Single-Voice Transformations

This study demonstrates how smooth voice leading in music can be effectively modeled using concepts from abstract algebra. Minute voice-leading displacements are explained as iterations of the basic operation, the single-semitone transformation (SST). The SST is a type of transformation in which only a single voice in a chord is transposed by a semitone. Unlike previous music theoretic studies, the SST model does not rely on twelve-tone operations on sets to determine voice-leading paths. SST-succession classes can then be defined; they allow SSTs to be generalized as parsimonious voice-leading relations between pair-ordered set classes. Voice leading between chords of different “sizes” can be obtained through split and fuse operations. Once a mathematical basis for smooth voice-leading is formalized, 3D graphical representations in the form of lattices of parsimoniously related chord types can be developed. The study compares the single-voice transformational model to transformational theories of atonal voice leading and to recent work in the emergent field of neo-Riemannian theory. The final chapter examines music from tonal, atonal, and “post-atonal” stylistic periods by Chopin, Scriabin, Webern, Paul Lansky, and John Adams, showing the new voice-leading model’s versatility as an analytical tool.