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Singular Homology of Hypergestures

  • Guerino Mazzola
  • René Guitart
  • Jocelyn Ho
  • Alex Lubet
  • Maria Mannone
  • Matt Rahaim
  • Florian Thalmann
Chapter
Part of the Computational Music Science book series (CMS)

Summary

In this chapter we interpret the basic cubic chain spaces of singular homology in terms of hypergestures in a topological space over a series of copies of the arrow digraph ò. This interpretation allows for a generalized homological setup. The generalization is (1) to topological categories instead of topological spaces, and (2) to any sequence of digraph pΓnqnPZ instead of the constant series of Ò. We then define the corresponding chain complexes, and prove the core boundary operator equation B2 “ 0, enabling the associated homology modules over a commutative ring R. We discuss some geometric examples and a musical one, interpreting contrapuntal rules in terms of singular homology.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Guerino Mazzola
    • 1
  • René Guitart
    • 2
  • Jocelyn Ho
    • 3
  • Alex Lubet
    • 1
  • Maria Mannone
    • 1
  • Matt Rahaim
    • 1
  • Florian Thalmann
    • 4
  1. 1.School of MusicUniversity of MinnesotaMinneapolisUSA
  2. 2.Institut de Mathématiques de JussieuUniversité Diderot Paris 7ParisFrance
  3. 3.Department of MusicUCLA Herb Alpert School of MusicLos AngelesUSA
  4. 4.School of Electronic Engineering and Computer ScienceQueen Mary University of LondonLondonUK

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