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Geometric and Transformational GENERIC (MOD-7) GEOMETRIC AND TRANSFORMATIONAL APPROACHES TO VOICE LEADING IN TONAL MUSIC Leah Frederick Submitted to the faculty of the University Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Jacobs School of Music, Indiana University May 2020 Accepted by the Graduate Faculty, Indiana University, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Doctoral Committee ______________________________________ Julian Hook, Ph.D. Research Director ______________________________________ Richard Cohn, Ph.D. ______________________________________ Andrew Mead, Ph.D. ______________________________________ Christopher Raphael, Ph.D. ______________________________________ Frank Samarotto, Ph.D. April 10, 2020 ii Copyright © 2020 Leah Frederick iii ACKNOWLEDGEMENTS There has been perhaps no better time than now, in our socially distanced society of April 2020, to acknowledge the individuals and communities who have provided support and encouragement over the past few years. First and foremost, I wish to thank my advisor, Jay Hook, for his meticulous attention to detail, for his mentorship on navigating academia, and for sharing his enthusiasm of both music and mathematics. Perhaps more than any of those, however, I’m grateful for the encouragement and confidence that he has offered during the many moments when I’ve doubted myself. I also wish to recognize the rest of my dissertation committee, Rick Cohn, Andy Mead, Frank Samarotto, and Chris Raphael, for their willingness to serve on my committee and for their helpful feedback. I further extend gratitude to the entire music theory faculty at Indiana University. During my four years in Bloomington, each and every faculty member contributed to shaping my values and ways of thinking about music. I also thank the theory department and the Jacobs School of Music for their financial support through a Jacobs Fellowship. This dissertation project was supported by the Society for Music Theory’s 2020 SMT-40 Dissertation Fellowship. I am grateful for that support and recognition. I further wish to extend gratitude to the many scholars in the Society who have responded to my work with feedback, encouragement, and advice at conferences over the past few years. I am excited to belong to this community for many years to come. One does not enter a doctoral program without the encouragement of prior teachers and mentors. I thank several musicians and scholars from Penn State University: Tim Deighton, for fostering my values about music and encouraging me to pursue this career; Eric McKee, for introducing me to the world of music theory; Mark Ferraguto, for sparking my interest in academic writing and scholarship; and Maureen Carr, for enthusiastically supporting every step of my academic career. iv Spaces have been important to this document in more ways than one. I’m thankful for comfortable chairs and many caffeinated beverages from Bloomington’s Crumble Coffee & Bakery and Oberlin’s Slow Train Cafe. Much of this document was also drafted in IU’s Cook Music Library, a resource-filled space that I already miss. I thank my new colleagues at Oberlin Conservatory, especially Megan Long, Brian Alegant, and Catrina Kim, for their encouragement and support over this past year as I’ve begun the transition from graduate student to faculty member. I am also grateful to the many fellow graduate students who read drafts of conference proposals, attended practice run-throughs of talks, and spent many hours talking about music theory, inside and outside of our coursework. This list includes Lauren Wilson, David Geary, Stephen Gomez-Peck, Rachel Rosenman, Emily Barbosa, Christa Cole, Jinny Park, Nathan Lam, and many others. Thanks also to Erin Trautmann and Curtis Rainey for always being just a phone call away. Finally, I thank my family, not only for enduring many years of piano plunking and viola screeches, but also for shaping my mathematical inclination from a young age, and, most importantly, for teaching me to always be curious. v Leah Frederick GENERIC (MOD-7) GEOMETRIC AND TRANSFORMATIONAL APPROACHES TO VOICE LEADING IN TONAL MUSIC This dissertation develops a variety of geometric and transformational spaces to describe voice leading in tonal harmonic progressions. Whereas existing mathematical approaches using geometric and transformational techniques draw on Forte’s (1973) mod-12 pitch-class set theory, the tools developed in this dissertation build upon Clough’s (1979) mod-7 diatonic set theory. These musical spaces are constructed from generic pitch space, where each element represents an equivalence class of registrally differentiated letter names with any number of accidentals attached. After a review of existing transformational and geometric approaches in Chapter 1, Chapter 2 reconstructs the geometric approach of Callender, Quinn, and Tymoczko (2008) using generic pitch space. It defines the OPTIC equivalence relations for mod-7 space and then presents generic versions of the geometric voice-leading spaces. Due to differences in the mathematical properties of generic pitch space and continuous pitch space, the mod-7 OPTIC spaces exist only as discrete graphs, unlike the continuous topological mod-12 versions. Chapter 3 draws on transformational techniques, specifically those described as “neo-Riemannian” (Cohn 1996, 1997, 1998). After examining challenges of using mod-12 transformations to describe functional progressions, it explores ways of defining parsimonious transformations on mod-7 triads. The main theoretical apparatus of the chapter is a transformation group that acts on the set of 21 generic triads differentiated by closed-position inversion. The group is generated by the “voice-leading” transformation, v1, which relates triads by ascending, single-step motion. A few extensions to this group are proposed, including one that combines geometric and transformational techniques to describe voice leading in non-triadic sonorities. Chapter 4 studies chromatic applications of these geometric and transformational systems by incorporating existing approaches to scalar voice leading with the generic tools introduced in Chapters 2 and 3. vi CONTENTS Acknowledgements ....................................................................................................................................................................... iv Abstract ........................................................................................................................................................................................... vi Table of Contents ........................................................................................................................................................................ vii List of Figures .................................................................................................................................................................................. x List of Tables ................................................................................................................................................................................. xv Introduction .............................................................................................................................................. 1 Chapter 1. Mathematical Approaches to Voice Leading ............................................................................. 6 1.1. Introduction: Foundational Pitch Spaces ......................................................................................................... 6 1.2. Transformational Theory ................................................................................................................................... 11 1.2.1. Generalized Interval Systems .......................................................................................................... 12 1.2.2. Algebraic Groups ............................................................................................................................... 16 1.2.3. IVLS as an Algebraic Group ........................................................................................................... 18 1.2.4. Transformation Groups .................................................................................................................. 19 1.2.5. Function Notation ............................................................................................................................ 22 1.2.6. Triadic Transformational Theory ................................................................................................. 23 1.2.7. Basics of Graph Theory .................................................................................................................... 27 1.2.8. Transformations and Tonal Music ............................................................................................... 29 1.3. Geometric Theory ................................................................................................................................................ 30 1.3.1. Graph-Theoretic Extensions of Neo-Riemannian Voice-Leading Relationships .............. 30 1.3.2. Atonal Voice Leading ....................................................................................................................... 37 1.3.3. Generalized Voice-Leading Spaces ................................................................................................ 39 1.3.4. Mathematical
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