A Generalized Dual of the Tonnetz for Seventh Chords: Mathematical, Computational and Compositional Aspects

Abstract

In Mathematical Music Theory, geometric models such as graphs and simplicial complexes are music-analytical tools which are commonly used to visualize and represent musical operations. The most famous example is given by the Tonnetz, a graph whose basic idea was introduced by Euler in 1739, and developed by several musicologists of the XIX th century, such as Hugo Riemann. The aim of this paper is to introduce a generalized Chicken-wire Torus (dual of the Tonnetz) for seventh chords and to show some possible compositional applications. It is a new musical graph representing musical operations between seventh chords, described from an algebraic point of view. As in the traditional Tonnetz, geometric properties correspond to musical properties and offer to the computational musicologists new and promising analytical tools.

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