Singular homology on hypergestures

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Abstract

In this paper, 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 ↑. This interpretation allows for a generalized homological setup. The generalization is (1) to topological categories instead of to topological spaces and (2) to any sequence of digraph (⊤ n) n∈ℤ instead of to the constant series of ↑. We then define the corresponding chain complexes and prove the core boundary operator equation δ 2 = 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.

Original languageEnglish (US)
Pages (from-to)49-60
Number of pages12
JournalJournal of Mathematics and Music
Volume6
Issue number1
DOIs
StatePublished - Mar 1 2012

Bibliographical note

Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

Keywords

  • counterpoint
  • homology
  • hypergestures

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