| dc.description |
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. |
|