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Generalized Tonnetz and discrete Abel-Jacobi map. (English) Zbl 1492.57015

To represent the classical tonal space, Euler in his work on music theory from 1739 introduced a lattice diagram called Tonnetz (tone network). This diagram has been interpreted as a triangulation of a torus with 24 triangles representing all the major and minor chords. Motivated by this, in this paper, a more general simplicial complex is presented and denoted as of Tonnetz type. The authors study various topological invariants of this space in detail. Apart from giving supporting examples and some geometrical considerations, the authors also extend the above definition to the limiting case. Eventually they prove that the generalized Tonnetz is also a triangulation of a torus.

MSC:

57Q15 Triangulating manifolds
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14H40 Jacobians, Prym varieties
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B70 Polyhedral manifolds
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
00A65 Mathematics and music
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References:

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