Some reformulations and extensions of the theory of rhythmic canons

The algebraic theory of periodic rhythmic canons was developed by the author of the present paper in connection with the study of a special class of rhythmic canons, nowadays referred as “Vuza canons”. A basic concept in the theory is the outer rhythm attached to a canon. In the original paper, for any given canon an auxiliary canon was constructed and the outer rhythm was expressed as an object attached to the latter canon. In the present paper we show that the outer rhythm can be expressed directly in terms of the given canon, without the need of an auxiliary construction. Strongly related to the outer rhythm is the canon category, which is a numerical measure of the periodic symmetry of the outer rhythm relative to the canon inner rhythm. We give a new definition of the category in terms of the stability groups associated to a canon. Many interesting results due to various authors have been obtained for Vuza canons, which by their definition must have maximal category. We show here that interesting facts can also be said about canons whose category is not maximal. We describe partitions of a canon into subcanons of minimal and maximal category and we discuss the relation between outer rhythm, category and a class of maps that can be regarded as natural morphisms between canons.