workstation and the Galois theory of concepts. Thirdly, we discuss the central category of local and global compositions with general Yoneda >addresses< i.e., domains of presheaves. This leads to Grothendieck topologies and presheaves of affine functions which are essential for classification purposes. This latter subject is dealt with in the fourth section. We discuss enumeration theory of musical objects and algebraic schemes the points of which parametrize isomorphism classes of global compositions. Based on a substantial isomorphism between harmonic and contrapuntal structures, we give a preview in the fifth section of what future research in mathematical music theory could (and should) envisage.

The fact that this report has been realized under the excellent organization of the Universidad Nacional Autónoma de México and Emilio Lluis-Puebla, president of the Mathematical Society of México, is also a sign that mathematical music theory has transcended its original Swiss roots and has attended international acceptance. At this point, I would like to acknowledge all my collaborators and colleagues for their continuous support and encouragement.

1 Models

1.1 What Are Models?

Basically, mathematical models of musical phenomena and their musicological reflections are similar to corresponding models of physical phenomena. The difference is that music and musicology are not phenomena of exterior nature, but of interior, human nature. To begin with, there is a status of music structures and corresponding conceptual fields, together with compositions in that area, and the modeler first has to rebuild this data in a precise concept framework of mathematical quality. Next, the historical material selection in music and musicology (scales, interval qualities, for example) has to be paralleled in the mathematical concept framework by a selection of instances. Here, the historical genesis is contrasted by the systematic definition and selection of a priori arbitrari instances. After this positioning act, the musical and/or musicological process type (such as a modulation or cadence or contrapuntal movement) has to be rephrased in terms of the mathematical concept framework. With this in mind, the historically grown construction and analysis rules of that determined process have to be modeled on the level of mathematics. This means that the formal process restatement must be completed by structure theorems (including the proofs, a strong change of paradigm!), and then, by use of such theorems, the grown rules must be deduced in the mathematical concept framework.

The typical property of mathematical models in music is this: To enable a quasi-automatic generalization to situations where the classical music theory for which the model was constructed has no answer. In the case of modulation which originally was modeled for major scales, the generalization extends to arbitrary 7-tone scales. This is due to the a priori systematic concept framework of mathematics. Once a bunch of concepts and structures has been set up, there is no reason whatsoever to stick to the historical material selection, the genericity of