A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords

TL;DR

This paper introduces a 5-dimensional geometric model called the Tonnetz for analyzing voice-leading between nearly symmetric hexachords, extending traditional models to more complex chord structures.

Contribution

It develops a novel 5-dimensional Tonnetz framework for nearly symmetric hexachords, expanding the geometric tools available for music theory analysis.

Findings

Derived Cartesian coordinates for the 5D Tonnetz grid

Established generalized coordinate collections for 5-simplices

Identified geometric nearest-neighbors for specific chords

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.