Castrillón López, Marco and Dominguez, , Manuel and Noll, Thomas
(2011)
*An extension of Christoffel duality to a subset of Sturm numbers and their characteristic words.*
Theoretical Computer Science, 412
(27).
pp. 2942-2954.
ISSN 0304-3975

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Official URL: http://www.sciencedirect.com/science/article/pii/S0304397510007681

## Abstract

The paper investigates an extension of Christoffel duality to a certain family of Sturmian words. Given an Christoffel prefix w of length N of an Sturmian word of slope g we associate a N-companion slope g(N)* such that the upper Sturmian word of slope g(N)* has a prefix w* of length N which is the upper Christoffel dual of w. Although this condition is satisfied by infinitely many slopes, we show that the companion slope g(N)* is an interesting and somewhat natural choice and we provide geometrical and music-theoretical motivations for its definition. In general, the second-order companion (g(N)*)(N)* = g(N)** does not coincide with the original g. We show that, given a rational number 0 < M/N < 1, the map g -> g(N)** has exactly one fixed point, phi(M/N) is an element of [0, 1), called odd mirror number. We show that odd mirror numbers are Sturm numbers and their continued fraction expansion is purely periodic with palindromic periods of even length. The semi-periods are of odd length and form a binary tree in bijection to the Farey tree of ratios 0 < M/N < 1. Its root is the singleton {2}, which represents the odd mirror number -1+root 8/2 = [0; (22) over bar]. The characteristic word c(phi M/N) of slope phi(M/N) remains fixed under a standard morphism which can be computed from the semi-period of phi(M/N). Finally, we prove that the characteristic word G(c(phi M/N)) is a harmonic word. As a minor open question we ask for the properties of even mirror numbers. A final conjecture provides a proper word-theoretic meaning to the extended duality for odd mirror number slopes: given a characteristic word c(phi M/N), the succession of those letters which immediately precede the occurrences of the left special factor of length N coincides - up to letter exchange - with the G-image of the dual word c(phi M/N)*.

Item Type: | Article |
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Uncontrolled Keywords: | Sturmian words; Christoffel duality; Well-formed scales; Characteristic words; Harmonic words |

Subjects: | Sciences > Mathematics > Geometry |

ID Code: | 14872 |

References: | Pascal Alessandri, Valérie Berthé, Three distance theorems and combinatorics on words, L’Enseignement Mathématique 44 (1998) 103–132. Jean Berstel, Aaron Lauve, Christophe Reutenauer, Franco Saliola, Combinatorics on Words: Christoffel Words and Repetition in words, CRM-AMS, 2008. Jean Berstel, Dominique Perrin, The origins of combinatorics on words, European Journal of Combinatorics 28 (2007) 996–1022. Valérie Berthé, Fréquences des facteurs des suites sturmiennes, Theoretical Computer Science 165 (1996) 295–309. Valérie Berthé, Aldo de Luca, Christophe Reutenauer, On an involution of Christoffel words and Sturmian morphisms, European Journal of Combinatorics 29 (2) (2006) 535–553. Norman Carey, Distribution Modulo 1 and Musical Scales, Ph.D. Thesis, University of Rochester, 1998. Norman Carey, David Clampitt, Aspects of Well-formed scales, Music Theory Spectrum 11 (2) (1989) 187–206. Norman Carey, David Clampitt, Self-similar pitch structures, their duals and rhythmic analogues, Perspectives of New Music 34 (2) (1996) 62–87. [9] Arturo Carpi, Aldo de Luca, Harmonic and gold sturmian words, European Journal of Combinatorics 25 (2004) 685–705. David Clampitt, Thomas Noll, Modes, the height-width duality, and divider incidence, Paper presented at the Society for Music Theory national converence, Nashville, TN, 2008. Aldo de Luca, Sturmian words: structure, combinatorics, and their arithmetics, Theoretical Computer Science 183 (1) (1997) 45–82. Manuel Domínguez, David Clampitt, Thomas Noll, Well-formed scales, maximally even sets and Christoffel words, in: Proceedings of the MCM2007, 2008. Manuel Domínguez, David Clampitt, Thomas Noll, Plain and twisted adjoints of Well-formed words, in: Proceedings of the MCM2009, 2009. R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley Longman Publishing Co., Boston, MA, USA, 1994. M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, Cambridge, 2002. M. Lothaire, Applied Combinatorics on Words, Cambridge University Press, Cambridge, 2005. [17] Thomas Noll, Ionian theorem, Journal of Mathematics and Music 3 (3) (2009). |

Deposited On: | 19 Apr 2012 08:23 |

Last Modified: | 18 Apr 2013 14:48 |

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