The continuous hexachordal theorem

Brad Ballinger, Nadia Benbernou, Francisco Gomez, Joseph O'Rourke, Godfried Toussaint

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationMathematics and Computation in Music
Subtitle of host publicationSecond International Conference, MCM 2009, John Clough Memorial Conference, Proceedings
Pages11-21
Number of pages11
DOIs
StatePublished - 2009

Publication series

NameCommunications in Computer and Information Science
Volume38
ISSN (Print)1865-0929

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)

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  • Cite this

    Ballinger, B., Benbernou, N., Gomez, F., O'Rourke, J., & Toussaint, G. (2009). The continuous hexachordal theorem. In 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