which has, as a practical consequence, the »mystical aura of pure form« (Roeder, 1993) of some mathematical theorems in contrast to the »mundanity« of their application to music. Criticism could be levelled against the potential competence of a mathematician expressing »in a very general way relations that only have musical meaning when highly constrained« (Roeder, 1993, p. 307). This essay is an attempt to discuss some general abstract group-theoretical properties of a compositional process based on a double preliminary assumption: the algebraic formalization of the equal-tempered division of the octave and the isomorphism between pitch space and musical time.

Historically there have been different approaches from Zalewski’s »Theory of Structures« (Zalewski, 1972) and Vieru’s »Modal Theory« (Vieru, 1980), to the American Set-Theory (Forte, 1973; Rahn, 1980; Morris, 1987), whose special case is the so-called diatonic theory, an algebraic-oriented ramification of Set-Theory which is usually associated with the so-called »Buffalo School« at New York (cf. Clough and Myerson, 1986; Clough, 1994). See Agmon (1996) for a recent summary in the theory of diatonicism.¹

The common starting point is that every tempered division of the octave in a given number of equal parts is completely described by the algebraic structure of the cyclic group   of order which is usually represented by the so-called »musical clock«. Three theorists/composers are responsable for this crucial achievement: Iannis Xenakis, Milton Babbitt and Anatol Vieru. They form what we could call a »Trinity« of composers for they all share the interest towards the concept of group in music.²

In Babbitt’s words, »the totality of twelve transposed sets associated with a given [twelve-tone set] constitutes a permutational group of order 12« (Babbitt, 1960, p. 249). In other words, the Twelve-Tone pitch-class system is a mathematical structure i.e. a collection of »elements, relations between them and operations upon them« (Babbitt, 1946, p. viii). Iannis Xenakis is sometimes more emphatic, as in the following sentence: »Today, we can state that after the Twenty-five centuries of musical evolution, we have reached the universal formulation for what concerns pitch perception: the set of melodic intervals has a group structure with respect to the law of addition« (Xenakis, 1965, p. 69-70).

But unlike Babbitt’s and Vieru’s theoretical preference for the division of the octave in 12 parts, Xenakis’ approach to the formalisation of musical scales uses a different philosophy. He considers the keyboard as a line with a referential zero-point which is represented by a given musical pitch and a unit step which is, in