Algebraic Models in Music Theory, Analysis and Compositio: Towards a Formalized Computational Musicology
Résumé
Algebraic models have a long tradition in systematic musicology. The paper aims to clarify the relevance of an algebraic-oriented perspective in the foundation of a structural and formalized approach in contemporary computational musicology. We firstly focus on the historical emergence and development of the concept of algebraic structure in Twentieth-Century music by showing how this process accompanied almost contemporaneously the constitution of modern music theory, as well in American Set Theory (from Ernst Krenek and Milton Babbitt’s axiomatization of the Twelve-Tone System to David Lewin’s transformational approach to music analysis) as in European and East European structural tradition (from Iannis Xenakis and Anatol Vieru’s formalized music to Guerino Mazzola’s category-oriented Mathematical Music Theory). We then present our mathematical environment (Math Tools) within OpenMusic graphical programming language, which enables the construction of algebraic models of music-theoretical, analytical and compositional processes. Its “paradigmatic” architecture, taking several different group actions as the basis of variable catalogues of musical structures, enables to give a formalized and flexible description of the notion of “musical equivalence”. This makes use of some standard algebraic structures (cyclic, dihedral, affine and symmetric groups) as well as more complex constructions based on the ring structure of polynomials. Algebraic models have both a pedagogical and a musicological interest for they enable the music theorist to visualize some structural musical properties in a geometric way and to test the relevance of different segmentations in music analysis. This could have a strong implication in the way to teach music theory, analysis and composition and raises, at the same time, new interesting philosophical questions about the duality between “objects” and “operations” in music and its possible cognitive and perceptual ramifications.