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.
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.
A slightly more exotic harmonic structure with only one cadence-set.
Not finished yet (latest version: 2001-03-20) and performed badly - but you can already hear the modulations.
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)|
- Mazzola, G.: Geometrie der Töne, Elemente der mathematischen Musiktheorie, Birkhäuser, Basel 1990.
- Muzzulini, D.: Musical Modulation by Symmetries, Research Report Mo. 90-02, Juli 1990, SAM ETH. Also appeared in: Journal of Music Theory, Volume 39
- Schönberg, A: Harmonielehre (1911). Universal Edition, Wien 1966.
- Mazzola, G.:The Topos of Music, Geometric Logic of Concepts, Theory, and Performance, Birkhäuser, Basel 2002.