## 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 language | English (US) |
---|---|

Pages (from-to) | 49-60 |

Number of pages | 12 |

Journal | Journal of Mathematics and Music |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2012 |

## Keywords

- counterpoint
- homology
- hypergestures