Periodic musical sequences and Lyndon words

Chemillier, Marc

doi:10.1007/s00500-004-0387-2

Cited by 20 publications

(13 citation statements)

References 7 publications

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“…A Lyndon word is a string that is lexicographically smaller than all of its proper suffixes. Despite the simplicity of its definition, Lyndon words have many deep and interesting combinatorial properties [36] and have been applied to a wide range of problems [36,43,35,15,5,19,34,4,27,41,18]. Lyndon words have recently been considered in the context of runs [12,13], since any run with period p must contain a length-p substring that is a Lyndon word, called an L-root of the run.…”

Section: Introductionmentioning

confidence: 99%

The “Runs” Theorem

Bannai¹,

Tomohiro²,

Inenaga³

et al. 2017

SIAM J. Comput.

142

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We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the "runs" conjecture (Kolpakov & Kucherov (FOCS '99)), which states that the maximum number of runs ρ(n) in a string of length n is less than n. The proof is remarkably simple, considering the numerous endeavors to tackle this problem in the last 15 years, and significantly improves our understanding of how runs can occur in strings. In addition, we obtain an upper bound of 3n for the maximum sum of exponents σ(n) of runs in a string of length n, improving on the best known bound of 4.1n by Crochemore et al. (JDA 2012), as well as other improved bounds on related problems. The characterization also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string. We also establish a relationship between runs and nodes of the Lyndon tree, which gives a simple optimal solution to the 2-Period Query problem that was recently solved by Kociumaka et al. (SODA 2015). * A preliminary version of this paper has appeared in [1].

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Section: Introductionmentioning

confidence: 99%

The “Runs” Theorem

Bannai¹,

Tomohiro²,

Inenaga³

et al. 2017

SIAM J. Comput.

142

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“…Word theory and mathematical music theory have proceeded in almost total ignorance of each other during this time. Notable exceptions include the rhythmic studies of Truchet 2003 andChemillier 2004. The authors would like to acknowledge a remark by Franck Jedrzejewski pointing to a talk by Christian Kassel (2004), joint work published in Kassel and Reutenauer 2007, which directed their attention to word theory.…”

Section: Introductionmentioning

confidence: 99%

Modes, the Height-Width Duality, and Handschin’s Tone Character

Clampitt

Noll

2011

MTO

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The theory of well-formed modes is a modal refinement of the theory of well-formed scales. The mathematical approach is based on various results from the subdiscipline of algebraic combinatorics on words. Section 1 provides anchors and motivations for this investigation both in music theory and in mathematics and traces some earlier cross-connections. An overview of the theory is presented in terms of a dichotomy between generic and specific levels of description. Section 2 presents a first group of basic theoretical results. Height-width duality mediates between scale step patterns and fifth-fourth folding patterns. Both are encoded through divided words, on two-letter alphabets, such as aaba|aab and yx|yxyxy. The letters a and b denote ascending whole and half steps, and the letters x and y denote ascending perfect fifths and descending perfect fourths, respectively. These words are well-formed words; i.e., in the language of word theory, they are conjugate to Christoffel words and inherit a duality that is akin to Christoffel duality. Qualitative differences between the modal varieties of the same underlying scale can be detected and formalized through word-theoretical arguments. For example, a property we refer to as divider incidence characterizes modes corresponding to standard words. Positive standard words generalize the ascending authentic Ionian mode. Sturmian morphisms provide a transformational meta-language for the study of well-formed modes. Section 3 revisits Jacques Handschin's concept of tone character and defends it on the basis of the mathematical results against two criticisms that had been raised by Carl Dahlhaus. Section 4 explores distinctions among the modes based upon considerations of word theory, especially divider incidence and concomitant properties that support major-minor tonality. A concluding section connects these arguments with other lines of investigation.

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“…Lyndon words have been widely studied in the combinatorics of words area [10]. However, only a few papers consider Lyndon words addressing issues in other areas than word algebra, e.g., [1,7,12].…”

Section: Introductionmentioning

confidence: 99%