to derive and explain music-theoretical concepts in terms of overtone relations. On the other hand, it is evident that such a lack of success must not be blindly projected to other attempts as well, which -- in pure mathematics -- would be like confusing a wrong proof with a proper counter-example. Pragmatically speaking, it is of course not necessary to wait for irrefragable counter-arguments in order to refrain from spending energy into scientific adventures. However, within our common sense argumentation we have to insist on this little side-remark. Who is able to distinguish between scientific adventures and true pioneering work? The latter is an indispensable part of science.

What are the subjects of mathematical investigations into music theory? As mentioned above, musical conventions became an important focus in 20th century musicology. The very fact that musical conventions develop and change as a part of cultural dynamics does -- of course -- not imply that these conventions can be considered as arbitrary snapshots of cultural processes. But it may indeed be necessary to include other disciplines apart from musicology and mathematics in order to reach a deeper understanding of the dynamics of the musical mentality of a culture. There is, however, a peculiarity in musical signs as compared to other sign systems of a culture. The absence of stable extra-musical signification in musical signs makes it reasonable to look for purely inner-musical motivations of musical conventions. Therefore, from a pragmatic point of view, a broader transdisciplinary project may benefit from the results of a narrowed mathemusical research.

The investigation of pure music-theoretical problems by purely mathematical means raises methodological questions. Several approaches share a common strategy which I tend to call >contrafactual experiments<. If a music-theoretical discourse subject enjoys a particular prominence through the interest of many musicians and music theorists (like the diatonic tone system, the consonance/dissonance dichotomy or the major and minor triads) one may ask whether there are also mathematical peculiarities to these structures which may motivate their choice among other (contrafactual) structures within a suitably chosen formal context. The contrafactual experiment is thus given through the definition of a formal context of parametrized structures such that the music-theoretically prominent structure can be seen as one parameter setting among the others. If the researcher is able to find reasonable predicates in such a way that the prominent structure becomes also mathematically prominent, he or she may try to derive explanatory power from these predicates. In the following section we consider three examples for investigations of this kind.