On group-theoretical methods applied to music: some compositional and implementational aspects
| Moreno Andreatta |
| Music Representation Team |
| IRCAM-Centre G. Pompidou, Paris |
| Moreno.Andreatta@ircam.fr |
This paper focuses on the group-theoretical approach to music theory and composition. In particular we concentrate on a family of groups which seem to be very interesting for a >mathemusical< research: the non-Hajós groups. This family of groups will be considered in relationships with Anatol Vieru’s »Theory of modes« as it has been formalised and generalised to the rhythmic domain by the Roumanian mathematician Dan Tudor Vuza. They represent the general framework where one can formalize the construction of a special family of tiling canons called the »Regular Unending Complementary Canons of Maximal Category« (RCMC-canons). This model has been implemented in Ircam’s visual programming language OpenMusic. Canons which are constructible through the Vuza’s algorithm are called Vuza Canons. The implementation of Vuza’s model in OpenMusic enables to give the complete list of such canons and offers to composers an useful tool to manipulate complex global musical structures. The implementation shows many interesting mathematical properties of the compositional process which could be taken as a point of departure for a computational-oriented musicological discussion. |
1 Introductory remarks on the role of group theory in music
»The question can be asked: is there any sense talking about symmetry in music? The answer is yes« (Varga, 1996, p. 86). By paraphrasing Iannis Xenakis previous statement, one could pose a similar question about groups and music: is there any sense talking about mathematical groups in music? With the assumption of the relevance of symmetry in music the answer follows as a logical consequence of this universal sentence: »Wherever symmetry occurs groups describe it« (Budden, 1972).
As Guerino Mazzola’s Mathematical Music Theory suggests, there are many reasons for trying to generalise some questions about symmetry in music. But the question needs to be asked as to whether new results could be musically relevant, or whether they represent purely mathematical speculations. A concept of »musical relevance« in a mathematical theory of music is one of the most difficult to define precisely. Inevitably there is a »tension« between mathematics and music