Synchronization of musical words

We study the synchronization of musical sequences by means of an operation defined on finite or infinite words called superimposition. This operation can formalize basic musical structures such as melodic canons, for which we provide a construction based on the superimposition of infinite words. In the case of circular canons, we present an enumeration algorithm involving Lyndon words, which appear to be a useful tool for enumerating periodic musical structures. We also define the superimposition of finite words, which is extended to the superimposition of languages and the iterated superimposition of a language. The latter is applied to the study of basic aspects of serial music. We conclude this paper with closure properties of rational languages of finite words under superimposition and iterated superimposition. The fact that the transduction associated with the superimposition is rational appears to be a powerful argument in the proof of these properties. Since the superimposition of finite words is the max operation of a semi-lattice, the last section address the link between the rationality of a semi-lattice operation and the rationality of the order relation associated with it.