Enumeration of non-isomorphic canons

Abstract

In this note we describe how to apply group actions in order to enumerate the isomorphism classes of rhythmic canons. The notion of a rhythmic canon was originally formalized by O. Messiaen. He proposed to study canonical musical forms in which the imitation between the different voices con-cerns only rhythmical aspects, independent from melody or harmony. The first math-ematical investigation of rhythmic canons goes back to Vuza’s papers [7, 9, 8, 10] in which he actually introduced even much more complicated families of canons, the “Reg-ular Complementary Canons of Maximal Category ” (in short RCMC-canons). Their definition will be given below. Before describing rhythmic canons in cyclic time, we view them as they may appear within a musical composition, in free linear time which has no cyclicity. Like in a melodic canon, one has several voices that may enter one after the other until all voices are present. As in the case of a melodic canon, all voices are just copies of a ground voice that is suitably translated in the time axis. For simplicity we suppose here that all

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