Generalized Tonnetz and discrete Abel-Jacobi map

Motivated by classical Euler's \(Tonnetz\), we introduce and study the combinatorics and topology of more general simplicial complexes \(Tonn^{n,k}(L)\) of "Tonnetz type". Out main result is that for a sufficiently generic choice of parameters the generalized tonnetz \(Tonn^{n,k}(L)\) is a triangulation of a \((k-1)\)-dimensional torus \(T^{k-1}\). In the proof we construct and use the properties of a "discrete Abel-Jacobi map", which takes values in the torus \(T^{k-1} \cong \mathbb{R}^{k-1}/\Lambda\) where \(\Lambda \cong \mathbb{A}^\ast_{k-1}\) is the permutohedral lattice.