Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

parameter space $EHLD... R$ . Given the mother performance field $Z$ , we have a new field

Z/\,Dir = Z - LZ/\.Dir,

(2)

where $L$ is the Lie derivative. By the method of characteristics in partial differential equations it can be shown (Mazzola and Zahorka, 1993-1995, Vol.I,p.214) that this type of operator englobes all known shaping operators in the implementations on the RUBATO $®$ workstation.

Summarizing, the mathematical model of performance is a canonical generalization of the very special, and musically too narrow, situation known for tempo. And even in that special case has musicology never achieved a valid definition of tempo which exceeds the medieval level of a locally constant velocity (!) (Mazzola, 2001b).

2 Concepts

The conceptual extension enforced by research in mathematical music theory is a dramatic process which led to new problems in musicology, knowledge representation theory, and mathematics.

2.1 Generalization of Common Structures

Ten years ago, the geometric approach to music theory was nothing more than a common mathematization of music(ologic)al objects in the sense that one dealt with categories of local and global compositions. A local composition is a pair $(K, M )$ , where $K$ is a (usually finite) subset of a module $M$ over a commutative ring $A$ , whereas a morphism $f : (K, M )-- > (L, N)$ between two local compositions is a set map $f : K --> L$ which extends to an affine homomorphism $F : M --> N$ , i.e., $F (m) = n + F0(m)$ , a translation by $n$ in the codomain plus a $A$ -linear homomorphism $F0 : M --> N$ . A global composition is defined via a finite covering of a set $K$ by charts $Ki$ which are in bijection with supports of local compositions $(Li,Mi)$ , including transition isomorphisms of local compositions induced by the pairwise chart intersections. Morphisms are the evident maps which are locally chart morphisms (Mazzola, 1990a).

But this setup was too special for two main reasons. Firstly, the development of data base management systems for music research software had to cover more general musical objects, not just local or global compositions. For instance, the objects had to carry names, had to be defined in a recursive way in order to enable hierarchical concepts, and had to admit completely heterogeneous types, such as products, coproducts, lists, etc. In this environment, local and global compositions turned out to be too tightly related to naive mathematical objects. Secondly, new constructions of musicological objects required more general points than just elements of modules: For instance, new developments in harmony (Noll, 1995) require local compositions $K$ where the elements of $K$ are affine morphisms $k : B --> M$ on a domain module $B$ instead of the classical case $B = 0$ which evidently covers the elements of the codomain module $M$ . Thirdly, the recursive constructions