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Mathematics and Music


Mathematical logic, musical logic in the XXth century.

Paris, Ircam, December 3-4, 1999.


Under the authority of the European Mathematics Society, a public conference on the theme " Mathematics and Music " will be held in parallel in Lisboa, Paris and Vienna on December the 3rd and 4th, 1999. The specific theme dedicated to the Parisian days is "mathematical logic, musical logic" in the XXth century. This event is organized by Gérard Assayag (Ircam) and Laurent Mazliak (University Paris VI), with the assistance of a scientific and musical committee including, in addition to the organizers, Marc Chemillier, Laurent Fichet, François Nicolas and Andre Riotte. It is supported by CNRS. The goal of this forum will be to present in prospect certain aspects of evolution of formalism in these fields and to seek some meeting points.



Abstracts and biographies

Information and registration



Mathematical logic, musical logic.
One notices a double evolution in the principle of the 20th century: logic has become more mathematical, thereby loosing its ancient status of philosophical discipline; the question of a musical logic has become explicit as a search for a "coherence" specific to music. This musical evolution is contemporary with the end of tonality and of thematism which were the principles that until then, ensured the coherence of musical works. Tonality and thematism certainly implicated specific logic, which could possibly be formalised, but which were often based on natural rather than axiomatic foundations: tonality was based on physics, and thematism was based on psychology. The "logical" dimensions therefore stayed, for a large part, dependant on ontological foundations, or foundations in terms of musical being (key and theme). In the 20th century, composers found themselves before an ontological void. They had to make decisions that didn't come from physics and psychology in an obvious way anymore, but which remained all the same the starting or articulation points of symbolical calculations, expressing an internal logic of the musical form and material. As for logic, it looks at progressions that are universally valid due to not being attached to such or such position of existence. By becoming more mathematical it has also acquired added calculation power, trying with success formal reasoning on itself, one emblematic result of which is the non-fulfilment theorem by Gödel. Mathematics (or its theories) itself could start where axioms of existence intervene, such as those of the theory of ensembles. We may then ask ourselves the following questions: do formalisms built on musical "reasoning" (the mixed ensemble of its rationality, decisions, aims, and of its calculations, deductions and progressions) have anything to do with such or such formalism of logic? What formal coherence, described in a logical way, can exist in a work, beyond the arbitrary of singular esthetical decisions? Can logic, expressed mathematically help musicians to clarify the specificity of their way of reasoning? Taking the problem the other way round, can musical logic be a source of inspiration for mathematicians?
From a text by François Nicolas.



- December 3rd, morning -
Historical perspectives
09:00 - 10:00
10:00 - 10:30
Opening session
Hugues Vinet, scientific director of Ircam.
Jean-Pierre Bourguignon, Director of the IHES, Professor at the Polytechnic School.
10:30 - 11:30
Formalization of Logic : the Issue of Meaning
Marie-Josée Durand-Richard, University of Paris 8.
11:30 - 12:30
Mathematical Methods for Music Analysis in the XXth Century.
Laurent Fichet, University of Toulouse.
- December 3rd, afternoon -
Implicit computation, unconscious computation
14:00 - 14:45
Where do the Structures Reside? The issue of levels in the study of cognition.
Daniel Andler, Paris IV University.
14:45 - 15:30
Ethnomusicology, Ethnomathematics. The Underlying Logic of Oral Artistical Practices.
Marc Chemillier, University of Caen.
15:30 - 16:15
Cognitive Musicology and the Logic of Musical Images
Marc Leman, director of IPEM, University of Ghent.
16:15-16:30 Coffee break.
International Round Table - Visioconference
16:30 - 18:30
Are the Relations Between Mathematics and Music Natural or Cultural ? In other words, is the presence of mathematics in music " gradually discovered" or on the contrary "gradually built" according to needs induced by historical context ?
- December 4th, morning -
Formal systems
9:30 - 10:15
Xenakis and Logic.
Mikhail Malt, Ircam.
10:15 - 11h00
Music and Lambda Calculus.
Yann Orlarey, scientific director, GRAME.
11:00 - 11:30
Coffee break
11:30 - 12:15
The Topos Geometry of Musical Logic.
Guerino Mazzola, University of Zurich.
12h15 - 13h00
Statistical Learning Applied to Music and Sound.
Shlomo Dubnov. Ben-Gurion University.
- December 4, afternoon -
The limits of formalization
14:00 - 15:00
Musical Logic and the Open Work : Solo by Stockhausen.
Benny Sluchin, InterContemporain Ensemble. (Benny Sluchin will perform Solo for melodic instrument with feedback by k. Stockhausen, in a trombone / live electronic version).
15:00 - 15:30 Coffee break
What is a Musical Logic?
François Nicolas, composer.
16:15 - 17:00
Can Mathematics Stimulate Creativity in Music?
Jean-Paul Allouche, Director of Research at CNRS.
17:00 - 17:30 Coffee break
17:30- 18:15
Logic and Coherence in Musical Creation.
Andre Riotte, composer.

