The Geometrical Groove: rhythmic canons between Theory, Implementation and Musical Experiment

During the second half of the twentieth century, algebraic methods have been increasingly recognised as powerful approaches to the formalisation of musical structures. This is evident in the American music-theoretical tradition as well as in the European formalised approach to music and musicology. We mention the mathematician and composer Milton Babbitt, the Greek composer Iannis Xenakis and the Roumanian theoretician and composer Anatol Vieru, that gave important impulses to the subject of our paper (Babbitt, 1960; Xenakis, 1971; Vieru, 1980). We also mention Gerald Balzano’s original contribution (Balzano, 1980) and Dan Tudor Vuza’s model of Vieru’s modal theory (Vuza, 1982-), as well the approaches of Guerino Mazzola (Mazzola, 1990), Harald Fripertinger (Fripertinger, 1991) and Marc Chemillier (Chemillier, 1990), who opened the path to a generalisation and implementation of algebraic properties of musical structures. This paper especially deals with the implementation of Vuza’s model of periodic rhythm in OpenMusic, an open source visual language for composition and music analysis developed by IRCAM. This has been done as a part of a specific OM package called Zn, entirely based on the algebraic properties of finite cyclic groups and their applications to music. A complete catalogue of intervallic structures (up to transposition) is the starting point for a classification of such structures by means of musically interesting algebraic properties (Olivier Messiaen’s limited transposition property, Milton Babbitt’s all-combinatoriality, Anatol Vieru’s partitioning modal structures...), their generalisation for any n-tempered system and reinterpretation in the rhythmic domain. In this article we extend the idea of ’Regular Complementary Canons of Maximal Category’ (Andreatta, Agon , Chemillier, 1999) to rhythmic canons of various kinds having the property of tiling musical space (See below). Rhythmic Canons Tiling the Space The present essay focuses on the implementation of a family of rhythmic canons having the property of tiling musical time space. Before describing them in terms of an abstract model of cyclic time, we view them as they may appear within a musical composition, in the ’free’ linear time, which has no cyclicity. Like in a melodic canon, one has several voices that may enter one after the other until all voices are present. As in the case of a melodic canon all voices are just copies of a ground voice that is suitably translated in the time axis. For simplicity but yet with respect to linear time we suppose here, that all voices are extended