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Smooth Voice-Leading Systems for Atonal Music Smooth Voice-Leading Systems for Atonal Music Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Yeajin Kim, M.M. Graduate Program in Music The Ohio State University 2013 Dissertation Committee: David Clampitt, Advisor Anna Gawboy Marc Ainger Copyright By Yeajin Kim 2013 Abstract In this dissertation, I introduce and develop a methodology to analyze atonal music from the view-point of voice-leading. I organize pitch-class sets having the same cardinality to demonstrate what is defined here as smooth voice-leading. I first deal with the historical study relating to voice-leading systems for atonal music, a heavily researched field, and I focus on the interval array in pitch-class series. This dissertation introduces a partition of Schoenberg‘s signature set, the 6–Z44 hexachord, into four systems (orbits), in close analogy with Cohn‘s four systems of maximally smooth cycles of hexatonic triads, and offers analytical examples of pairs of adjacent sets from this perspective. Selected analyses demonstrate its applicability. Moreover, I suggest relatively smooth voice-leading systems covering Allen Forte‘s all pitch-class sets. The relatively smooth voice-leading systems are organized into groups according to criteria of shared interval-string contents from the pitch-class sets‘ arrays. In groups of pitch-class sets sharing a cardinality and sharing interval multiplicities in their interval string arrays, I propose smooth voice-leading systems. Voice leading within pitch-class sets in the same group could be considered relatively smooth compared to other voice leadings, either those defined by Cohn to be maximally smooth, or between pitch-class sets in different groups according to the interval-string content criteria, which ii are not smooth at all. Definitions of these concepts are provided in the dissertation. iii Dedication Dedicate to My Mother iv Acknowledgment First, I really want to thank earlier scholars who have studied atonal music since their numerous papers encouraged me to consider various ideas concerning analytical methodologies for atonal music, as well as made me have enthusiasm for searching for the latent logic in atonal music. I also thank my family for their support during the preparation of this dissertation. Especially, I would like to thank to my husband, Seung-ho and my daughter, Sarah. They waited for me and encouraged me with love throughout my doctoral studies so that I could finish my doctoral program. Special gratitude from deep in my heart goes to my mother in heaven who gave me plenty of love and support. In addition, I wholeheartedly appreciate my advisor, Professor David Clampitt who offered his encouragement, criticism, comments and suggestions. Finally, I want to send my highest appreciation to God who makes it all possible. v Vita 2004……………………………………….………….……..B.M. Music Composition Yonsei University, Seoul, South Korea 2007……………………………………………….………..……M.M. Music Theory Yonsei University, Seoul, South Korea 2008–2009………………………………...…………….Graduate Teaching Associate, School of Music, The Ohio State University 2011 to present………….…………..…………………………………………Lecturer Yonsei University, Seoul, South Korea 2012 to present....…………………...………………………………………….Lecturer Korean National University of Arts, Seoul, South Korea 2012 to present……...………………..……………..…………………………Lecturer University of Seoul, Seoul, South Korea vi Presentations An analytic Method for Atonal Music based on Cognitive Considerations, Society for Music Perception and Cognition (SMPC), Indiana University-Purdue University, Indiana, USA, 2009 Voice-Leading Systems of Schoenberg‘s Signature Hexachord, Music Theory Mid- West (MTMW) (with David Clampitt) , 21st Annual Conference, Miami University, Ohio, USA, 2010 Fields of Study Major Field: Music Area of Specialization: Music Theory vii Table of Contents Abstract………………………………...……………………………………………...….ii Dedication…………………………………………………………………..…………….iv Acknowledgments……………………………………………………………………...…v Vita……………….………………………………………………………………...…….vi List of Figures…........……………………………………………………………..…...….x List of Tables…………..…………………………………………………………..……xvi List of Examples……………………..…………………………………….…….…..…xvii Introduction …………….……………..………………………………………………….1 Chapter 1: Historical Literature Review…………………………………….…………….3 Chapter 2: An Appropriate Analytic Methodology for Atonal Music……….……..…….17 Chapter 2.1: Consideration of motive …………..……………………………………….17 Chapter 2.1.1: Pattern-completion and Prerequisite conditions………………………….19 Chapter 2.2: Latent axis theory…………………………………………………………..29 Chapter 2.2.1: Analysis of Webern‘s op. 16, no. 2 ………………………………………40 Chapter 3: Theoretical Exposition of Smooth Voice-Leading Systems for Atonal Music ……………………………………………………………………………………………61 Chapter 3.1: Voice-leading systems within the Schoenberg hexachord…………………61 Chapter 3.1.1: Examination of 6-Z44 hexachord systems……………………………….74 Chapter 3.2: Relatively Smooth Voice-Leading Systems……………….