G within

two cyclic contexts: the former presents G within the pentachordal subset <G-F-E-D-C#>

of the C/C# octatonic collection, while the latter presents G within the movement’s main

4-cycle {G-Eb-B} emphasized on the offbeats of beats three and four. The final cyclic

contextualization of G within the 5-cycle OS-set is initially hinted at in the

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accompanimental chords that enter in mm. 3 and 4, in which the violin I G occurs

harmonized in the later measure by {B-C-D#-E}. The resultant pentachord, a member of

[01458], presents the focal G both as a member of the main 4-cycle collection and paired

with the cello C3 in an initial expression of the OS-set. These two cyclic associations, G

within the main 5- and 4-cycle sets, form the basis of the theme’s closure in mm. 13-14,

which is depicted in Figure 5.1c. This chord, itself a member of [0124589], clearly arises

from the conglomeration of two members of [0148], the 4-cycle+ collections {C-G-Eb-

B} in the cello and viola and its T7 transposition {G-D-Bb-F#} in the viola. Thus, the

theme’s final chord expresses the 5-cycle OS shading {C-G-D} in the accompanimental

voices’ bottom strings and links it to 4-cycle sets {G-Eb-B} and {D-F#-Bb}. Ultimately,

the cyclic contexts for the focal G are a fundamental structural feature of the remaining

three variations and help to clarify the role the movement plays in the entire quartet.

[Example 5.2]

Although this dissertation focuses on Ginastera’s first and second quartets, his

(final) String Quartet No. 3 also occasionally features musical constructions that can be

92

modeled by the analytic techniques developed in Chapter 2.91 Example 5.2 presents the

first seven measures of Movement III, which slowly assembles the 3-cycle trichord {E-G-

Bb} in mm. 6-7 through the unfolding of source trichord <E-F#-G> in the viola (mm. 2-

4) into <G-Ab-Bb> (mm. 4-6) in violin II (both are sc[013], although they are

inversionally related) via a single U(ff) transformation. Ultimately, the techniques

developed in this dissertation are of limited utility in the third quartet, but they do reveal

similar constructions among all the quartets and potentially provide an analytic point of

departure for the final quartet.

[Examples 5.3a, b, c and d]

The methodology developed in this dissertation is also applicable to other of

Ginastera’s works. Examples 5.3a, b, c and d depict the beginnings of analyses of the first

four phrases of the composer’s Sonata for Guitar, illustrating significant passages

comprised of interval cycles (Example 5.3a), extended U-chains (Ex. 5.3b), TTO cycles

(Example 5.3c), and a device I will describe as K-net “hyper cycles” which derive from

K-net interpretations of extended passages featuring TTO cycles (Example 5.3d).

Example 5.3a depicts the First Movement’s first phrase, which begins with a presentation

of the guitar’s open strings (the “guitar chord”), followed by a rubato passage comprised

91 The third string quartet contains five movements for string quartet and soprano voice in a work reflecting the composer’s affinity for Schoenberg’s Op. 10. The texts for the quartet are by 20th-Century Spanish poets Juan Ramón Jiménez (La Música, Movement I and Ocaso, Movement V), Federico García Lorca (Canción de Belisa, Movement III), and Rafael Alberti (Morir al Sol, Movement IV). Whereas the first two quartets are neo-classic in style, the third clearly reflects the experimentalism associated with the 1960’s and thus lies outside the scope of this dissertation. Of the third quartet and its context within his contemporaneous works, the composer notes “As in my most recent works…I have made use of a technique based on the interplay of fixed and variable structures, and on the creation and organization of multidimensional space wherein develop infinite phenomena – and corresponding resonance – of the ever-changing universe of sound.” See “Composer’s Note” (1973) in Boosey & Hawkes score, copyright 1977.

93

exclusively of 5-cycles in an exposition of one of the sonata’s fundamental sounds.

