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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.

Reviewer: Ashish K. Upadhyay (Patna)

### 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 |

### Keywords:

generalized Tonnetz; discrete Abel-Jacobi map; permutohedral lattice; simplical complexes; polyhedral combinatorics; triangulated manifolds
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\textit{F. D. Jevtić} and \textit{R. T. Živaljević}, Topol. Methods Nonlinear Anal. 57, No. 2, 547--567 (2021; Zbl 1492.57015)

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