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

the role of musical manifolds. It relies on two constructions: coefficient systems of affine functions and resolutions of global compositions. From the musicological point of view, this is one of the most difficult chapters of mathematical music theory since the relation between classification and musicology, in particular: esthetics, is quite implicit. Therefore classification is highly controversial in musicology. Here are three prominent reasons:

It is commonly believed that classifying musical objects on whatever level is contrarious to the individual expressivity of compositions as they are cultivated since the Renaissance.
Classification is misunderstood as a purely bureaucratic activity of list compilation.
Due to a catastrophical lack of technical tools, traditional musicology has only rarely been able to control the variety of their objects.

The third point shows a disdain of detailed technical work which is psychologically comprehensible but must scientifically be blamed for a major retard even with respect to other humanities such as linguistics.

We have already vaporized these mystifications on several occasions, e.g. Mazzola and Hofmann (1989); Mazzola (1990a), and we will stress one main argument again here: Classification is nothing else than the task of totally understanding an object. This is Yoneda’s lemma in its full philosophical implication, in fact, the isomorphism class of an object $X$ is equivalent to that of its contravariant Yoneda functor $Hom( - ,X)$ . The latter boils down to the synopsis of all perspectives $Y --> X$ (morphisms) under which $X$ may be >observed<. Such a result is in complete harmony with Adorno’s, Valéry’s, and Bätschmann’s insights in the theory of arts. They state that understanding works of art means a synthesis of all their interpretative perspectives, see Mazzola (1990a, 2001b) for details. Results and methods regarding classification of musical structures have already been applied to the theory of the string quartet, composition, and performance (Mazzola, 1990a, 1994, 1990b; Mazzola and Zahorka, 1993-1995).

4.1 Enumeration Theory

A first aspect of classification deals with enumeration, i.e., calculating the number of isomorphism classes of determined musical structures, such as local or global composition, if this number is finite. The representative case is that of local and global objective compositions at finite addresses $A$ and living in finite ambient modules $M$ , such as the classical situation of $A = 0$ and $M = Z 12$ , the case considered by the American Set Theory for pitch classes modulo 12, and $A = M = Z 12$ , the case of self-addressed pitch classes investigated by Thomas Noll (Noll, 1995). The most complete enumeration results have been obtained by Harald Fripertinger by use of Pólya-de-Bruijn enumeration theory (Fripertinger, 1991b,a, 1993a,b, 1996, 1999, 2001).

Let us give a short overview of the main (some results being negleced, no doubt; the author apologizes for this incompleteness) historical landmarks in enumeration