Periodic Sequences and Persistent Homology: theoretical foundations and new result

We face math-music research problems with algebraic tools. In the first part, we study modular periodic sequences and their properties. We present the state of the art of Vieru’s sequences and, through the link with modular binomial coefficients, we prove new results on the period and the coefficients of the sequences when the finite sum operator is applied. In the second part, we use persistent homology to study harmonic complexity in music. The musical pieces are modeled as sequence of chords, which are used to construct filtration of simplicial complex, whose homological information is used to reconstruct the musical properties of the piece.