CCCG 2008, Montreal, Quebec, August 1315, 2008

A Pumping Lemma for Homometric Rhythms

Joseph ORourke Perouz Taslakian Godfried Toussaint

Abstract

Homometric rhythms (chords) are those with the samehistogram or multiset of intervals (distances). Thepurpose of this note is threefold. First, to point outthe potential importance of isospectral vertices in apair of homometric rhythms. Second, to establish amethod (pumping) for generating an infinite sequenceof homometric rhythms that include isospectral vertices.And finally, to introduce the notion of polyphonic homo-metric rhythms, which apparently have not been previ-ously explored.

1 Introduction

Both chords of k notes on a scale of n pitches, andrhythms of k onsets repeated every n metronomicpulses, are conveniently represented by n evenly spacedpoints on a circle, with arithmetic mod n, i.e., in thegroup Zn. This representation dates back to the 13th-century Persian musicologist Safi Al-Din [Wri78], andcontinues to be the basis of analyzing music throughgeometry [Tou05] [Tym06]. Such sets of points on acircle are called cyclotomic sets in the crystallographyliterature [Pat44] [Bue78]. It is well-established thatin the context of musical scales and chords, the inter-vals between the notes largely determine the aural toneof the chord. An interval is the shortest distance be-tween two points, measured in either direction on thecircle. This has led to an intense study of the inter-val content [Lew59]: the histogram that records, foreach possible interpoint distance in a chord, the num-ber of times it occurs. This same histogram is studiedfor rhythms [Tou05] and in crystallography.Of special interest are pairs of noncongruent

chords/rhythms/cyclotomic sets that have the same his-togram: the sets are homometric in the terminology ofLindo Patterson [Pat44], who first discovered them. Incrystallography, such sets yield the same X-ray pattern.In the pitch model, they are chords with the same in-terval content. One of the fundamental theorems inthis area is the so-called hexachordal theorem, whichstates that two non-congruent complementary sets with

Department of Computer Science, Smith College, Northamp-ton, MA 01063, USA. orourke@cs.smith.edu

School of Computer Science, McGill University, Montreal,Canada. perouz@cs.mcgill.ca

School of Computer Science, McGill University, Montreal,Canada. godfried@cs.mcgill.ca

k=n/2 (and n even) are homometric, whose earliestproof in the music literature is due to Milton Babbittand David Lewin [Lew59].Henceforth we specialize to the rhythm model, with

each (n, k)-rhythm specified by k beats and nk restson the Zn circle; and we specifically focus on the struc-ture of homometric rhythms. Figure 1 shows a pair ofhomometric rhythms with (n, k)=(12, 5).

0

1

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11 1

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6 2

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0

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11 1

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56

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45

23

1

1 2 3 4 5 6

1

2

Figure 1: Homometric (n, k)=(12, 5) rhythms:(0, 1, 2, 4, 7) and (0, 1, 3, 5, 6). Vertices 0 and 5in the first and second rhythms (respectively) areisospectral.

2 Isospectral Vertices

Let P and Q be two different rhythms, with p Pand q Q vertices (onsets) in each. The vertices pand q are called isospectral1 if they have the same his-togram of distances to all other vertices in their respec-tive rhythms. In Figure 1, vertex 0 in the first rhythm,

1The term is used in the literature on Golomb rulers.