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 T onn n,k (L) of "Tonnetz type". Out main result is that for a sufficiently generic choice of parameters the generalized tonnetz T onn 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 torusk−1 is a permutohedral lattice.

Automatic Identification of Some Vietnamese Folk Songs Cheo and Quanho Using Convolutional Neural Networks

“…The tonnetz concept was first described by Leonhard Euler in 1739 [36,37]. It is a layout of notes in music (which can be called a grid or a graph of notes), which shows the relationship between notes and is often used in harmony.…”