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Set-theoretic solution for the tuning problem
arXiv:2506.13969v2 Announce Type: replace
Abstract: In this paper I want to suggest a new solution to the problem of musical tuning. On one hand, I see it as a generalization of Just Intonation (JI) to inharmonic timbers, on another, as a unification of spectral interference and harmonicity contributions to consonance within a single framework. The main achievement of the work is the ability to mathematically quantify the phenomenon of musical consonance using set theory. That quantification is done by defining two measures of consonance: affinity and harmonicity. These measures naturally generate sets of intervals that can be used as dynamic tuning systems. The paper is aimed at a broad audience of people who may not be skilled in music and tuning theory or mathematics. Thus, I attempt to give as much details and explanations as I can, while keeping the number of pages as low as possible.
Abstract: In this paper I want to suggest a new solution to the problem of musical tuning. On one hand, I see it as a generalization of Just Intonation (JI) to inharmonic timbers, on another, as a unification of spectral interference and harmonicity contributions to consonance within a single framework. The main achievement of the work is the ability to mathematically quantify the phenomenon of musical consonance using set theory. That quantification is done by defining two measures of consonance: affinity and harmonicity. These measures naturally generate sets of intervals that can be used as dynamic tuning systems. The paper is aimed at a broad audience of people who may not be skilled in music and tuning theory or mathematics. Thus, I attempt to give as much details and explanations as I can, while keeping the number of pages as low as possible.
Anonymous
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