such an argumentation should deal with non-European tonalities. So there is the mathematical modeling enterprise as described above, on the level of musicological theory. Besides that, the model must also be tested on the corpora of compositions where there is a certain chance to recognize such modulation processes. But let us get off on the theoretical level first and comment on the experimental work later.

In the first steps, one makes the concepts of »tonality«, »degree«, »cadence« precise. Then, one should model the modulation mechanism, and last, one has to prove theorems which yield the pivotal degrees in process part B. Since this model has been described on several occasions (Mazzola, 1981, 1985, 1990a; Muzzulini, 1995), we shall be very sketchy and only mark the cornerstones of the modeling operation¹

. For the tonalities, one takes a seven-element scale of pitch classes and covers by seven triadic degrees which are three-element subsets with each an intermediate pitch class between the first and second, and between the second and third degree pitch. For the -major scale , this gives us the classical triadic degrees. By definition, a tonality is a scale , together with its covering by triadic degrees. For the given modulation problem, we consider the translation orbit of the -major tonality . For a given couple , the modulation mechanism is the datum of a symmetry , i.e., a translation or an inversion on the ambient space which carries the first tonality onto the second. The cadence concept is grasped by minimal subsets of triadic coverings such that only the respective scales contain these degrees as their degree subsets. In , there are five such minimal cadential sets, i.e., . So finally, a modulation from to in is a quatruple where is a modulation symmetry, and is one of the five minimal cadential sets for the target tonality.

The last point of this model is the calculation of the pivotal degrees. This is achieved by what we call a »modulation quantum«. This is a subset such that

1. is an inner symmetry of the quantum;
2. the quantum contains all degrees of the cadence ;
3. is rigid, i.e., has no translation or inversion symmetry as inner symmetry and is covered by degrees of ;
4. is minimal with properties 1. and 2.

So a modulation quantum >materializes< the modulation symmetry (much like quanta in physics materialize forces), contains enough elements to express a cadence for the target tonality, has its trace covered by target tonality degrees and determines uniquely its associated symmetry (this follows from rigidity) and is a minimal such candidate (economical condition). If such a quantum exists, we shall (by definition!) recover the pivotal degrees from the triadic covering of the trace by degrees of .