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Publications (112)
Pitch spaces allow pitch relations to be expressed through geometrical representations for many different purposes. The Tonnetz is a well-known pitch space in the field of music theory; equivalent representations have been described in the field of cognitive science, especially Krumhansl's model of perceived triadic distance. Despite her empirical...
The hexachordal theorem is an intriguing combinatorial property of the sets in \Z/12\Z , discovered and popularized by the musicologist Milton Babbitt (1916–2011). It has been given several explanations and partial generalizations. Here we enhance how this phenomenon can be understood by giving both a geometrical and a probabilistic perspective.
After presenting the general framework of `mathemusical' dynamics, we focus on one music-theoretical problem concerning a special case of homometry theory applied to music composition, namely Milton Babbitt's hexachordal theorem. We briefly discuss some historical aspects of homometric structures and their ramifications in crystallography, spectral...
The “colored Cube Dance” is an extension of Douthett’s and Steinbach’s Cube Dance graph, related to a monoid of binary relations defined on the set of major, minor, and augmented triads. This contribution explores the automorphism group of this monoid action, as a way to transform chord progressions. We show that this automorphism group is of order...
This paper deals with the computational analysis of musical structures by focusing on the use of morphological filters. We first propose to generalize the notion of melodic contour to a chord sequence with the chord contour, representing some formal intervallic relations between two given chords. By defining a semi-metric, we compute the self-dista...
Many Western art music composers have taken advantage of tabulated data for nourishing their creative practices, particularly since the early twentieth century. The arrival of atonality and serial techniques was crucial to this shift. Among the authors dealing with these kinds of tables, some have considered the singular mathematical properties of...
The `colored Cube Dance' is an extension of Douthett's and Steinbach's Cube Dance graph, related to a monoid of binary relations defined on the set of major, minor, and augmented triads. This contribution explores the automorphism group of this monoid action, as a way to transform chord progressions. We show that this automorphism group is of order...
This library is a complementation of the mathtools developed by Carlos Agon and Moreno Andreatta. It has that aim to make more easy the use of the Xenakis' sieves in OM#.
A pitch-class set complex is a multidimensional object that spatially represents a collection of pitch-class sets and the intersections between them. If we consider the pitch classes within short time slices a piece can be divided into, we can evaluate for how long some combinations of pitch-classes sound simultaneously and then filter the piece ac...
In this paper we focus on Anatol Vieru’s periodic sequences that we approach with the formalism of the theory of cellular automata. After extending previous results about the action (in the image direction) of one particular cellular automaton on periodic sequences we show the existence of a second one which is its complementary (or dual). The main...
Klumpenhouwer networks (K-nets) and their recent categorical generalization, poly-Klumpenhouwer networks (PK-nets), are network structures allowing both the analysis of musical objects through the study of the transformations between their constituents, and the comparison of these objects between them. In this work, we propose a groupoid-based appr...
Despite a long historical relationship between mathematics and music, the interest of mathematicians is a recent phenomenon. In contrast to statistical methods and signal-based approaches currently employed in MIR (Music Information Research), the research project described in this paper stresses the necessity of introducing a structural multidisci...
In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-nets) constitute a useful framework with connections in both group theory and graph theory. Recent attempts at formalizing K-nets in their most general form have evidenced a deeper connection with category...
In the context of mathematical and computational representations of musical structures, we propose algebraic models for formalizing and understanding the harmonic forms underlying musical compositions. These models make use of ideas and notions belonging to two algebraic approaches: Formal Concept Analysis (FCA) and Mathematical Morphology (MM). Co...
Transformational music theory, pioneered by the work of Lewin, shifts the music-theoretical and analytical focus from the "object-oriented" musical content to an operational musical process, in which transformations between musical elements are emphasized. In the original framework of Lewin, the set of transformations often form a group, with a cor...
