the so called »Leitton-Wechselklang«, a-minor-triad whose root is a major third above the root of the first triad. The diatonic scales are also connected through such a neighbourhood relation. A diatonic collection differs from its fifth transposition just by a single semitone shift.

In the formal context of atonal set theory one could formulate this property as a solidarity between transformation and voice leading parsimony. Richard Cohn performed the following contrafactual experiment (c.f. Cohn, 1996):

If in a set of tones (i.e. »pitch classes«) a single element is shifted by a semitone up or down then -- in general -- the result will neither be a transposition nor an inversion of the original tone set. In general, the result will be a set from another tone set class. Only in special cases does a tone set have two different semitone-neighbours which at the same time are transpositions or inversions of . One can easily verify the following fact: If a tone set class has the solidarity property, then corresponding class their complements has this property too. In other words, apart from the major and minor triads and the diatonic collections we already encountered there are two other classes with the solidarity property: the pentatonic collections and the 9-tone-sets being complements of major and minor triads. The surprising result of Cohn’s experiment is the following: Among all 208 tone set classes with sets of at least 3 and at most 9 tones there are no other classes except from these four with the solidarity property.

3 Metalanguage and Internal Logics

This short section cannot provide an introduction into the complicated issues of metalanguage in Mathematical Music Theory. Its intention is to highlight a particular aspect and to explain it on the basis of a musical example. However, it is hoped that the reader may arrive at other ideas and directions from it. Mazzola (2002) presents a highly elaborate metalanguage ⁷

, whose very design is the consequence of a mathematically natural answer to several problems. One epistemologically and pragmatically important problem is this:

On the one hand, there is a desire to construct complex objects from simpler ones, and any metalanguage should support such constructions as flexibly and uniformly as possible. On the other hand, transformations of musical spaces did become a central issue for mathematical music theory (c.f. Mazzola, 1985; Lewin, 1987; Mazzola, 1990). The Lewin-school even coined the term Transformational Theory. In other words, there is also a desire to include the effects of transformations on and within these complex objects into the metalanguage.

Consequently, Mazzola introduces Topos Theory, its geometric logics and its techniques into his proposed language of forms and denotators. The objects in our example are still very simple, namely just chords and tones of a study by Alexander Scriabin. But even in this simple example it will become evident that the consideration of transformations leads to a non-classical intuitionistic logics.