Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 009, 19 pages      arXiv:2402.00507      https://doi.org/10.3842/SIGMA.2024.009
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Taking Music Seriously: on the Dynamics of 'Mathemusical' Research with a Focus on Hexachordal Theorems

Moreno Andreatta a, Corentin Guichaoua b and Nicolas Juillet c
a) IRMA-CNRS-CREAA-University of Strasbourg and IRCAM, Paris, France
b) Independent Researcher, SMIR Project, France
c) IRIMAS, Université de Haute-Alsace, France

Received July 01, 2023, in final form January 11, 2024; Published online January 25, 2024

Abstract
After presenting the general framework of `mathemusical' dynamics, we focus on one music-theoretical problem concerning a special case of homometry theory applied to music composition, namely Milton Babbitt's hexachordal theorem. We briefly discuss some historical aspects of homometric structures and their ramifications in crystallography, spectral analysis and music composition via the construction of rhythmic canons tiling the integer line. We then present the probabilistic generalization of Babbitt's result we recently introduced in a paper entitled ''New hexachordal theorems in metric spaces with probability measure'' and illustrate the new approach with original constructions and examples.

Key words: mathemusical research; homometric sets; distance-sets; metric measure spaces; ball volume; scalar curvature; Patterson function.

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