Configurations, Tessellations and Tone Networks

TL;DR

This paper explores a geometric and graph-theoretic representation of the tonnetz, a musical tessellation, as a configuration of points and lines in the plane, offering new tools for musical composition and analysis.

Contribution

It introduces a novel geometric realization of the tonnetz as a  configuration, expanding its applications in music theory and composition.

Findings

Realized the tonnetz as a  configuration in  plane geometry.

Connected the graph-theoretic and geometric representations of the tonnetz.

Proposed new methods for musical analysis and composition based on these configurations.

Abstract

The tonnetz, which is commonly represented as a tessellation of the plane by a triangular network of tones, can also be represented as a bipartite graph of degree three with twelve vertices denoting major triads and twelve vertices denoting minor triads. We show that this Levi graph can be realized geometrically as a system of twelve points and twelve lines in $\mathbb R^2$ with the property that three points lie on each line and three lines pass through each point, in a configuration $\{12_3\}$ of Daublebsky von Sterneck type D222. This tonnetz configuration, alongside various generalizations thereof, can be used as a new basis for the composition and analysis of music.