construction. In the composer’s words: »Let us assume that we have such a tree in the pitch versus time domain. We can rotate (transform) it; the rotation can be treated as groups. We can use traditional transformations of the melodic pattern: we can take the inverse of the basic melody, its retrograde and its retrograde inverse. There are of course many more possible transformations because we can rotate the object at any angle« (Varga, 1996, p. 89). And, more recently: »This is the Klein group. But we can imagine different kinds of transformations, as a continuous or non continuous rotation of any angle. This gives new phenomena, new evenements, even by starting with a melody, for a simple melody becomes a polyphony« (Delalande, 1997, p. 93).

In section 2 we will show in detail how an old problem in number theory has recently taken shape in an algebraic theory of musical canons. The theory has been developed by the Roumanian mathematician Dan Tudor Vuza independently from the solution of Minkowski’s conjecture by the Hungarian mathematician G. Hajós. As far as I know my own work on Hajós Groups, Canons and Compositions (Andreatta, 1996) was the first attempt to dicuss Vuza’s theory from the perspective of the Minkowski/Hajós problem. For although Hajós Groups had been previously referred in connection with music, the context were completely different