Interlocking and Euclidean rhythms

Gómez-Martín, Francisco; Taslakian, Perouz; Toussaint, Godfried T.

doi:10.1080/17459730902916545

Cited by 6 publications

(4 citation statements)

References 8 publications

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“…We refer the curious reader to Bjorklund’s work for the proof of Euclidean rhythms’ equal-distribution properties [1]. Many other theoretical properties of Euclidean rhythms have been described by Gomez-Martin et al [5].…”

Section: Determining Euclidean Rhythmsmentioning

confidence: 99%

On the Euclidean Algorithm: Rhythm Without Recursion

Morrill

2022

Bull. Aust. Math. Soc.

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A modified form of Euclid’s algorithm has gained popularity among musical composers following Toussaint’s 2005 survey of so-called Euclidean rhythms in world music. We offer a method to easily calculate Euclid’s algorithm by hand as a modification of Bresenham’s line-drawing algorithm. Notably, this modified algorithm is a nonrecursive matrix construction, using only modular arithmetic and combinatorics. This construction does not outperform the traditional divide-with-remainder method; it is presented for combinatorial interest and ease of hand computation.

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Section: Determining Euclidean Rhythmsmentioning

confidence: 99%

On the Euclidean Algorithm: Rhythm Without Recursion

Morrill

2022

Bull. Aust. Math. Soc.

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“…We can, for example, establish three possible dynamic levels for any onset of a given rhythm: 1 (default or "flat"), 2 ("ghost" note) or 3 (accented). 8 In this sense, the intention of adding a phenomenal accent to the third onset of the tresillo of Figure 3 would result in a new code: 2 1 × 3 0 × 5 0 × 7 1 × 11 0 × 13 0 × 17 3 × 19 0 = 68782. Therefore, with each unit mapped by a prime number, it is possible to assign different sets of informational values, depending on the level of detail that one intends.…”

Section: Rhythmic Encodingmentioning

confidence: 99%

“…A distinctive trait of this approach is the use of phylogenetic analysis for depicting similarity relations between the rhythmic varieties. Francisco Gómez-Martín, Perouz Taslakian, and Godfried Toussaint [8] address two rhythmic categories, namely interlocking and Euclidean rhythms. While the latter category corresponds to "rhythms where the onsets are distributed among the pulses as evenly as possible" (p.18, italics in the original), the notion of interlocking rhythms refers to interaction of two or more rhythmic lines resulting into complex textures (as in African-Cuban music or in a heavy-metal groove), an aspect to be explored in the present study.…”

Section: Introductionmentioning

confidence: 99%

Prime Decomposition Encoding: An Analytical Tool by the Use of Arithmetic Mapping of Drum-Set Timelines

Mathias¹,

Almada²

2021

MusMat

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This paper introduces an original proposal intended to encode timelines as univocal integers, by the use of arithmetic mapping, exploring inherent properties of prime numbers. A group of algorithms were developed in order to encode drum-set timelines (as well as to retrieve the original rhythms from the codes), either individually or taken together, forming grooves. Additional parameters (dynamic levels and timbre) are also included in the encoding process. Geometrical representation of the grooves, adopting terminology and methodology proposed by Gottfried Toussaint [19], is also provided. Some practical application, addressing analysis and composition, are suggested at the last section of the study.

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“…Moreover, it is exactly the lower mechanical word of slope k/n, and thus Euclidean strings are closely related to Christoffel words, a notion that plays a considerable role in music as we shall soon see. Research on rhythmic oddity and more generally on asymmetric rhythms is not represented in this issue, but their connection with Euclidean strings is described in two articles previously published in the Journal of Mathematics and Music by Gómez-Martín, Taslakian, and Toussaint (2009a;2009b).…”

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confidence: 99%

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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Interlocking and Euclidean rhythms

Cited by 6 publications

References 8 publications

On the Euclidean Algorithm: Rhythm Without Recursion

On the Euclidean Algorithm: Rhythm Without Recursion

Prime Decomposition Encoding: An Analytical Tool by the Use of Arithmetic Mapping of Drum-Set Timelines

Music and combinatorics on words: a historical survey

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