On some theoretical and computational aspects of Anatol Vieru's periodic sequences
This article develops some aspects of Anatol Vieru's compositional technique based on finite difference calculus of periodic sequences taking values in a cyclic group. After recalling some group-theoretical properties, we focus on the decomposition algorithm enabling to represent any periodic sequence taking values in a cyclic group as a sum of a reducible and a reproducible sequence. The implementation of this decomposition theorem in a computer aided music composition language, as OpenMusic [ 1], enables to easily verify if a given periodic sequence is reducible or reproducible. In this special case, one of the two terms will be identically zero. This means that every periodic sequence has in itself a certain degree of reducibility and reproducibility. We also suggest how to use this result in order to explain some regularities of the distribution of numerical values in the case of the finite addition process and how to generalize the decomposition theorem by means of the Fitting Lemma. This opens the problem of the musical relevance of a generalized module-theoretical approach in Vieru's theory of periodic sequences.
periodic sequences decomposition algebraic formalization finite difference calculus on groups