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.
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BibTeX
@misc{lascabettes2026construction,
title = {{Construction, Enumeration and Duality of Perfectly Balanced Musical Sets of Order $2pq$}},
author = {Paul Lascabettes and Sidney Ali and Alexandre Popoff and Moreno Andreatta},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05441},
eprint = {2611.05441},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05441},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Construction, Enumeration and Duality of Perfectly Balanced Musical Sets of Order 2pq
AU - Paul Lascabettes
AU - Sidney Ali
AU - Alexandre Popoff
AU - Moreno Andreatta
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05441
UR - https://arxiv.org/abs/2611.05441
AB - 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.
ER -