Generalized Tonnetze
Algebraic Topology
2019-01-07 v1
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.
Keywords
Cite
@article{arxiv.1612.03519,
title = {Generalized Tonnetze},
author = {Michael J. Catanzaro},
journal= {arXiv preprint arXiv:1612.03519},
year = {2019}
}