Block-Designs en composition et analyse musicales

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

Programme et résumés (PDF)

• 14h30 - 14h45 Moreno Andreatta : Welcoming and Overview of the block-designs session
• 14h45 – 16h00 Tom Johnson : Composing with Block Designs (avec la participation d’Adam Weisman, percussionniste)
• 16h00 – 16h45 Franck Jedrzejewski : Groupes et t-designs

• 17h00 - 17h45 Reinhard Laue : Creating and Visualizing Designs by Groups (pdf)
• Discussion

Composing with Block Designs (avec la participation d’Adam Weisman, percussionniste)

Since Kirkman's Ladies (2003), Tom Johnson has written several pieces withcombinatorial designs, a relatively recent mathematical discipline that is good for constructing groups of chords or rhythms in a rigorous manner. He will speak especially of the system (11,4,6), used to compose 55 Chords, an organ piece, but also about Block Design for Piano, Septet, Twelve, and the Vermont Rhythms, which were just premiered in Cambridge by the Dutch ensemble Klang.

Groupes et t-designs

Après une introduction générale sur les t-designs et leur intérêt pour la composition, on étudie les liens de leurs groupes d'automorphismes avec les transformations musicales issues des groupes cyclique, diédral, affine et du groupe des permutations. Les représentations graphiques des t-designs que nous proposons semblent mieux adaptées à l'utilisation de ces éléments pour l'analyse et la composition musicale que la représentation usuelle, mais posent des problèmes mathématiques nouveaux. Enfin, les relations avec la classification des échelles et accords de Forte sont étudiées pour les block designs de petite dimension

Creating and Visualizing Designs by Groups

A t-design on a point set V represents a selection of k-element subsets of V , that covers each t-element subset of V the same number of times. It is thusan approximation of all k-element subsets which is not biased with respect to t-element subsets of V . Many prominent examples have additional interesting properties, in particular nice symmetry groups. We have developed at the University of Bayreuth a program system DISCRETA that constructs t-designs with prescribed symmetries. For cases where these symmetries are also symmetries of a graph we can show these symmetries of the designs by patterns on the graph. These patterns then may reveal further properties of the designs. We present DISCRETA in the talk and show several examples of graphical visualizations of designs. The symmetries of a design may be used for representations of a design on further media, hopefully by some experienced composers of music, to give a new flavour of these pearls of discrete mathematics.