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arXiv · 2505.08752

Configurations, Tessellations and Tone Networks

Jeffrey R. Boland·Lane P. Hughston
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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.

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Jeffrey R. Boland, Lane P. Hughston. 2025-05-13. Configurations, Tessellations and Tone Networks. https://arxiv.org/abs/2505.08752

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