Example 5.3b presents the movement’s second phrase, in which a second statement of the

guitar chord is followed by a series of ascending chords that alternate between members

of [015] and [037] respectively (excluding the pedal A); the topmost voice <F-E-G-F#-A-

G#-B-Bb-C#> is an extended U-chain, while the trichords including the notes of the

upper extended U-chain form a pattern well-modeled by a pattern of K-net hyper

transformations. Example 5.3c presents a similar passage in its depiction of phrase three,

which illustrates a T-cycle (T3 in the present case) arising from the patterned motion of

the passage’s upper voices. Finally, Example 5.3d presents the movement’s fourth

phrase, in which another replicated pattern results in an upper-voice U-chain atop a

succession of tetrachords which, when interpreted as K-nets in a manner similar to

Example 5.3b, form a predictable cycle of hyper T-transformations, resulting in a pattern

of hyper T-transformations that I will call “hyper cycles.” Certainly, Examples 5.3a

through 5.3d are very speculative and are included here to suggest potential applications

of this dissertation’s methodology.

[Example 5.4]

Although this dissertation’s methodology and analytic approach arise in response

to some particularities of the tonal language, thematic and motivic process in Ginastera’s

first two quartets, similar peculiarities exist in the music of other 20th-Century composers

whose music the composer is known to have appreciated. The second song of Berg’s Op.

4 (“Sahst du nach dem Gewitterregen” from the Altenberg Lieder) provides a prime

94

example of music literature that displays such particularities.92 Example 5.4 provides a

reduction of two significant passages from the beginning of the song. The leftmost

measure marked “Gesang” features the first two measures of the vocal part, while the

rightmost measure presents a rising figure in the horn and bassoons in mm. 3-4

immediately following the initial vocal intonation. Beneath the example appear three

analytic diagrams indicating the Tn-type ([034]) formed by the initial vocal gesture <Bb-

B-G>, the U-cell <Eb-B-A>-<E> ([0167]) formed by the trailing end of the initial vocal

gesture, and the extended U-chain <B-D#-D>-<F#-F-A-Ab> formed by four U(st)

transformations of the source trichord <B-D#-D> in the horn and bassoon ascent. As

demonstrated in Headlam 1996, members of [014] and [016] play a vital role in the song;

presently, passages making significant use of members of [014] and [016] are included

merely to identify some bedrock methodological constructions of this dissertation within

significant passages of Berg’s song and demonstrate the viability of the present

approach.93

In addition to demonstrating promising potential for further analytic research, the

methodology also suggests a few areas for further theoretic research as well. The

following section identifies three main ideas contained previously in this dissertation and

expounds briefly on how these ideas could potentially be developed into viable theoretic

research. The first discussion revisits the useful concept of shading and leads to a second

(and related) discussion of “contextual transposition.” The final suggestion revisits the

92 See Headlam 1996, pp. 86-94 for a detailed analysis of the song from a cyclic perspective. 93 A second [016]-based U-cell occurs at the climax of the song in m.8, where it appears in a dramatically high and soft vocal subphrase on the text “Siehe Fraue.”

95

notion of extended tertian sonorities as verticalized U-cells and their extensions and

suggests some potentially relevant music literature for such an inquiry.

At its core, the concept of shading is designed to easily accommodate contextual

shifts among various subsets of a background cyclic collection without invoking a

specific TTO. For example, the shift in contextual emphasis from the subset {CGD} in

the First Theme of the Second Quartet (see Chapter 4) to the subset {DAE} in the Second

Theme is most simply understood as a change of focus within a sui generis referential 5-

/7-cycle {CGDAE} formed by the open strings of a string quartet rather than the result of

either T2 or I4 of the initial collection. Since such an understanding privileges the notes of

the cyclic collection over the repeated transposition operation that creates the collection,

shading could prove a useful concept in the analysis of any music that features prominent

background cyclic collections, such as the music of Berg, Stravinsky, Bartòk, and

Scriabin.

