SLIDE, the transformation that exchanges triads sharing the same third, is a pervasive feature of Schubert’s mature works. In giving support for prolongational analyses of chromatic and neo-tonal music, this study provides a case for tonality-based approaches to post-functional harmony. After using group theory to explore the relationship of diatonic scale theory and tuning theory to transformational and prolongational analysis, excerpts from Wolf, Wagner, and Ravel are analyzed using mod-7 transformations. This study explores a number of specific graphing techniques, including the diatonic lattice (Jones 2002), the just- intonation Tonnetz, and mod-12/mod-7 prolongational networks. Prolongational analyses of music may be represented by Schenkerian notation or transformational networks based on Lewin's Generalized Musical Intervals and Transformations (1987). In this regard, acoustical measures of stability, motivic connections, and chord equivalence all may form a part in determining the structural harmonies. The main obstacle for prolongational views of extended tonality is finding sufficient conditions for establishing that certain harmonies are structural in the absence of traditional harmonic function. While both tonal and post-tonal theory have been extended in various ways to address this music, the use of tonal theory for analysis of this repertoire has not been completely formalized. Most analytical methods are designed to address either tonal music or atonal music, but no single method completely illuminates this body of extended-tonal music. Many musical compositions from the end of the nineteenth century and the beginning of the twentieth century retain some elements of functional tonality but abandon others. These passages exhibit a harmonic syntax that is often difficult to analyze as anything other than “tonally unstable” or “transitional.” This study seeks to analyze these passages in terms of what they are, rather than what they are not.
The viability of the theoretical model is examined in analyses of passages from the repertoire of Frédéric Chopin.
It allows for the descriptive “mapping” or prescriptive “navigation” of harmonic paths through a defined space. These geometric realizations are based on the organization of the neo-Riemannian Tonnetz, but they expand and apply the organizational principles of the Tonnetz to seventh sonorities. Rather than emphasizing mathematical proofs, as a number of approaches have done, this study relies on two- and three-dimensional geometric visualizations and spatial analogies to describe pitch-class and harmonic relationships. Some scholars have applied these principles to seventh chords, laying the groundwork for this study, which strives toward a reasonably comprehensive, usable model for musical analysis. Parsimonious voice leading is a term, first used by Richard Cohn, to describe non-diatonic motion among triads that will preserve as many common tones as possible, while limiting the distance traveled by the voice that does move to a tone or, better yet, a semitone.