Geometry, iterated quantization and filtered voice-leading spaces

Clifton Callender

pp. 257-266

in: Tom Collins, David Mérédith, Anja Volk (eds), Mathematics and computation in music, Berlin, Springer, 2015

Abstract

A recent special issue of the Journal of Mathematics and Music on mathematical theories of voice leading focused on the intersections of geometrical voice-leading spaces (GVLS), filtered point-symmetry (FiPS) and iterated quantization, and signature transformations. In this paper I put forth a theoretical model that unifies all of these approaches. Beginning with the basic configuration of FiPS, allowing the n points of a filter or beacon to vary arbitrarily yields the continuous chord space of n voices ((T^n/S_n)). Each point in the filter space induces a quantization or Voronoi diagram on the beacon space. The complete space of filter and beacon is a singular fiber bundle, combining the power and generalization of GVLS with the central FiPS insight of iterated filtering by harmonic context. Additionally, any of the sixteen types of generalized voice-leading spaces described by Callender, Quinn, and Tymoczko can be used as filters/beacons to model different contexts.