Music and combinatorics on words: a historical survey

Abstract

sense, that is to say, sequences of elements taken from a set, and its natural framework is the free monoid over an alphabet. It brings together various domains that have grown separately within several branches of mathematics (Berstel and Perrin 2007). The emergence of this field is related to a group of former students of a historical figure in the development of computer science, Marcel-Paul Schützenberger (1920-1996). Following the tradition of Nicolas Bourbaki, the name Lothaire was chosen as a collective nom de plume for researchers writing the different chapters of a series of books (Lothaire 1990, 1997, 2002, 2005). The third author of the present survey is a contributor to Lothaire's books. At the end of the 1970s, automata, formal languages, and grammars had been intensively used in musicology and computer music for at least two decades. As early as 1961 and 1962, composers Iannis Xenakis (1922-2001) and Pierre Barbaud (1911-1990) presented their ideas on the subject during seminars at the laboratory which later became the CAMS at EHESS (Centre d'Analyse et de Mathématique Sociales) (see Chemillier 2011). At that time, the influence of ideas from combinatorics on words in music was almost completely non-existent. In his 1979 paper "Grammars as representations for music," Curtis Roads gave an important survey of the use of formal languages to represent music structures (Roads 1979), whereas there were very few articles on the combinatorics on words side. One can nevertheless mention a paper by Antonio Bertoni and others dealing with repetitions in a musical sequence (Bertoni et al. 1978). Although not in touch with the Lothaire group, their paper was one of the first attempts to study music from a combinatorics on words perspective. At the beginning of the 1980s, Jean-Paul Allouche, who is another contributor to Lothaire's books, was teaching at the Ecole Normale Supérieure de Fontenay-aux-Roses. The second author of this survey was a student there and was looking for a PhD subject mixing computer science and music. Allouche sent him to the LITP in Paris (Laboratoire d'Informatique Théorique et de Programmation), the laboratory with which Schützenberger and other Lothaire members were affiliated. He began a thesis under the direction of Dominique Perrin. His basic idea was to use words for representing the succession of events in a musical sequence, to add an operation of superimposition for representing simultaneity, and to study the resulting algebraic structure that was coined solfege (Chemillier 1990, 2003). In the conclusion of his 1988 joint paper with the late Dan Timis (1954-2009) at the ICMC Conference in Cologne, they wrote: "Further developments will include algorithms for complex pattern recognition, rewriting rules, combinatorics on words, and code theory" (Chemillier and Timis 1988). Another great figure of the LITP, the

Symbolic dynamical scales: modes, orbitals, and transversals

“…There are many references that address the theory of musical scales that are relevant to our work, like the fundamentals [17,40], from the point of view of mathematics inclusive [26,32], several of which are related to combinatorics on words [2,14,13,1].…”

“…We observe that when s = •, α(f (1) ) = f (5) , α −1 (f (6) ) = f (8) , and the set T • {f (1) , f (2) , f (3) , f (4) , f (7) , f (8) } is independent, and when s = •, α −1 (f (11) ) = f (13) and the set T • {f (9) , f (10) , f (11) , f (12) } is independent. Thus, both T • and T • are transversals of F • and F • , respectively, and since T • and T • are mutually independent, we obtain T F T • ∪ T • as an admissible transversal of Fibonacci scales, its elements are shown in Figure 4.…”

“…TM7.4 Observe that O α (m (1) ) = {m (1) , m (4) , m (7) , m (10) , m (14) , m (16) } O α (m (2) ) = {m (2) , m (5) , m (8) , m (11) , m (3) , m (6) }, and also that the set (9) , m (12) , m (13) , m (15) , m (17) , m (18) } is independent. Therefore, the transversals of M (contained in M ) are the sets T formed by M \ O α (m (1) ) ∪ O α (m (2) ) and one element from each of the two α-orbits O α (m (1) ) and O α (m (2) ).…”

“…Observe that g (3) ∈ O α (g (1) ) with (g (1) ) = 5, g (4) , g (5) , g (6) ∈ O α (g (2) ) with (g (2) ) = 4, (g (7) ) = 3, g (10) ∈ O α (g (8) ) with (g (8) ) = 7, g (11) , g (12) , g (13) , g (15) , g (16) , g (18) , g (19) ∈ O α (g (9) ) with (g (9) ) = 8, and g (17) , g (20) ∈ O α (g (14) ) with (g (14) ) = 9. All are aperiodic except for g (7) and g (14) that are 3-periodic.…”

Axiomatic scale theory

“…The connection between interval words and wellformed scales was first described by Clampitt, Domìnguez, and Noll (2007) and Domínguez, Clampitt, and Noll (2009). Further studies on this topic include for example Castrillon Lopez and Domìnguez Romero (2016) and a recent special issue of the Journal of Mathematics and Music (Brlek, Chemillier, and Reutenauer 2018).…”