Generalized tonnetz and discrete abel-jacobi map

Abstract

Motivated by classical Euler’s Tonnetz, we introduce and study the combinatorics and topology of more general simplicial complexes Tonnn k (L) of Tonnetz type. Out main result is that for a sufficiently generic choice of parameters the generalized Tonnetz Tonnn,k (L) is a triangulation of a (k-1)-dimensional torus Tk-1. In the proof we construct and use the properties of a discrete Abel-Jacobi map, which takes values in the torus Tk-1=ℝk-1/Λ where Λ=A*k-1 is the permutohedral lattice.