PITCH AND INTERVAL STRUCTURES IN GEORGE PERLE'S THEORY OF WLVE-TONE TONALITY Gretchen Foley Faculty of Music Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy Faculty of Graduate Studies The University of Western Ontario London, Ontario December 1998 8 Gretchen Foley 1999 National Library Bibliotheque nationale B*m of Canada du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellington Street 395. rue Wellington Ottawa ON KIA ON4 Ottawa ON KIA ON4 Canada Canada Your Me Vofre reference Our &I None refdrencs The author has granted a non- L'auteur a accorde me licence non exclusive licence allowing the exclusive pennettant a la National Library of Canada to Bibliotheque nationale du Canada de reproduce, loan, distribute or sell reproduire, preter, distribuer ou copies of this thesis in microform, vendre des copies de cette these sous paper or electronic formats. la forme de microfiche/filrn, de reproduction sur papier ou sur format electronique. The author retains ownership of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts fiom it Ni la these ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent Stre imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract George Perle, American composer and theorist, has authored an innovative theory called Twelve-Tone Tonality (1977; znded. l996), a compositional system based on the conjunction of interval cycles and inversional symmetry. Perle's theory is the result of his search for an atonal analogue to the precornpositional structures and hierarchical organization of tonal music. The theory is not widely known in the theoretical community, partly due to its compositional purview and idiosyncratic terminology. This dissertation explicates Perle's theory in a reorganized, accessible format, and includes the most recent developments to the theory that appear in the second edition of Twelve-Tone Tonaliv (1996). The study also explores the properties of the cyclic sen, the fimdamental entities of the theory, outside the context of twelve-tone tonality. Through the process of imbrication the cyclic sets generate close associations of pitch class set classes identified as ICS families (for -'imbricated cyclic set"). The set classes in these families share a number of structural properties, including inversional symmetry, transpositional combination (as developed by Richard Cohn l987), and equivalence in other modular universes. The study also introduces an original simiiarity relation, the Rsvnfrela&ion,to reflect the symmetrical nature of the intervdlic similarity between pairs of set classes in the ICS families. The dissertation provides detailed analyses of two etudes from Perle's Six E~zrdesfor Piano (1973-76). The analyses differentiate between the abstract dimension of twelve- tone tonal constructs and their concrete realization at the musical surface, and show both local and long-range structure. Further, although the analyses are guided primarily by Perle's theory of twelve-tone tonality, they are supplemented by observations from the perspective of pitch class set theory, as developed by Allen Forte (1973) and extended by Robert Moms (1995b). This multifaceted analytical approach highlights distinctive features of the etudes. Through the presentation of the tenets of twelve-tone tonality, the theoretical exploration of the cyclic sets, and the analysis of selected works, the dissertation aims to show the depth and potential of the theory, both within and outside its own context. Keywords: atonality, axis of symmetry, Richard Cohn, equivalence classes, Allen Forte, IcVSIM, interval cycles, inversional symmetry, Eric Isaacson, modular equivalence, Robert Morris, George Perle, pitch-class set theory, similarity relations, transpositional combination, twelve-tone tonality To Brian, Michael, and Alison Acknowledgements I wish to offer my deepest gratitude to my advisor Catherine Nolan, whose thought- provoking questions and criticisms have been a constant source of inspiration. The generous amounts of time and encouragement she has given me have been greatly appreciated. I would also like to offer my sincere thanks to Richard Parks, who has been involved in this project fiom the outset, offering perceptive and timely advice. To George Perle, who kindly sent me some preliminary analytical materials pertaining to his Six Etudes for Piano, and to Elliott AntokoIetz, who first introduced me to the theory of twelve-tone tonality, I extend my thanks. With appreciation I acknowledge GunMar Music Inc., the copyright holders of George Perle's Six Etudes for Piano (0l977), who granted me permission to reproduce Etudes No. 1 and No.4 in this dissertation. I would like to thank my friends Betty Anne Younker and Dale Inder, whose warm hospitality during the