Abstracts and Biographies

Formalization of Logic : the Issue of Meaning
Marie-Josée Durand-Richard, Université Paris 8
Marie-José Durand-Richard (1944 -) is "maître de conférence" (lecturer) in epistemology and history of science at the University of Paris 8. Her work mainly deals with the study of the emergence and effective historical role of the symbolic approach to algebra in England at the end of the Industrial Revolution, an approach marked by the inventions of Charles Babbage (1791-1871) concerning the logical functions of a computer, and those of George Boole (1815-64) aiming at an algebra of logic.
The movement of the mathematisation of logic occurs with the work of George Boole (1815-64) on what he regarded as the laws of thought. It is rooted in a reflexion carried out in England in the first half of the 19th century, from the mathematics side at Cambridge, and within logic at Oxford, on the following issue: Which of these two disciplines constitutes the true base of all knowledge, and which of them makes it possible to integrate new knowledge? With these questions, the idea of a radical separation between a logic of operations, whose nature is strictly symbolic, and its possible signifiant interpretations was posed and made functional. Consequently, there was to be a ceaseless renewal of the question of the place and nature of meaning between the defenders of blind calculation and those of a subjacent ontology, a debate whose outlines we will follow up to the present day.
Mathematical Methods for Music Analysis in the XXth Century.
Laurent Fichet, I.U.F.M., Toulouse.
Laurent Fichet is a qualified teacher and Doctor in musicology. He has published some articles and works on scientific theories of music, from the 17th to the 20th century, and given lectures on these subjects at the University of Paris IV-Sorbonne. He is at present in charge of the Technologies of Information and Communication in teaching at the I.U.F.M. of Toulouse.
In the 20th century, several methods of analysis of music are widely inspired by mathematical processes. Using those that seem more likely to give interesting results, we put forward the analysis of part of a piece (2nd sonata by Boulez) which seems to allow such a method of reasoning. The comparison between what these mathematical analyses and a more intuitive analysis might bring will give a balanced view of the links between musical logic and mathematical logic.
Where do the Structures Reside? The issue of levels in the study of cognition.
Daniel Andler, University of Paris IV.
Trained first of all in mathematical logic (theory of models) in Paris and then in Berkeley, Daniel Andler taught mathematics in different Parisian universities before officially moving on to philosophy. He is at present a professor in philosophy at the University of Paris-Sorbonne (Paris IV), and a member of the CREA (Research Centre in Applied Epistemology). He practises the philosophy of science, and centres his research on cognitive sciences, being interested amongst other things in their foundations and their effects on the philosophy of knowledge and on philosophical anthropology.
When Chomsky offered to study the linguistic capacities of Homo sapiens by the possession of a particular competence called universal grammar, a lot of people wondered about the psychological reality of such an entity. Generally speaking, the explanatory diagrams developed within cognitive science are often interpreted in an instrumental way: it is often thought the calculations and the representations it puts forward are just convenient analysis tools, striped of their psychological or neurophysiological existence. In the same way, von Neumann's computer, or neural nets also, could only at best give momentarily convenient metaphors for characterising the brain. This is not however, a unanimous point of view, nor even a dominant one amongst practitioners of this discipline. This issue gets more complicated when we consider the production by the cognitive system of entities, which are pretty obviously equipped with structures and dynamics, such as language, mathematics or music. Is there a relationship between the form (static or dynamic) of the productive system and the form of its products? At what level of description are we more likely to understand this relationship?
Ethnomusicology, Ethnomathematics. The Underlying Logic in Oral Tradition Arts
Marc Chemillier
Marc Chemillier is a former student in Ecole Nationale Supérieure, Fontenay-aux-roses, and the Conservatoire National Supérieur de Musique de Paris, post-graduated in music and doctor in computer science. His research relates to the modeling of musical structures. He is co-editor with François Pachet of the collection devoted to music and computers at Hermès Editions. He is more particularly interested in the African tradionnal music and went several times to Africa for recording sessions and ethnomusicological prospects.
Oral tradition musics sometimes show complex structures (for example the Central African polyphonies studied by Simha Arom). It is outstanding that these structures appear in societies without writing. Other activities in these societies, such as graphic arts, rituals or games, also showing very elaborated structures, gave rise to a current in history of mathematics named ethnomathematics.
Music shares with these activities one common point: the human mind freely explores a logic of rules (rules of the strategy games, generative rules for graphical design, rules for polyrythmical combinations, etc). Starting by some examples, borrowed from the Vanuatu "drawings on the sand" and from the corpus of African music, we will consider several questions: how to articulate the description of these examples with the reconstitution of the cognitive activity that produced them ? What is there in common between these productions, and the ones experienced in the occidental culture under the name of "mathematics" ? Should we consider that they are variations of the same universal scheme, that is the playful prolongation of the elementary rationality with which human beings are provided for their survival ? In the case of music, what relation can be found between these formal combinatory games and perception ?
Cognitive Musicology and the Logic of Musical Images
Marc Leman, Université de Gand.
Marc Leman is research leader at the Fund for Scientific Research and Professor at the University of Ghent. He is director of the Institute for Psychoacoustics and Electronic Music, head of the Research Society for the Foundations of Music Research (sponsered by FWO), and editor-in-chief of the Journal of New Music Research (published by Swets & Zeitlinger, the Netherlands). His research interest is focused on the epistemological and methodological foundations of cognitive and systematic musicology.
