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.