In a sense, a change in the shading of a particular cyclic pc collection is a type of

contextual transformation performed on the larger collection’s subsets. Since the shaded

subcollection is both inversionally and transpositionally symmetrical, a change in shading

could be considered either a contextual inversion or transposition operation. In

transformational music-analytic literature, especially in the works of David Lewin, the

concept of contextual inversion is well established and employed. However, the notion of

contextual transposition is at present unexplored. Presently, the motion from {CGD} to

{DAE} in the above example can be understood in two basic ways; the motion either

arises from shading of the OS set or results from performing a TTO (T2 or I4). If the a

priori construct of the governing 5-/7-cycle OS set is accepted as a structural entity, then

96

the motion from the former to the latter trichordal subset could be considered a contextual

transposition T2n, with n representing the cyclic interval.94 Ultimately, the use of both

contextual and non-contextual transpositions could enrich the analyst’s understanding of

a cyclic piece’s structure by separating transpositions within a cyclic structural entity

from transpositions outside that entity, potentially providing a more nuanced analysis.

[Example 5.5]

Finally, Example 2.7b in Chapter 2 illustrates a remarkable connection between

collections created by U-transformations of triadic source trichords and extended tertian

“third stacks” common to jazz, popular music, and occasionally Western music theory

and repertoire. Example 5.5 recreates part of Headlam’s (1996) Example 1.13b, which

depicts the beginning of a chord progression appearing in a famous passage in the first

several measures of Berg’s Op. 2, no. 2. Headlam’s point in the example is to illustrate

how a specific alignment of descending 1-cycles and an ascending 5-cycle creates a

musical surface in which the vestiges of late tonal practice (the ideas of falling fifths and

“Fr+6” chords, members of [0268]) and an emergent cyclic atonal language coexist

almost indistinguishably. The present example reinterprets the first three chords of

Headlam’s example with standard jazz/pop symbols for each chord. Appearing below the

chord symbols are diagrams indicating the potential interpretation of each of these chords

as U-cells of the given source trichords. Although the Berg passage is unequivocally in

the Western Art tradition, it employs chords found in jazz and popular traditions. Thus,

the potential for a cyclic-based understanding of jazz and popular music theory exists,

94 In the present formulation, the collection is considered a 7-cycle, with 2n equaling 2 (14 mod12).

97

and aspects of the methodology developed in this dissertation could prove useful in such

an inquiry.

Finally, the methodology employed is used to begin to answer a broader question

of the place and nature of Ginastera’s String Quartets in his oeuvre. In response to some

scholars problematic “style period-based” view of Ginastera’s work, this dissertation

engages the debate from two angles, one theoretic and one epistemic. In the former, the

emergence of the structural 4-cycle in the Second Quartet marks a distinct departure from

the octatonic/diatonic 3- and 5-cycle tonal language of the First Quartet, yet the

methodology demonstrates consistency in the use of 3-, 4- and 5-cycle elements. This

demonstrated consistency informs the latter, epistemic angle that questions the general

efficacy of a rigid period-based understanding of the composer’s work. Indeed, the

remarkable rhetorical and stylistic consistency across the composer’s work suggests that

a more comprehensive investigation of similarity, if not unity, in his output would

provide a clearer understanding of Ginastera’s music and its role in the 20th-Century. As

discussed previously, the “period-based” perspective on the composer’s music remains

problematic for the preeminent Ginastera scholar Kuss, and the approach developed in

this dissertation supports her general position by demonstrating an overarching

consistency between the first two quartets, works composed ten years apart.

Fundamentally, the broad question considered here lies at the intersection of a

20th-Century composer’s work and the issues it raises for the practice of music analysis.

As is the case with any music-analytic project, the careful study of the music leads the

analyst through a select group of concerns of seemingly divergent natures; the analyst

must define and contend with numerous questions regarding the composer’s background

98

and aesthetics, scholars’ engagement with the composer’s life and music, the suitability

of existing analytic tools and, as is the case in the present study, the panoply of issues that

arise in the building of novel theoretic and analytic tools in response to significant aspects

of the music’s construction. Scholars’ engagement of Ginastera has produced a body of

work marked by varying degrees of quality and success and has resulted in an appreciable

degree of understanding of the composer and his music. The analytical framework of this

dissertation does provide some insight into a few technical aspects of the composer’s

music and offers a viable path to an increased understanding of how the work of this

remarkable composer opens a dialogue with the work of other composers who receive

considerably more attention from the scholarly community.