Cognitive musicology aims at understanding the nature of musical information processing and imagination in composing, listening and performing. Of particular relevance is the choice of a proper description system. The cognitivist approaches of the seventies took formal (proposition and predicate) logic as a basic system for representational descriptions. Although its assumptions fit quite well with the "note" or "score" &endash;based ontology of classical music theory, this approach has been criticized [1,2] for making unrealistic claims about the nature of musical information processing. The naturalist approach is grounded on a physical and physiological theory of human information processing [2,3]. We argue that a musical logic can be developed in terms of a logic of musical images. The latter can be conceived of as representations of neuronal activation. Formal logic can be used as a meta-level description system for a clarification of the underlying processing of such images. We present a framework that incorporates a logic of musical images useful for our understanding of music perception. If perception is a basis for composition and performing then this logic should also be relevant for musical creation.
[1] M. Leman, Adequacy criteria and models of musical cognition, in J.N. Tabor (ed.), Otto Laske: Navigating new musical horizons, Westport, CT: Greenwood Publ. Comp., 1999, (ISBN 0-313-30632-X)
[2] M. Leman, Naturalistic approaches to musical semiotics and the study of causal musical signification, in I. Zannos (ed.), Music and Signs -- Semiotic and Cognitive Studies in Music, ASCO Art & Science, Bratislava, 1999 (ISBN 80 - 88829 - 15 -4)
[3] M. Leman (ed.) Music, Gestalt, and Computing -- Studies in Systematic and Cognitive Musicology, Springer-Verlag, 1997 (ISBN 3-540-63526-2).
Music and Lambda Calculus.
Yann Orlarey, GRAME, Lyon.
Composer and researcher, director of the Research Department of Grame, Lyon, France.
Although developed initially in the thirties as a study of the general properties of the mathematical functions, Lambda-Calculation constitutes today one of the theoretical bases of computer science. Since work of McCarthy and Landin in the sixties, Lambda-Calculation had a considerable influence on the design and the implementation of the programming languages, particularly for the functional languages.
In the field of the computer music, if language LISP was at the basement of many works, Lambda-Calculation, up to one recent period, was not, unless few, exploited directly.
The talk proposes to show how the Lambda-Calculation, and particularly the concepts of abstraction and of application, can play a part in the description of musical concepts, formal structures and compositional process, and how it can be at the base of new computer music tools for aid to the musical composition.
The Topos Geometry of Musical Logic
Guerino Mazzola, University of Zurich.
Guerino Mazzola was born 1947 near Zurich and studied mathematics, theoretical physics, and crystallography at the University of Zurich. After postdoctoral studies in Paris and Rome habilitation in algebraic geometry and representation theory in 1980. 1980-1989 development of mathematical music theory and composition software PRESTO. 1984-1986 director of the Darmstadt symmetry exposition. 1986-1988 SNSF grant on Depth EEG and music perception. 1992-1996 SNSF grant on the generic music platform RUBATO.
1996-1999 associate professor at University Laval/QuÈbec. Lecturer at ETH Zurich and at the Universities of Zurich and Vienna. Scientific consultant at TU Berlin, web consultant at Swiss universities and expert at the Swiss Science Council. Author of over 80 scientific papers and eight books on arithmetic, topology, category theory, algebraic geometry, mathematical music theory, brain research, computer graphics, performance theory, music informatics, science - and web policy. As a working contemporary jazz
pianist, Mazzola has published ten LPs and CDs. Presently, he is finishing the book <The Topos of Music> with 13 collaborators.
The logic of musical composition, representation, analysis, and performance share important basic structures which can be described by Grothendieckís functorial algebraic geometry and Lawvereís topos theory of logic. We give an account of these theoretical connections, discuss, and illustrate their formalization and implementation on music software. Three issues are particularly interesting in this context: First, the crucial insight of Grothendieck that "a point is a morphism" carries over to music:
Basically, musical entities are transformations rather than constants.
Second, it turns out that musical concepts share a strongly circular character, meaning that spaces for music objects are often defined in a self-referential way. Third, the topos-theoretic geometrization of musical logic implies a progressively geometric flavour of all rational interactions with music, in particular when implemented on graphical interfaces of computer environments.
Statistical Learning Applied to Music and Sound
Shlomo Dubnov, Lecturer, Ben-Gurion University of the Negev Beer-Sheva, Israel.
Shlomo Dubnov studied piano and composition in the Rubin Academy in Jerusalem. He did his PhD in Computer Science at the Hebrew University and during the years 1996-98 he worked as an invited researcher at IRCAM. He won an ICMC best paper award for his research on statistical analysis of timbre. Dubnov's music works were presented at JIM, ICMC and at Portes Ouverts at IRCAM. Today, S.Dubnov is in charge of Multimedia
Program at Communication Engineering Department in Ben-Gurion University, Israel.
Many aspects of musical structure are impossible to define formally. Nevertheless, music and sound exhibit a great amount of structure and redundancy. In the talk I will describe the use of information theory for discovering these hidden structures, specifically considering statistical relations and dependencies that exist among musical parameters in existing musical and sound material.
It is important to note the close link that exists between redundancy, compression, prediction and classification. If we know how to discover redundant patterns, we can both represent the music more compactly and also create new music sequences or sounds with similar typical redundancy. In the talk I will present applications of statistical learning methods for classification and generation of musical and sound structures. Diverse results, such as melodic and polyphonic voice modelling, jazz chord sequence generation and statistical learning of granular synthesis
(sound textures) will be demonstrated.
Musical Logic and the Open Work : Solo by Stockhausen.

