Variety and multiplicity for partitioned factors in Christoffel and Sturmian words

Abstract

The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equals Variety (CV) and Structure yields Multiplicity (SM), correspond to statements about Christoffel words. These results may be stated and proved in a purely word-theoretic environment by means of the concept of a partitioned factor. In this setting, the generalized circle of fifths of Clough and Myerson is replaced by the lexicographic order of word theory, and their notion of the generic description of a musical line is replaced by that of a partitioned factor of a circular Christoffel word. Sturmian words are the infinite counterparts to Christoffel words, characterized as aperiodic but of minimal complexity, i.e. words such that for all $n\in \mathbb{N}$ there are n+1 factors of length n. Berthé showed that the factors of a given length have at most three frequencies (probabilities). Drawing on previous work, we extend to results on factors under a fixed partitioning (decompositions of factors of length n into concatenations of words whose lengths are given by a composition of n into k parts). Any factor of a Sturmian word thus partitioned into k components belongs to one of k+1 types (varieties). We show how to compute the frequencies of the varieties. These results recapture CV and SM in Sturmian words, and we suggest musical interpretations, parallel to those in the finitary case.

Keywords:

• Balance
• Christoffel word
• circular word
• factor
• frequency
• mechanical word
• Myhill's property
• composition
• partitioned factor
• Sturmian word

2010 Mathematics Subject Classification:

• 68R15

Dedication

We dedicate this article to the fond memory of John Clough (1930–2003).

Acknowledgments

The authors wish to thank the readers and the editors for their insightful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Norman Carey  http://orcid.org/0000-0002-6417-0952

David Clampitt  https://orcid.org/0000-0001-5218-7186

Notes

1 Defined as ‘balanced${}_1$’ in Berstel et al. (2009, 41)