# Structures algébriques et symétries en musique

La Journée est organisée en collaboration avec l'Escola Superior de Musica de Catalunya, le CEA et le Laboratoire disciplinaire «Pensée des sciences» de l'ENS.

## Programme

- 10h - 13h : Matinée pédagogique. The Z/12Z-Story - A Mathemusical Tutorial. [pdf]

Séance animée par**Thomas Noll**, mathématicien et théoricien de la musique, ESMUC.

- 15h-16h :
**Andrei Rodin**(ENS) - Pensée categorielle comme pensée de l’irréversible - 16h-17h :
**Franck Jedrzejewski**(CEA) - Analyse nodale des textes littéraires et musicaux - 17h-18h : Discussion

## Résumés

#### Thomas Noll

**The Z/12Z-Story - A Mathemusical Tutorial**

This tutorial provides an introduction to basic concepts of structural mathematics: groups, monoids, categories, group-actions, monoid-actions, functors, and finally functor categories as instances of toposes. We unfold this small world of concepts by motivating them in terms of music-theoretical applications throughout and by departing from families of musical objects that are related to the 12-tone system in various ways. Once these mathemusical objects are defined, it is not difficult to leave the narrow Z/12Z-world and to transfer ideas from atonal and tonal theory to other domains.

#### Andrei Rodin

**Pensée categorielle comme pensée de l’irréversible**

The usual way to think about translation (between given languages) is in terms of equivalences: given expression x of language X one looks for an equivalent expression y in language Y. Now if languages are thought of as sets of possible expressions this approach may be formalized as follows. Consider union U=XUY and equivalence relation T on U coupling each expression x from X with its equivalent y from Y (and of course each element of U with itself). By Frege's suggestion this allows for introduction of new abstract objects ("meanings") corresponding to equivalence classes by T: expressions x and y "have the same meaning" iff x and y are T-equivalent. (A musical example could be a melody played in different tonalities.)

Let us account to the same situation differently. Consider transformation f of X into Y, and transformation g of Y into X (these transformations can be thought of as translation programs). Suppose that one translates a given x from X by f into language Y and then translates the result back into X by g. The ideal translation programs would bring us x back. Moreover the symmetric operation with any y from Y (y is translated into X by g and then back to Y by f) would also bring back what one starts from. Provided above conditions are met f, g are called *reversible* (one is the reverse of the other) or *isomorphisms*. Suppose (more realistically) that this indeed doesn't happen, and translating x there and back we get another expression x' (of X). The equivalence-based attitude can be probably saved by supposing that x and x' are indeed equivalent in X. To account for it we would need to allow different expressions x, x' of the same language X to have the same meaning (which is quite plausible). In other words we would consider another equivalence relation P (equivalence "up to periphrases") defined on X. Or to put it into the language of transformation we would consider isomorphisms of X to itself (periphrases). Isomorphisms of a given object form an algebraic structure called *group*. This alone suggests an interesting research program in linguistics analogous to Erlangen program in geometry: to study languages from the point of view of (groups of) available periphrases. (Notice that a program able to periphrase expressions in given language would certainly need to comprise all the grammatical rules of this language!). However real advantages of redescription of translation in terms of properties of translation programs (rather than element-wise correspondences) are revealed in the case when the above way out is not available, that is, when x and x' have, intuitively speaking, different meanings. One might think that in this case our translation programs f,g simply don't work, and we get nothing but a mess. Of course this is always up to us to decide what is a good result and what is the mess but the following is the case. In the general case when translations between languages are not reversible we get a structure called a *category* which allows for further neat specifications. Given category of languages might have very "good" universal properties without being trivialised into a set of distinct objects (with or without periphrasing structures). So translations between languages may work well without being reversible. Are there interesting examples of non-reversible transformations also in music?

#### Franck Jedrzejewski

**Analyse nodale des textes littéraires et musicaux**

Dans cette conférence, je montre comment construire des structures de noeuds, de tresses ou d'entrelacs sonores à différents niveaux d'un texte musical, d'un corpus d'oeuvres ou d'un ensemble d'éléments, en imitant la manière dont F. de Saussure cherchait à mettre en évidence par prélèvements successifs des anagrammes, qui contenaient à l'état de germe la totalité d'un texte ou d'une phrase. Aujourd'hui, la possibilité d'interprétation des structures textuelles et musicales d'une œuvre par la description des entrelacements et des enchevêtrements pluriels qu'elle renferme, ouvre de nouvelles perspectives d'analyse et pose les premiers principes d'une linguistique nodale.