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

International audienceIn 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

Gray Codes in Music Theory

In the branch of Western music theory called serialism, it is desirable to construct chord progressions that use each chord in a chosen set exactly once. We view this problem through the scope of the mathematical theory of Gray codes, the notion of ordering a finite set X so that adjacent elements are related by an element of some specified set R of involutions in the permutation group of X. Using some basic results from the theory of permutation groups we translate the problem of finding Gray codes into the problem of finding Hamiltonian paths and cycles in a Schreier coset graph of the permutation group generated by the involutions R. Having made this translation we can use known results about Hamiltonian paths in Schreier (and Cayley) graphs of groups to generate serialism-like chord progressions. We illustrate the method by examining two theorems from the literature on Hamiltonian paths, due to Conway, Sloane, and Wilks (Graphs Combin. 5 (1989), no. 4, 315–325), and to Eades and Hickey (J. Assoc. Comput. Mach. 31 (1984), no. 1, 19–29). We give proofs of these theorems that complement the published proofs by filling in some details and clarifying some potentially confusing points, and we then use the algorithms extracted from these proofs to produce chord progressions

On the Group of Transformations of Classical Types of Seventh Chords

This paper presents a generalization of the well-known neo-Riemannian group PLR to the classical five types of seventh chord (dominant, minor, half-diminished, major, diminished) considered as tetrachords with a marked root and proving that it is isomorphic to the abstract group $S_5 \ltimes \mathbb{Z}_{12}^4$. This group includes as subgroups the PLR group and several other groups already appeared in the literature