https://doi.org/10.1140/epjs/s11734-026-02505-6
Regular Article
Quantum interval ratios and applications to tunings
CEA, Paris-Saclay, 91191, Paris, France
a
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Received:
13
January
2026
Accepted:
25
June
2026
Published online:
5
July
2026
Abstract
Quantum mathematics has studied the deformations of algebraic or combinatorial objects and has developed a quantum calculus (or q-calculus) that addresses these deformations controlled by a real parameter q. Recently, Morier-Genoud and Ovsienko [1] defined q-deformed rational numbers, which they called q-numbers. We use them to introduce quantum analogs of musical interval ratios and facilitate tuning deformations. We construct these new interval ratios from the quantum Stern–Brocot tree and study their matrix properties. These quantum ratios locally deform temperaments and tunings continuously and allow the introduction of micro-deformations into common tunings. They open the way to new compositional transformations.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
