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FOURTH DIDEROT
MATHEMATICAL FORUM
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.
-
- Argument
Schedule
Abstracts and
biographies
Information and
registration
-
Argument
-
- 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.
-
-
Schedule
-
-
- - December 3rd,
morning -
- Historical
perspectives
-
-
- 09:00 - 10:00
- Registration
-
- 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
-
- 15:30-16:15
- 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.
-
- References:
- [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
-
Information
-
-
- Ircam, Scientific
direction.
-
- Florence Quilliard : 01 44 78 48
09
- Florence.Quilliard@ircam.fr
-
Registration
- Registration fee : 100 FF
- By e-mail : send name, 1st-name, postal Adress, e-mail,
telephone to
Florence.Quilliard@ircam.fr
- By phone : Florence Quilliard, 33 1 44 78 48 09 / 01 44
78 48 09
-
- The registration form will be sent to you by regular mail.
-
-