Reinterpreting and Extending Anatol Vieru’s Periodic Sequences Through the Cellular Automata Formalisms

Paul Lanthier (1) , Corentin Guichaoua (2, 3) , Moreno Andreatta (4, 2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 IRMA - Institut de Recherche Mathématique Avancée
3 USIAS - Institut d’Etudes Avancées de l’Université de Strasbourg - Institute for Advanced Study
4 IRCAM - Institut de Recherche et Coordination Acoustique/Musique

In this paper we focus on Anatol Vieru's periodic sequences that we approach with the formalism of the theory of cellular automata. After extending previous results about the action (in the image direction) of one particular cellular automaton on periodic sequences we show the existence of a second one which is its complementary (or dual). The main idea of the paper is that the study of preimages of one of those two automata is the study of the images of the other one and vice versa. By using the duality, we have been able to show explicitly the evolution of the period and the form of the preimages for both automata. In order to illustrate the theoretical constructions, a musical composition is presented using the two automata both in the image and preimage directions.

Domaines: Systèmes dynamiques [math.DS]

Paul Lanthier, Corentin Guichaoua, Moreno Andreatta. Reinterpreting and Extending Anatol Vieru’s Periodic Sequences Through the Cellular Automata Formalisms. Mathematics and Computation in Music, 7th International Conference, MCM 2019, Madrid, Spain, June 18-21, 2019, pp.261-272, 2019, ⟨10.1007/978-3-030-21392-3_21⟩. ⟨hal-02372515⟩