-say--between 1730 and 1830) just because of a tuning practice in non-equal-temperament. A simple counter-argument we could consider homogeneity of a tone system as a necessary precondition for the fine psychological effects of key characteristics. If we open the discussion towards other musical genres there are other controversial arguments as well: On the one hand the number of tones per octave is apparently not limited to twelve, but on the other hand enharmonic identification always uses the same possibilities as in keyboard music: Why there are no identifications in vocal music scores? Our short propedeutical remarks lead to the conclusion, that it is useful to simultaneously develop and to investigate alternative music theoretical approaches and to compare them in analytical experiments. The present paper intends to introduce a concrete small portion of mathematical music theory which is motivated by immediate consequences of the idea of homogeneity.

1.2 Chords as Tone-Sets

A simple way to describe chords is to model them as sets of tones. But this implies the serious danger the violate basic assumptions of certain harmonic theories. Hugo Riemann, for example, considers major and minor triads as prime chords being the elementary building stones of his harmonic system. To him tones are always understood as occupying consonant or dissonant roles with respect to tonally interpreted prime chords of reference. To use sets (i.e. unordered collections of their elements) as models for these primary objects of harmonic theory is therefore arguable. In voice-leading studies, however, it is natural to consider chords as secondary coincidences of voices. The deep theoretical problems in the understanding of the relation between hamony and voice-leading is reflected in the celebrated »either/or«-debate concerning the primacy of counterpoint and voice-leading over harmony (or vice versa) on various levels (theory, a given piece, a given situation in a piece). Again we prefer to dismiss a-priori arguments in favor (or against) set models. Instead it is interesting to explore the explanatory power of a concrete model and to eventually compare this with the explanatory power of alternative ones.

The present study offers a partial contribution to this broad problem. But the mathematical sharpening of some concepts may eventually help to bring new aspects into this debate. Our study aims at experimentally transferring selected aspects from Riemannian functional harmony to a model where chords are modeled sets of tones. The morphological approach partially compensates this violation of Riemann’s assumptions by embedding the problematic notion of elementship into a transformational framework. But besides, there are fundamental harmonic aspects which are not properly grasped by the morphological approach, such as the domain of root progressions.

As mentioned, we restrict to the twelve tone system, which is mathematically modeled by of the module of residue classes of integers . Within this setup we simply use the term tone in order to denote an element, or--geometrically as well as semiologically speaking--a position, of this homogeneous twelve tone