2.  Transforms in the musical space and hierarchical composition

2.1.  Some philosophical questions

In spite of great technological advances, the question whether mathematics does (or should) play any important role in music still seems to be controversial. A basic analogy between music and mathematics is that both have to do with structure (but not only, of course). Mathematics (among other sciences) helps to understand structures. Thus, if we are able to find relevant musical structures that are analogous to certain mathematical structures, then a mathematical analysis of theses structures is likely to lead to important insights regarding musicological questions. In particular, this can help to obtain a logical foundation of music theory, extensions of traditional music theory and new compositional techniques and styles. Naturally, in all this, one should bear in mind that each mathematical structure is only a simplified representation of certain structural aspects of music. Thus, in a mathematical analyis only certain aspects are discussed. Different mathematical tools may be needed for different aspects. Moreover, abstract mathematical models always have to be validated, i.e. their musical relevance has to be checked empirically. Without empirical evidence, mathematical models in music would indeed remain purely abstract constructions without any concrete impact on musical theory and practice.

In particular, empirical validation needs to answer the following questions:

1. Analysis: Can one find the postulated mathematical structures in a) existing scores; b) performances of these scores; c) improvised music?
2. Composition: Can one derive tools for musical composition from mathematical structures? Are the resulting compositions still music?

A partial “objective” answer to 1 is given below. (The answer to 2 is, by its very nature, much more “subjective”, and can only be decided by listening to the corresponding compositions.) In the following, the basic theory used for the composition of the piano concert is summarized very briefly. For a complete account see Mazzola (1990) (also Beran and Mazzola 1999a,b).

2.2.  Representation of basic events (notes) of a composition:

Mazzola (1990) defines a local composition to be a set C in module M = T ×P ×D ×L. A global composition is defined by “joining” local compositions using the mathematical definition of manifolds. Which module is used depends on the aspects considered. The large variety of aspects in music motivates to use the flexible definition of algebraic modules, instead of spaces with predifined coordinates such as R^k,Z^k etc. Similarily, the very general definition of manifolds is needed to encompass a sufficiently vast variety of structural aspects.