A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords

Abstract

The standard 2-dimensional Tonnetz describes parsimonious voice-leading connections between major and minor triads as the 3-dimensional Tonnetz does for dominant seventh and half-diminished seventh chords. In this paper, I present a geometric model for a 5-dimensional Tonnetz for parsimonious voice-leading between nearly symmetric hexachords of the mystic-Wozzeck genus. Cartesian coordinates for points on this discretized grid, generalized coordinate collections for 5-simplices corresponding to mystic and Wozzeck chords, and the geometric nearest-neighbors of a selected chord are derived.