LR property of non-well-formed scales. Issue 1 (2nd January 2016)

• Record Type: Journal Article
• Authors: Castrillón López, Marco; Domínguez Romero, Manuel
• Is Part Of: Journal of mathematics and music. Volume 10:Issue 1(2016)
• Page Start: 18
• Page End: 35
• Publication Date: 2016-01-02
• Subjects: well-formed scales -- non-well-formed words -- Three Distance Theorem -- Lyndon words -- Christoffel words -- generated scales -- standard factorization; 00A65 -- 68R15; Combinatorics on words; Music theory -- Periodicals; Music -- Mathematical models -- Periodicals; Music -- Mathematics -- Periodicals; 781.2
• DOI: 10.1080/17459737.2016.1164907
• Languages: English
• ISSNs: 1745-9737

Abstract

Abstract : 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.