writing of this dissertation eased the whole process immeasurably. My parents Mark Schoenberg and Alice Vassallo have been constant in their moral support, encouragement, and empathy. I offer them my heardelt gratitude. My children Michael and Alison deserve special mention. Their patience, understanding, and gentle gestures of love have shown them to be mature beyond their years. My husband Brian has been my pillar of stren=gth in every way. His belief in me has never wavered; he has supported me at every turn, and has consistently but quietly created the environment necessary for me to work toward my goals. This dissertation is the result of a true partnership. Table of Contents Page Certificate of Examination .. 11... Abstract i11 Ackno wledgernents v i Table of Contents vii List of Examples is List of Figures xii List of Tables xvi Chapter One Introduction Perle 's analylical studies of hventieth-century works Other analytical investigations of interval cycles in twentieth-century works Theoretical investigations of interval cycles Analytical studies of Perle 's [heory of twelve-tone tonality Two The Theory of Twelve-Tone Tonality Cyclic sets Arrays and array segments Transposition and inversion The cognate relation Secondary d'erences and sums Modes Keys Synoptic arrays Tonality Modulation vii Three Structural Properties of Cyclic Sets The partitions of the intend-class cycles imbrication of cyclic sets The ICS families Transpositional combination in ICS families Similarity relations within KSfamilies Modzrlrrr equivalence between lCS families Four Analysis of Etude No. 1 and Etude No.4 fiom Si=c Etudes for Piano by George Perle The annotated scores Analysis of Etde Noo.I from the perspective of twelve-tone tonulily Analysis of Etude No. ilfrorn the perspective of twelve-tone tonality Set-theoretical perspectives Five Conclusions The propagution of the lheory of twelve-tone tonali~ Critical reception of Perle 's theory of twelve-lone tonality Szrggestions for firrher study Appendix One The Cyclic Sets of All Cyclic Intervals (Mod- 12) Two Set Class Membership in ICS Families Three Annotated Score of Etude No. I from George Perle's Six Etudes for Piano (1973-76) Four Annotated Score of Etude No. 4 from George Perle's Six Etudes for Piano (1973-76) Five Glossary of Terms from Perle's Theory of Twelve-Tone Tonality 199 Select Bibliography Vita List of Examples Example Description Page Scriabins's master scales (a), interval2 cyclic segment created 4 by altered pc within the 7-3 1 subset of the octatonic collection (b), and the interval-1 and interval-4 cyclic segments created by addition of non-octatonic pc to 7-3 1 subset (c) Interval-cycle segments in Berg's Op.2, No2 Unfolding interval-cycle segments in first ten measures of first movement, Berg's String Quartet, Op.3, No.3 Basic cells in Berg's Wozzeck (a) and Ltrlu (b) Series of chords generated by combination cycles as an Ives Omnibus Melodic sequences that illustrate "serial" properties in their component diatonic interval cycIes Interval cycles Cyclic sets of interval 1 (a) and interval 7 (b) Correspondence between axis-dyad chord structure and notation in mrn.27-3 1, Etude No. 1 Corresponding passages in mrn.5-7 of A-section (a) and mrn. 13- 1 5 of B-section (b), Etude No. 1 Transitional passage of sum tetrachords, m.25, Etude No. 1 Dyadic rotation in alternating tetrachords, m.5, Etude No. 1 Linear gesture, m. 1, Etude No. 1 Example Description Page 4.6. Dyadic sum patterns in melodic gesture of array ptpehei8, mm.3 -4, Etude No. 1 4.7. Melodic gesture, m.8, Etude No. 1 4.8. Crossing interval- 1 and interval-9 cycles, mm.9- 10, Etude No. l 4.9. Mutated array segments in the repeat of episode A'-b', Etude No.4 4.10. Mutation of individual voices of mm. 1 and 10 in m. 16, Etude No.4 4.1 I. Mutation of rn. 16 in m. 17, Etude No.4 4.12. First phrase of phrase group B-e (rnm.39-40), derived f?om first phrase of phrase group B-d (mrn.32-33) through mutation, Etude No.4 4.13 - First phrase of phrase group B-f (mm.47-48), derived from linking passage (m.29) through mutation, Etude No .4 4.14. Opening of linking passage between B- and A'-sections (m.59) derived from opening of linking passage between A- and B-sections (m.29), Etude No.4 4.15. Mutated repeat of subject A'-b' (derived from subject statements in mrn.63-64 and rnm.72-73), rnm.78-79, Etude No.4 4.16. Realization of sum tetrachords as passing motions in mutated repeated subject statements of A'-c' subsection, rnrn.84, 86, 88, and 89, Etude No.4 4.17. Embedded passing and neighbour gestures in vertical and linear dyads, mrn.42-47, Etude No.4 4.18. Corresponding gestures embedded in treble and bass parts, mm.44-47, Etude No.4 4.19. Embedded passing gestures, mrn.52-55, Etude No.4 4.20.