20 November 2025 — Padova

Periodic Sequences (from a joint paper with Luisa Fiorot and Riccardo Gilblas)

Alberto Tonolo, University of Padova

Abstract

The Romanian composer Anatol Vieru in his ``Book of Modes’’ explores a composition technique based on periodic sequences with values in \(\mathbb{Z}/12\mathbb{Z}\). Manipulating the sequence \(V:= (2,1,2,4,8,1,8,4)\) Messiaen’s second mode of limited transposition, he noticed some unexpected phenomena and asked the mathematical music community three questions. These three questions stimulated the interest of some mathematicians obtaining some partial results. In the absence of a definitive answer to the questions raised by Vieru, Luisa Fiorot ,Riccardo Gilblas, and I decided to tackle the problem. Through new recurrence relations for certain binomial coefficients modulo a power of a prime, we obtained new results in the evolution of the iterated anti-differences of periodic sequences modulo an integer. These new results have enabled us to obtain a complete solution to the questions posed by Vieru. Thanks to music applications, I will try not only to present our results, but also to let you hear some of them.