Structural properties of Euclidean rhythms

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

doi:10.1080/17459730902819566

Cited by 11 publications

(5 citation statements)

References 15 publications

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“…Another type of two-interval generated Zeitnetz can be constructed on what Osborn (2014) calls Euclidean rhythms, following Gómez-Martín, Taslakian, and Toussaint (2009) and Toussaint (2013), which include many rhythms common in popular music. These are defined as the beat-class sets that can be generated by permuting the adjacency intervals of a maximally even rhythm.…”

Section: Uniform Tonnetzementioning

confidence: 99%

GeneralizedTonnetzeandZeitnetze, and the topology of music concepts

Yust

2020

Journal of Mathematics and Music

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The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces, and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the twelve pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. The present article examines how these duplications can relate to important musical concepts such as key or pitch-height, and details a method of removing such redundancies and the resulting changes to the homology the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth-chords, scalar tetrachords, scales, etc., as well as Zeitnetze on a common types of cyclic rhythms or timelines. Their different topologies -whether orientable, bounded, manifold, etc. -reveal some of the topological character of musical concepts.

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Section: Uniform Tonnetzementioning

confidence: 99%

GeneralizedTonnetzeandZeitnetze, and the topology of music concepts

Yust

2020

Journal of Mathematics and Music

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“…Each rhythm has a length, and each symbol occurring in the rhythm is either a note or a rest . Rhythms are treated cyclically; for example, is equivalent to [3, 4].…”

Section: Introductionmentioning

confidence: 99%

“…This is achieved by recursively concatenating rhythms of shorter length according to back-substitution. At a conceptual level, Bjorklund’s algorithm resembles Euclid’s, with subsequences and symbol replacement taking the place of integer division and back-substitution [4]. These two algorithms are in fact equivalent [3].…”

Section: Introductionmentioning

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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“…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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Structural properties of Euclidean rhythms

Cited by 11 publications

References 15 publications

GeneralizedTonnetzeandZeitnetze, and the topology of music concepts

GeneralizedTonnetzeandZeitnetze, and the topology of music concepts

On the Euclidean Algorithm: Rhythm Without Recursion

Music and combinatorics on words: a historical survey

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