A generalized system for musical modulation

The thoughts and compositions on this page are inspired from Schönberg's pattern of the modulation process [3]. Guerino Mazzola's mathematical model of musical modulation [1] [4] formalizes this pattern and yields an actual formula to calculate the chords that are to be used in modulations - with a surprisingly high coincidence with Schönberg's theoretical writings. But a really exciting property of this model - in fact already of Schönberg's model - is the fact that it is not limited to classical music. In [2], Daniel Muzzulini presents a generalization of Mazzola's model to arbitrary 7-element pitch class sets in 12-tone equal temperament and tries to deduce a particular position of the most common scales used in classical music - with arguments, however, that are not always convincing to me.

One possible way to find hints to answers of questions raised in Muzzulini's report such as: what exactly distinguishes the diatonic major scale from any arbitrary one, or: what exactly is the musical relevance of Mazzola's mathematical structures, is to apply the theory. So I started defining harmonic structures differing more or less from classical theory and investigating them - partly with mathematical methods, partly in form of musical experiments. Some results are listed in the table below.

Ma	Mu (Terms of use)
On-To-Sû, So-Na-Ta A composition (sounding a little old-fashioned) based on a structure that differs only slightly from classical harmony. With this piece as an example, the underlying theory is explained in detail.	Midi File Score (PDF)
Scale 57 A slightly more exotic harmonic structure with only one cadence-set.	57_1.mid Not finished yet (latest version: 2001-03-20) and performed badly - but you can already hear the modulations.
Whole-tone scale Yet another harmonic structure, quite bizarre this time: based on the whole-tone scale, but with similar properties as harmonic minor.
Whole-tone scale in 19-tone equal temperament A first example for the natural application odf the theory in alternative tunings.	Gon Dance (MP3)

Literature:

1. Mazzola, G.: Geometrie der Töne, Elemente der mathematischen Musiktheorie, Birkhäuser, Basel 1990.
2. Muzzulini, D.: Musical Modulation by Symmetries, Research Report Mo. 90-02, Juli 1990, SAM ETH. Also appeared in: Journal of Music Theory, Volume 39
3. Schönberg, A: Harmonielehre (1911). Universal Edition, Wien 1966.
4. Mazzola, G.:The Topos of Music, Geometric Logic of Concepts, Theory, and Performance, Birkhäuser, Basel 2002.