Benny Sluchin, InterContemporain Ensemble.

Benny Sluchin studied music at the Conservatory of Tel Aviv, his home town, and at the Jerusalem Academy of Music. In parallel, he studied mathematics and philosophy at the University of Tel Aviv and was awarded a Master of Science degree with distinction. He began playing with the Israel Philharmonic Orchestra for two years, before holding the position of co-soloist with the Jerusalem Symphony Orchestra for four years (Radio Orchestra). A grant from the German government took him to Cologne, where he worked with Vinko Globokar, and was awarded his artist degree with distinction. He has been a member of the Ensemble Intercontemporain since 1976. He has played the most representative contemporary works for trombone and has taken part in numerous first performances of solo works (Iannis Xenakis, Vinko Globokar, Gérard Grisey, Pascal Dusapin, Frédérick Martin, ElliottCarter, Luca Francesconi, Marco Stroppa, James Wood). In parallel, he has taken part in Ircam's acoustic research and finished a thesis for a Doctorate in mathematics in 1982. He is the author of several articles and educational works in particular Contemporary Trombone Excerpts and et Jeu et chant simultanés sur les cuivres (European Musical Editions), was awarded the 1996 SACEM education production price. Professor in Academy of Levallois and teacher at the Academy of Paris (computer musical notation), Benny Sluchin is giving master-classes and conferences in the whole world. Among its recordings : The Contemporary Trombone, French Bel canto Trombone (Musidisc), Xenakis -Keren (Erato) Berio - Sequenza V (complete series of sequenzae at DGG).

Solo for melodic instrument with feedback, composed in 1996 by Stockhausen, exploits the concept of feedback, by which a musician transforms what it hears, and that it possibly even produced himself, according to instructions given to him. It is a polyphony for monodic instrument in which musical patterns related to memory, and produced by recording, transformation and re-injection in real time, are superimposed and intermingled. It is not then any more an object, but a structural development process which is given to perceive. The compositional logic integrates (and constrains) the logic of performance. Preparation by the soloist of a new version of Solo induces complex combinatorial problems, because it must at the same time take account of the formal constraints indicated by Stockhausen, and of the memory effects caused by the re-injections.

