LR property of non-well-formed scales



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Castrillón López, Marco and Domínguez Romero, Manuel (2016) LR property of non-well-formed scales. Journal of Mathematics and Music, 10 (1). pp. 18-35. ISSN 17459737

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This article studies generated scales having exactly three different step sizes within the language of algebraic combinatorics on words. These scales and their corresponding step-patterns are called non well formed. We prove that they can be naturally inserted in the Christoffel tree of well-formed words. Our primary focus in this study is on the left- and right-Lyndon factorization of these words. We will characterize the non-well-formed words for which both factorizations coincide. We say that these words satisfy the LR property and show that the LR property is satisfied exactly for half of the non-well-formed words. These are symmetrically distributed in the extended Christoffel tree. Moreover, we find a surprising connection between the LR property and the Christoffel duality. Finally, we prove that there are infinitely many Christoffel–Lyndon words among the set of non-well-formed words and thus there are infinitely many generated scales having as step-pattern a Christoffel–Lyndon word.

Item Type:Article
Uncontrolled Keywords:Christoffel words; generated scales; Lyndon words; non-well-formed words; standard factorization; Three Distance Theorem; well-formed scales
Subjects:Sciences > Mathematics
ID Code:38182
Deposited On:23 Jun 2016 08:25
Last Modified:01 Feb 2021 14:36

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