This article provides a first introduction to some formal and computational models applied in the analysis and generation of popular music (including rock, jazz, and chanson). It summarizes the main philosophy underlying the project entitled “Modèles formels dans et pour la musique pop, le jazz et la chanson”, which constitutes one of the research...
In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-Nets) constitute a useful framework with connections in both group theory and graph theory. Recent attempts at formalizing K-Nets in their most general form have evidenced a deeper connection with category...
We propose a spatial approach to musical analysis based on the notion of a chord complex. A chord complex is a labelled simplicial complex which represents a set of chords. The dimension of the elements of the complex and their neighbourhood relationships highlight the size of the chords and their intersections. Following a well-established traditi...
This article proposes some thoughts on formal and computational models in and for popular music by focusing on Beatles songs. After a brief presentation of some systematic approaches in the analysis of musical form and of some theoretical tools used in the geometric representation of musical structures and processes (the Tonnetz and other Neo-Riema...
In this article, we present a set of musical transformations based on the representations of chord spaces derived from the Tonnetz. These chord spaces are formalized as simplicial complexes. A musical composition is represented in such a space by a trajectory. Spatial transformations are applied on these trajectories and induce a transformation of...
Formal concept analysis associates a lattice of formal concepts to a binary relation. The structure of the relation can then be described in terms of lattice theory. On the other hand Q
-analysis associates a simplicial complex to a binary relation and studies its properties using topological methods. This paper investigates which mathematical inva...
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Cette contribution se propose de présenter certains aspects théoriques et discuter quelques enjeux épistémologiques des recherches menées par l’auteur dans le domaine des rapports entre mathématiques et musique. Après une introduction générale sur le contexte de la recherche « mathémusicale » à l’Ircam et la place du projet misa (Modélisation infor...
We represent chord collections by simplicial complexes. A temporal organization of the chords corresponds to a path in the complex. A set of n-note chords equivalent up to transposition and inversion is represented by a complex related by its 1-skeleton to a generalized Tonnetz. Complexes are computed with MGS, a spatial computing language, and ana...
The article discusses the application of Formal Concept Analysis to the algebraic enumeration, classification and representation of musical structures. It focuses on the music-theoretical notion of the Tone System and its equivalent classes obtained either via an action of a given finite group on the collection of subsets of it or via an identifica...
This paper presents an approach for the analysis of musical pieces, based on the notion of computer modeling. The thorough analysis of musical works allows to reproduce compositional processes and implement them in computer models, opening new perspectives for the their exploration. Computer models enable the simulations and generation of variation...
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In this article we analyse different types of representation of music, both from a cognitive and a computational point of view. Whereas mental representations of music are the objects of the musical mind, and are therefore by definition a matter of cognitive psychology and philosophy, it can be argued that also mathematical representations of music...
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This paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientif...
This paper describes phase-retrieval approaches in music by focusing on the particular case of the cyclic groups (beltway problem). After presenting some old and new results on phase retrieval, we introduce the extended phase retrieval for a generalized musical Z-relation. This concept is accompanied by mathematical definitions and motivations from...
This paper aims at discussing the polynomial approach to the problem of tiling the (musical) time axis with trans-lates of one tile. We show how this mathematical con-struction naturally leads to a new family of rhythmic tiling canons having the property of being generated by cyclo-tomic polynomials. We discuss the polynomial approach as an extensi...
This book constitutes the refereed proceedings of the Third International Conference on Mathematics and Computation in Music, MCM 2011, held in Paris, France, in June 2011. The 24 revised full papers presented and the 12 short papers were carefully reviewed and selected from 62 submissions. The MCM conference is the flagship conference of the Socie...
In western tradition, mathematics and music have been deeply connected for more than 2000 years. Despite this long history of the relationship between mathematics and music, the professional interest of mathematicians in this domain is a relatively new phenomenon. Whilst the power of applying mathematics in the description of music has been acknowl...
