A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords
This is an incremental contribution specific to music theory, addressing geometric representations for specialized chord types.
The paper tackles the problem of modeling parsimonious voice-leading connections for nearly symmetric hexachords in music theory by presenting a 5-dimensional Tonnetz geometric model, deriving coordinates and nearest-neighbors for mystic-Wozzeck genus chords.
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.