Generalized Tonnetz and discrete Abel-Jacobi map

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