Construction, Enumeration and Duality of Perfectly Balanced Musical Sets of Order $2pq$

Abstract

In this paper, we present a comprehensive study of perfectly balanced musical sets, also known in the literature as vanishing sums of roots of unity, in the particular case of order $2pq$. Starting from the well-established result concerning the decomposition of such sets as an integer linear combination of regular polygons, we first provide an algebraic formalization that allows to explicitly construct all these sets. We then propose six different decompositions of each set into regular polygons. Moreover, by investigating their combinatorial enumeration, we derive a formula for the exact number of these sets for a given order $2pq$, and describe their distribution and cardinality for the particular case of order $6p$. One of the main results concerns the duality that can be established between these sets. In particular, we prove that every non-periodic minimal perfectly balanced set of order $2pq$ can be combined with another one to produce a self-complementary $2p$-periodic set, and the reciprocal also holds. Finally, we demonstrate that the modulus of the discrete Fourier transform coefficients of dual sets are related. All these results shed new light on the distribution of perfectly balanced sets of order $2pq$ and suggest new mathematical questions in a more general case.