Approche fonctorielle en informatique musicale

Ircam, Salle I. Stravinsky
1, place I. Stravinsky 75004 Paris
(Entrée libre dans la mesure des places disponibles)

Programme (PDF)

• 14h30 - 14h45 Moreno Andreatta & Carlos Agon – Présentation de la séance
• 14h45 – 15h30 Gérard Milmeister - Rubato Composer - History and Concepts
• 15h30 – 16h15 Florian Thalman – Gestural Realtime Manipulation of Denotators for Music Composition
• Discussion (en vidéoconférence avec Guerino Mazzola, School of Music, University of Minnesota)

• 17h15 - 18h00 Thomas Noll – Functors in Music Theory and Analysis
• Discussion finale

Rubato Composer - History and Concepts

Both modern mathematical music theory and computer science are strongly influenced by the theory of categories and functors. One outcome of this research is the data format of denotators, which is based on set-valued presheaves over the category of modules and diaffine homomorphisms. The functorial approach of denotators deals with generalized points in the form of arrows and allows the construction of a universal concept architecture. This architecture is ideal for handling all aspects of music, especially for the analysis and composition of highly abstract musical works. This talk provides an historical and conceptual introduction to the theory of module categories and the theory of denotators, as well as the design of the « Tubato Composer » functorial programming language.

Gestural Realtime Manipulation of Denotators for Music Composition

Inspired by the gestural aspect and the immediacy of human improvisation, the music composition software Rubato Composer has been extended with gesture-driven realtime composition functionality. The software's complex mathematical operations on denotators, e.g. wallpapers of morphisms, multi-dimensional alterations and macro-transformations, can now be applied using simple mouse gestures to create or transform musical compositions. The intermediate gestural positions of any transformational movement are constantly visualized from different perspectives of an innovative modularity and are also immediately played back. This new approach encourages a highly spontaneous and explorative way of composing.

Functors in Music Theory and Analysis

In a previous talk [5] I presented an investigation into a small topos of monoid actions, which I called the "Topos of Triads". I presented analytical experiments with Music of Scriabin and Messiaen in the MaMuX session of May 23, 2004 [6]. In my talk I will discuss two ramifications of these earlier investiagtions: (1) Theoretical ramification. In "Topos of Triads" [7] I used the symbols P, L and R for the designation of three elements of the subobject classifier Omega of this topos. However, in Neo-Riemannian music theory these symbols designate well-known Triadic Transformations (P = Parallel, L = Leading Tone exchange, R = Relative). Therefore I still owe an explanation for my notational choice from 2004. My explanation is given in terms of a categorial redefinition of the "classical" T/I- and S/W-groups as groups of endofunctors on a suitably chosen subcategory of monoid actions. In particular I propose a method for constructing subgroups and associated actions from Lawvere-Tierney topologies. Generalizations of the classical octatonic and hexatonic subgroups <P,R> and <P,L> as well as the group <P> can be defined in this way. (2) Analytical ramification. Revisiting my 2004-analysis of Scriabin's Op. 65 No.3 I will present some reflections on functorial analysis.

• [1] G. Milmeister, The Rubato Composer Music Software: Component-based Implementation of a Functorial Concept Architecture, Springer, 2009.
• [2] G. Mazzola, The Topos of Music. Geometric Logic of Concepts, Theory, and Performance, Birkhäuser, 2002.
• [3] G. Mazzola, G. Milmeister, J. Weissmann, Comprehensive Mathematics for Computer Scientists (2 volumes), Springer, 2004. Online version available at : http://math.ifi.uzh.ch/book/
• [4] G. Mazzola, La vérité du beau dans la musique (en collaboration avec Yun Kang Ahn), Collection « Musique/Sciences », Ircam/Delatour France, 2007.
• [5] Th. Noll, « Transformational Logics in Harmonic and Metric Analysis », séance Théorie des groupes, des catégories et des topoï en musique et dans les arts plastiques : aspects théoriques et perspectives philosophiques, séminaire MaMuX, 20 mars 2004.
• [6] Th. Noll, « Experiments within Triadic Logics », séance Outils informatiques en analyse musicale. Logique et théories transformationnelles en musique, séminaire MaMuX, 23 May 2004.
• [7] Th. Noll, « The Topos of Triads », Colloquium on Mathematical Music Theory, H. Fripertinger, L. Reich (eds.), Grazer Math. Ber., Nr. 347, p. 103-135, 2005. http://user.cs.tu-berlin.de/~noll/ToposOfTriads.pdf