用集合论量化音乐协和性，实现动态调音系统。
Set-theoretic solution for the tuning problem
- 用集合论定义协和度与亲和度两个指标
- 可生成适应不同音色的动态调音方案
- 适合对音乐理论感兴趣的非专业读者
本文提出一种解决音乐调音问题的新方法。一方面，将纯律（Just Intonation, JI）推广至非谐波音色；另一方面，在统一框架中整合谐波成分与频谱干涉对协和性的贡献。核心成果是首次通过集合论数学化描述音乐协和性，定义了协和度（harmonicity）与亲和度（affinity）两个度量，自然生成可用于动态调音系统的音程集合。论文面向非专业背景读者，力求在有限篇幅内提供充分解释与细节。
原文摘要 · Abstract (English)
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.
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