Synchronization of musical words

Chemillier, Marc

doi:10.1016/s0304-3975(03)00309-8

Cited by 9 publications

(6 citation statements)

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“…Un autre exemple de structures musicales d'Afrique centrale qui révèle certaines propriétés remarquables concerne les formules de harpe jouées par les poètes-musiciens nzakara de République centrafricaine, sur lesquelles nous avons mené des recherches dans les années quatre-vingt-dix, et dont les propriétés surprenantes sont à l'origine de notre intérêt pour l'ethnomathématique. Les résultats ont été décrits dans plusieurs publications [Chemillier, 1995[Chemillier, , 1996[Chemillier, , 1997[Chemillier, , 1999[Chemillier, , 2003], et prolongés depuis par d'autres travaux que nous résumons ci-après. La première propriété remarquable de la formule est sa structure de canon.…”

Section: Formules De Harpe Nzakara 131 Structure En Canonunclassified

Mathématiques de tradition orale

Chemillier¹

2007

msh

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MATHÉMATIQUES DE TRADITION ORALE Marc CHEMILLIER 1 RÉSUMÉ-L'ethnomathématique met en évidence des structures mathématiques dans certaines activités spécialisées de sociétés de tradition orale. L'une des difficultés qui se posent dans cette approche est de mettre en relation ces structures formelles, étudiées « en laboratoire », avec les modes de penser locaux des personnes concernées, telles qu'ils peuvent être observés « sur le terrain ». Cet article expose les difficultés rencontrées à partir de différents exemples : arts visuels, musique, divination. Dans ce dernier cas, on présente les résultats d'enquêtes de terrain à Madagascar qui ont permis d'accéder aux véritables représentations mathématiques autochtones.

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Section: Formules De Harpe Nzakara 131 Structure En Canonunclassified

Mathématiques de tradition orale

Chemillier¹

2007

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“…At that time the second invited editor of this special issue was doing ethnomusicological fieldwork in Central African Republic studying harp music from the Nzakara people. He discovered curious patterns based on a canon-like structure, the upper part played by one hand being reproduced with a delay in the lower part played by the other (Chemillier 1995(Chemillier , 2003. Since these harp formulas are repeated as a loop, one cannot distinguish them from other sequences in the same conjugacy class, i.e.…”

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Music and combinatorics on words: a historical survey

Brlek

Chemillier

Reutenauer

2018

Journal of Mathematics and Music

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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

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“…As one can see, the upper voice is reproduced, with just a few exceptions, in the lower voice, with a delay as indicated by the two broken lines. This formula has the structure of a two-voice melodic ''canon'' (see [2,5,7] for more details, and [3,4] for audio samples of these harp canons).…”

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“…Actually, one can verify that Du has period r ¼ 6. The following proposition is taken from [7]. It has a converse part which is restricted to a very special case, and the interesting fact is that Nzakara harp formulas precisely fall into this case.…”

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“…A canon formula from Nzakara harp music Thus we want to compute a cross-section of the set R m ¼ R \ Per G (m) of infinite words u with period m ¼ card(G) · r, where r is the period of Du. Then f(R m ) is included in Per G (r), and f(R m ) ¼ S r , where S r is the subset of infinite words v of Per G (r) satisfying the following condition (C): the sum of the elements occurring in the prefix of length r of v is equal to an element of order card(G) (see [7] for details). Note that if this condition is true, then it is true for any conjugate infinite word.…”

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