GENERIC (MOD-7) GEOMETRIC AND TRANSFORMATIONAL APPROACHES TO VOICE LEADING IN TONAL MUSIC

Abstract

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, $v_1$, 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.