Generalized Tonnetz and discrete Abel-Jacobi map
Keywords
Generalized Tonnetz, discrete Abel-Jacobi map, permutohedral lattice, simplicial complexes, polyhedral combinatorics, triangulated manifoldsAbstract
Motivated by classical Euler's {\em Tonnetz}, we introduce and study the combinatorics and topology of more general simplicial complexes Tonn$^{n,k}(L)$ of {\em 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 {\em 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.References
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