TY - GEN

T1 - The continuous hexachordal theorem

AU - Ballinger, Brad

AU - Benbernou, Nadia

AU - Gomez, Francisco

AU - O'Rourke, Joseph

AU - Toussaint, Godfried

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=67649999139&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649999139&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-02394-1_2

DO - 10.1007/978-3-642-02394-1_2

M3 - Conference contribution

AN - SCOPUS:67649999139

SN - 9783642023934

T3 - Communications in Computer and Information Science

SP - 11

EP - 21

BT - Mathematics and Computation in Music

ER -