precise concepts and theorems enables a broader perspective which pure historicity cannot offer.

The property of extensibility of a mathematical model relocates the existing music theory (which it models) in a field of potential, fictitious theories. This puts the historically grown facticity into a relation with the potential >worlds of music<. The purely historic justification of existing modulation rules, for example, does not give us reasons for this choice, and this makes the purely historical approach a poor knowledge basis: We know that something is the case, but not why, and why other possibilities are not. In contrast, the mathematical approach gives us a field of potential theories wherein the actual one can be asked for its possible special properties with respect to non-existing variants. This differentia specifica is a remarkable advantage of mathematical methodology against the historical approach of musicology which cannot embed the facts in a viariety of fictions and thereby understand the selection of what is against what is not.

This evokes Leibniz’ idea that the existing world is the best of all possible worlds: Is the existing music theory the best possible choice? Or is it at least a distinguished one? In cosmology, this idea has been restated under the title of the »anthropic principle« (Barrow and Tipler, 1986). It says that the physical laws are the best possible for the existence of humans, more precisely (and less radically), it is the theorem stating that a slight variation of the fundamental constants, such as the gravitational constant, or the electric charge of electrons and protons, would make any higher molecular complexity as it is necessary for the carbon-based biochemistry impossible.

1.2 Modulation

The historically first model in mathematical music theory dealt with tonal modulation, more precisely: with Arnold Schönberg’s model of a tripartite modulation process from tonality to tonality , as it is described in the classical treatise on harmony (Schönberg, 1966). The process parts are the following, exemplified for -major, -major:

A.

Neutralization of the old tonality , neutral degrees of are presented, for example .

B.

The pivotal root progression degrees (German: »Fundamentschritte der Modulation«) are played to enforce the turning movement towards the new tonality, for example degrees .

C.

The new tonality (-major in our example) is evidenced by a set of cadence degrees, for example .

In Schönberg (1966), such transition processes are described for a set of tonality couples, but not for all possible couples: These omitted couples are dealt with by a chain of at least two successive modulations through intermediate tonalities. Also is the construction of the core steps, i.e., the pivotal degrees, not independent of the specific constellation, it is rather an ad hoc argumentation. Moreover, the concepts are quite fuzzy, as usual in musicology. Finally, one cannot infer, how