Generalized Tonnetze

TL;DR

This paper generalizes the classical Tonnetz to N-tone scales and various triads, classifying the resulting spaces and revealing diverse topologies including tori, simplices, and Klein bottles.

Contribution

It introduces a comprehensive classification of generalized Tonnetze for all N-tone scales and triad types, expanding understanding of their topological structures.

Findings

Most spaces are tori as N increases

Identifies new topologies like tetrahedra and Klein bottles

Classifies spaces containing tritones as circle of tetrahedra boundaries

Abstract

We study a generalization of the classical Riemannian Tonnetz to N-tone equally tempered scales (for all N) and arbitrary triads. We classify all the spaces that result. The torus turns out to be the most common possibility, especially as N grows. Other spaces include 2-simplices, tetrahedra boundaries, and the harmonic strip (in both its cylinder and Mobius band variants). The final and most exotic space we find is something we call a `circle of tetrahedra boundaries'. These are the Tonnetze for spaces of triads which contain a tritone. They are closely related to Peck's Klein bottle Tonnetz.