From a text by B. Sluchin.


What is Musical Logic?
François Nicolas, composer.
Former student at the École Polytechnique and qualified in philosophy, François Nicolas is a composer. After having been editor for the Entretemps magazine, he founded the edition of the same name and joined forces with the Revue de Musicologie. He is at present a composer in research at Ircam.
Some of his latest publications: Les enjeux du concert de musique contemporaine (CDMC), Quelle unité pour l'oeuvre musicale? Une lecture d'Albert Lautman (Horlieu), La Singularité Schoenberg (Ircam l'Harmattan).
Some of his recent works: Dans la distance (commissioned by Ircam), Transfiguration (commissioned by the University of Corfou), La Ballade de Maldoror (commissioned by the group X-Musique), Veränderung (commissioned by Radio France).
I. If we remember three philosophical meanings of what a mathematical logic is (a grammar / a tautology factory / a consistency in the appearance), we can in this way identify their resonance in the musical field: musical logic could either be the syntax of musical language, the coherence of large musical forms, or the consistency of what means in music the fact of hearing.
II. We will assert that in music, logic is traditionally said dialectic, and it then has two deployment areas:
- the "world" of music in its entirety: from this stems the logic of the musical topos which will be called dialectic of the eye and of the ear (more technically: dialectic of writing and of perception);
- the area that each musical work delimits in its own way: from this stems the logic of the work, which articulates the statements of the work with a specific duty of statements (more technically: dialectic of development, of deployment, and of variations), and which on the one hand structures the listening of the singular form of a work, and on the other hand - in a less "totalising" approach of the work - the hearing of its singular subjectivity (its strategy, its stakes, and its processes).
III. With these different hypotheses, we will explore a few important types of musical dialectic having been used in the 20th century, making sure they dialogue with diverse orientations of logic expressed mathematically.
Can Mathematics Stimulate Creativity in Music?
Jean-Paul Allouche, CNRS.
Jean-Paul Allouche was educated at the École Normale Supérieure de Saint-Cloud. He is "agrégé" in mathematics and "Docteur d'État" in mathematics. He is presently "Directeur de Recherche (2e classe)" at the CNRS in the Orsay Computing Research Laboratory (LRI, University Paris XI). He is working on problems between number theory and language theory.
Whereas the traditional assertions - in general by mathematicians who are not musicians - on the alleged close connections between mathematics and music seem to me common places without depth nor truth, I am interested in the work of composers who use mathematical tools. Those whose work I tried to understand or with whom I collaborated are divided essentially into two categories. For some of them mathematics yield metaphors. Inspired by some mathematical concept, their creations can get free of it if necessary, and mathematics are a framework and a catalyst but never a yoke. For others, sometimes explicitly minimalist, the logic of the piece is the algorithmic tool that helped (ruled?) its composition. We will try to illustrate this classification by alluding to the work of composers like Marcel Frémiot and Tom Johnson, and to outline a reflection towards the risks of perversion of the scientific language (à la Sokal and Bricmont).
Logic and Coherence in Musical Creation.
André Riotte, composer.
An engineer, André Riotte received a dual education, musical and scientific. He studied composition with André Jolivet and analysis with Olivier Messiaen and Jean Barraqué. He worked for ten years (1961-71) in Italy, in a European research centre where he also sustained the cultural life for the 1200 scientists. He has taught the formalisation of musical structures at the University of Paris 8, and then for the DEA in Musicology of the 20th century. Some of his works are Anamorphoses, for baritone and chamber ensemble, Multiple, quartet with multiple solutions, and La Bibliothèque de Babel for choir and orchestra after Borgès. He is Vice-president of the Société Française d'Analyse Musicale and editor of the Musurgia magazine.
No theory on form will be able to cover the entirety of works to come. Depending on their training and their preferences, creators oscillate between rigor and liberty; any rigidity that isn't lived in the reasoning of the composer restrains the significance of his works. We will try and define coherence as a criteria larger in quality, and non-contradictory with the notion of liberty.

Fourth Forum Mathematics Diderot - Paris, Ircam 3-4 December 1999


Ircam, Scientific direction.
Florence Quilliard : 01 44 78 48 09



Registration fee : 100 FF
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