……………….86 viii Conclusion…………………………………………………………………………..…129 Bibliography….……………………………………………………………………..…130 Appendix A: Forte, 1973, Prime Forms and Vectors of Pitch-Class Sets……….……..136 ix List of Figures Figure 1.1. Two musical subjects and the interval from s to t ………………………..…..5 Figure 1.2. Hyer‘s tonnetz………….………………………………………………….…..8 Figure 1.3. Richard Cohn‘s Hyper-Hexatonic System..……………...……...……….........9 Figure 2.1. Bartók, Fourth String Quartet, mvt. I, X-Y-Z progressions in exposition and development…………………………………………….………….…………………….31 Figure 2.2. Alignment of two inversionally complementary semitonal cycles intersecting at a dual axis of symmetry of sum 3………………………………………..……………32 Figure 2.3. George Perle‘s interpretation of diatonic collection………….……….……..35 Figure 2.4. Latin version and English version…………………...………………………40 Figure 2.5. Sum 0 (=12) axis : C (or F#)………………………………...………………43 Figure 2.6. G Axis between clarinet and voice lines in mm.1–2...……………..………..43 Figure 2.7. Webern‘s Op.16, No. 2………………………………………………………45 Figure 2.8. Voice-leading in pitch space………………………….……………………..46 Figure 3.1. Schoenberg‘s Signature Hexachord…………………………………………61 Figure 3.2. Construction of the 6–Z44 Partition………………………………………...63 Figure 3.3. J and K transformation………………………………………...…………....63 Figure 3.4. prime X-forms and inverted y-forms of 6–Z44 hexachord……………….....64 Figure 3.5. J- transformation……………………………………………………………65 x Figure 3.6. K-transformation…………………………………………………………….66 Figure 3.7. K-of-J-of-X…………………..….....................................................................66 Figure 3.8. A General Voice-Leading System of the 6-Z44 Hexachord…………………68 Figure 3.9. Western System (a=0) ……………………………………………………….69 Figure 3.10. Partition of 6-Z44 into 4 Orbits……….……………………………………70 Figure 3.11. A General Voice-Leading System of the 6-Z44 Hexachord………….……..71 Figure 3.12. The Schoenberg signature group G and the hexatonic triadic group H…...72 Figure 3.13. The dual commuting groups, for a = 0, 1, 2, 3…………………..…………73 Figure 3.14. Row Structure of Op. 27, II…………………………………….….……….75 Figure 3.15. Transformation T4 of 4-3…………………………………………..……….87 Figure 3.16. I4 relationship……………………………………………………..………..88 Figure 3.17. Chrisman‘s interval array permutation……………………………..………90 Figure 3.18. ………….……………………………………………………….….………91 Figure 3.19. Particular intervallic arrays………………………………………….……..92 Figure 3.20.………………………………………………………………….….………..93 Figure 3.21. Voice leading from pcs 6-22 to 6-Z24…………………………..………….95 Figure 3.22. Hexachord voice-leading system having two semitones, three whole-tones, one interval class four……………………………………………………………………96 Figure 3.23. 7-note voice-leading system having three semitones, three whole-tones, one interval class three…………………...……………………………………..……….97 xi Figure 3.24. Isolated symmetrical sets……………………………………………...……98 Figure 3.25. Voice-leading system of hexachord having three ic 1, three ic 3…..………98 Figure 3.26. 7-note voice-leading system having four interval class 1, three interval class 2, one interval class 3………………….…………………..…………………..…………99 Figure 3.27. 8-note voice-leading system having four interval class 1, four interval class 2 …………………………………….…………………………………………………….100 Figure 3.28. 9-note voice-leading system having six interval class 1, three interval class 2 ……………………………………………………………………….………………….101 Figure 3.29. Three-note Groups………………………..………………….……………103 Figure 3.30. Tetrachord having three ic 1, one ic 9…………………………………….104 Figure 3.31. Tetrachord voice-leading system having two ic 1, one ic 2, one ic 8……..104 Figure 3.32. Tetrachord voice-leading system having two semitones, one ic 3, ic 7……104 Figure 3.33. Tetrachord voice-leading system having two ic 1, one ic 4, one ic 6……..105 Figure 3.34. Tetrachord voice-leading system having two ic 1, two ic 5………………105 Figure 3.35. Tetrachord voice-leading system having one ic 1, two ic 2, one ic 7……..105 Figure 3.36. Tetrachord voice-leading system having one ic 1, one ic 2, one ic 3, one ic 6 …………………………………………………………………………………………..105 Figure 3.37. Tetrachord voice-leading system having one ic 1, one ic 2, one ic 4, one ic 5 ……………………………………………………………………………………….….106 Figure 3.38. Tetrachord voice-leading system having one ic 1, two ic 3, one ic 5….....106 xii Figure 3.39. Tetrachord voice-leading system having one ic 1, two ic 4, one ic 3……..107 Figure 3.40. Tetrachord having three ic 2, one ic 6…………………………………….107 Figure 3.41. Tetrachord voice-leading system having two ic 2, one ic 3, one ic 5……..107 Figure
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