Appendix One

Appendix One provides a detailed account of how each of the six U-

transformations transforms the given source trichord <265>, the first three notes of the

First Quartet’s first movement (see Example 2.3b). In so doing, Appendix One illustrates

two essential pieces of information: 1) the reason behind each operation’s order and 2)

the direct link between the interval from the first to the third terms in the source trichord

and the predominant interval cycle in the pc collection resulting from multiple

applications of the particular transformation. The overall progression of transformations

in Appendix One is from the smallest order (2) to the largest (8). It is essential to note

that the order of each transformation is dependent upon the interval between the first and

third terms of the source trichord.

Given source trichord <265>, f=2, s=6 and t=5; s-f=4, t-s=e, and t-f=3.

Order 2: U(ft)p and U(ss)

p

U(ft)p: <265> - <215> - <265>

U(ft) is an RI operation (sum 7) which maps <265> into <215> and back to <265>. By

definition, 2 maps into 5 (f into t) and 5 into 2; 6 maps into 1. <265> inverts to <512>,

which is in turn retrograded to <215>. Since the domain and the image have the same pcs

in the same ops, the second transformation returns the original source trichord, hence the

order is 2.

U(ss)p: <265> - <76t> - <265>

U(ss) is an RI operation (sum 0) which maps <265> into <76t> and back to <265>. By

definition, 6 maps into itself (s into s); 2 maps into t and 5 maps into 7. <265> inverts

into <t67>, which is in turn retrograded to <76t>. Since the domain and range have the

same pc in s, the operation must be an involution, hence the order is 2.

Order 4: U(ff)p and U(tt)

p

U(ff)p: <265> - <et2> - <80e> - <548> - <265>

U(ff) is an RI operation which in this case alternates between sums 4 and t, depending on

the identity of f in each trichord. The first operation maps 2 into itself (f into f) via sum 4;

6 maps to t and 5 maps to e. The retrograde of the image reverses the ops of the first and

third terms. Since the first term of the domain inverts into the last term of the image and

vice-versa, the ic between f and t in both domain and image is preserved. However, since

the relationship between the two is inversion, a partial interval cycle based on the ic from

f to t results; 2 and 5 (f and t of the domain) invert into 2 and e via sum 4, which is then

retrograded to e and 2. The resulting union of both domain and image f and t is {e25}, a

member of [036]. Since by definition s never becomes f or t, the chain returns the original

source trichord after the 3-cycle closes with the fourth transformation. The total pc

content of the chain, {024568te}, is a member of [0124678t], and the complete 3-cycle

subset is {258e}.

U(tt)p: <265> - <548> - <80e> - <et2> - <265>

U(tt) operates identically to U(ff), except it produces a retrograde.

Order 8: U(fs)p and U(st)

p

U(fs)p: <265> - <326> - <e32> - <0e3> - <80e> - <980> - <598> - <659> - <256>

U(fs) is an RI operation which is substantially different from the above in its exchanging

of outer and middle terms. The first transformation inverts <265> into <623> via I8

(f+s=8), which in turn is retrograded into <326>. Since the operation is a retrograde, the

dyad <26>, occupying fs of the first domain, moves into st of the image. The second

transformation completes the position shift of the initial pc 2 from f of the initial source

trichord, to s of the second, and finally to t of the third, taking a second step to

accomplish what U(ff) (order 4) did in one. However, like the U(ff) example, the partial 3-

cycle {e25} remains in f and t of the original source trichord and its second

transformation. The difference in the orders of both results from the additional

transformation required to map the pc in f into the pc in t. The total pc content of the

chain is {235689e0}, a member of [0134679t] (octatonic).

U(st)p: <265> - <659> - <598> - <980> - <80e> - <0e3> - <e32> - <326> - <256>

U(st) operates identically to U(fs), except it produces a retrograde.

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