"Music and Mathematics: From Topos Theory To Music Software"

Prof. Guerino Mazzola

Universität Zürich ­ Duebendorf Institut für Informatik/Multimedia Lab Switzerland

We give an overview of mathematical music theory and first describe and exemplify topoi of denotators. These are special set-valued functors on the category of modules; they realize a general concept of music objects which meets requirements of mathematics and of data base management systems.

We then outline the theory of transformation processes of symbolic music objects, as they are codified in traditional scores, into objects of physical reality, as they appear in artistic performance. Such processes rely on performance vector fields and involve mathematical music analyses of the given symbolic data, as well as performance grammars to shape performance fields as an expression of analytical facts. We discuss the theory, known results, and related problems.

This theory is illustrated by an exposé of the present object-oriented implementation in the RUBATO software, including sound and video samples.

We conclude with an outlook on future research in mathematical music theory.