Two approaches for characterising scales are presented and compared in this paper. The first one was proposed three years
ago by the musician and composer Pierre Audétat, who developed a numerical and graphical representation of the 66 heptatonic
scales and their 462 modes, a new cartography called the Diatonic Bell. It allows sorting and classifyi...
cote interne IRCAM: Andreatta09c
Pierre Boulez introduced the concept of creative analysis (Boissière 2002) in the late 1980s suggesting that the aim of analysis should be the production of new pieces. Marcel Mesnage and André Riotte followed this path in their work on computer-aided analysis and composition (Mesnage and Riotte 1993). Our current study focuses on Ligeti’s analysis...
cote interne IRCAM: Andreatta09a
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The mathematical study of the diatonic and chromatic uni-verses in the tradition of David Lewin (9) and John Clough (6) is a point of departure for several recent investigations. Surprisingly, Lewin's original idea to apply finite Fourier trans-form to musical structures has not been further investigated for four decades. It turns out that several...
The paper aims at clarifying the pedagogical relevance of an algebraic-oriented perspective in the foundation of a structural and formalized approach in contemporary computational musicology. After briefly discussing the historical emergence of the concept of algebraic structure in systematic musicology, we present some pedagogical aspects of our M...
cote interne IRCAM: Andreatta06a
Xenakis' tone sieves belong to the first examples of the-oretical tools whose implementational character has con-tributed to the development of computation in music and musicology. According to Xenakis' original intuition, we distinguish between elementary sieves and compound ones and trace the definition of sieve transformations along the sieve co...
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This article develops some aspects of Anatol Vierus compositional technique based on finite difference calculus of periodic sequences taking values in a cyclic group. After recalling some group-theoretical properties, we focus on the decomposition algorithm enabling to represent any periodic sequence taking values in a cyclic group as a sum of a re...
This article develops some aspects of Anatol Vieru's compositional technique based on finite difference calculus on periodic sequences taking values in a cyclic group. After recalling some group-theoretical properties, we focus on the decomposition algorithm enabling to represent any periodic sequence taking values in a cyclic group as a sum of a r...
We present computer models of two works for solo instrument by Iannis Xenakis: Herma for piano (1962) and Nomos Alpha for cello (1965). Both works were described by the composer (in his book Formalized Music) as examples of "symbolic music." Xenakis' detailed description of formal aspects in his compositional process makes it possible to implement...
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Alain Poirier (directeur)
Guerino Mazzola (rapporteur)
John Rahn (rapporteur)
Gérard Assayag (examinateur)
Marc Chemillier (examinateur)
Jean Petitot (examinateur)
cote interne IRCAM: Andreatta03g
In this paper we present the main ideas of the algebraic approach in the field of the representation of musical structures. In this perspective, well-known theories, as American Pitch-Class Set Theory, can be considered as a special case of the mathematical concept of group action. We show how the change of the group acting on a basic set enables t...
Le présent article se veut une introduction aux concepts théoriques à la base de la Set Theory. Il ne présuppose aucune connaissance préalable de ces théories. Un aperçu de leurs applications analytiques est donné avec comme but principal de mettre en évidence la diversité de leur nature. Celles-ci concerne plus particulièrement, la démarche « clas...
cote interne IRCAM: Andreatta03a
This paper aims at presenting an application of algebraic methods in computer assisted composition. We show how a musical problem of construction of rhythmic canons can be formalized algebraically in two equivalent ways: factorization of cyclic groups and products of polynomials. These methods lead to the classification of some musical canons havin...
Algebraic methods have been currently applied to music in the second half of the twentieth-century (see M. Andreatta [Group-theoretical methods applied to music, unpublished dissertation, (1997)], M. Chemilier [Structure et méthode algébraiques en informatique musicale. Thèse de doctorat, L. I. T. P., Institut Blaise Pascal (1990)] and G. Mazzola e...
cote interne IRCAM: Andreatta01d
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