Autosimilar melodies and their implementation in OpenMusic

Autosimilar melodies put together the interplay of several melodies within a monody, the augmentations of a given melody, and the notion of autosimilarity which is well known in fractal objects. From a mathematical study of their properties, arising from experimentations by their inventor, composer Tom Johnson, we have generalized the notion towards the dual aspects of the group of symmetries of a periodic melody, and the creation of a melody featuring a set of given symmetries. This is now a straightforward tool, both for composers and analysts, in OpenMusic visual programming language. I. INTRODUCTION A. History Autosimilar melodies have been conceptualized and intensively used by composer Tom Johnson and sev- eral american fellows from the 1980's. They were first rigorously defined in the last chapter of his book (5) under the label 'selfRep melodies' (the word 'autoSim- ilar' being used in a much fuzzier sense in the whole book). We will restrict the meaning of 'autosimilar' to the notion developed thereafter, because it is closer to the common mathematical usage. Also, while keeping close to this traditional meaning, it will be generalized much further than Johnson used it, to musical objects invariant under the action of a given subgroup of the affine automorphisms of some cyclic ring Zn. This lends itself particularly well to implementation in OpenMusic. It must be stressed that examples of autoSimilar melodies crop up in many different musical styles, from Mozart's classical music to New Orleans Jazz. Also the degree of control that our mathematical work entails into the software is a contrapuntist's dream, enabling subtle interplay between a melody and itself at several different tempos. B. Notations