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 and serial counterpoint. In the case of circular canons, we introduce the superimposition of infinite words, and 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, the superimposition of languages, and the iterated superimposition of a language, which is applied to the study of basic aspects of serial music. This leads to the study of closure properties of rational languages of finite words under superimposition and iterated superimposition. The rationality of the transduction associated with the superimposition appears to be a powerful argument in the proof of these properties. Since the superimposition of finite words is the max operation of a sup-semilattice, the last section addresses the link between the rationality of a sup-semilattice operation and the rationality of the order relation associated with it.

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