### Abstract

The Hexachordal Theorem may be interpreted in terms of scales, or rhythms, or as abstract mathematics. In terms of scales it claims that the complement of a chord that uses half the pitches of a scale is homometric to-i.e., has the same interval structure as-the original chord. In terms of onsets it claims that the complement of a rhythm with the same number of beats as rests is homometric to the original rhythm. We generalize the theorem in two directions: from points on a discrete circle (the mathematical model encompassing both scales and rhythms) to a continuous domain, and simultaneously from the discrete presence or absence of a pitch/onset to a continuous strength or weight of that pitch/onset. Athough this is a significant generalization of the Hexachordal Theorem, having all discrete versions as corollaries, our proof is arguably simpler than some that have appeared in the literature. We also establish the natural analog of what is sometimes known as Patterson's second theorem: if two equal-weight rhythms are homometric, so are their complements.

Original language | English (US) |
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Title of host publication | Mathematics and Computation in Music |

Subtitle of host publication | Second International Conference, MCM 2009, John Clough Memorial Conference, Proceedings |

Pages | 11-21 |

Number of pages | 11 |

DOIs | |

State | Published - 2009 |

### Publication series

Name | Communications in Computer and Information Science |
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Volume | 38 |

ISSN (Print) | 1865-0929 |

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

*Mathematics and Computation in Music: Second International Conference, MCM 2009, John Clough Memorial Conference, Proceedings*(pp. 11-21). (Communications in Computer and Information Science; Vol. 38). https://doi.org/10.1007/978-3-642-02394-1_2