We develop aspects of music theory related to harmony, such as scales, chord formation and improvisation from a combinatorial perspective. The goal is to provide a foundation for this subject by deriving the basic structure from a few assumptions, rather than writing down long lists of chords/scales to memorize without an underlying principle. Our approach involves introducing constraints that limit the possible scales we can consider. For example, we may impose the constraint that two voices cannot be only a semitone apart as this is too dissonant. We can then study scales that do not contain notes that are a semitone apart. A more refined constraint avoids three voices colliding by studying scales that do not have three notes separated only by semitones. Additionally, we require that our scales are complete, which roughly means that they are the maximal sets of tones that satisfy these constraints. As it turns out, completeness as applied to these simple two/three voice constraints characterizes the types of scales that are commonly used in music composition. Surprisingly, there is a correspondence between scales subject to the two-voice constraint and those subject to the three-voice constraint. We formulate this correspondence as a duality statement that provides a way to understand scales subject to one type of constraint in terms of scales subject to the other. Finally, we combine these constraint ideas to provide a classification of chords.

翻译：本文从组合数学的视角探讨音乐理论中与和声相关的若干方面，如音阶构成、和弦形成及即兴创作。研究目标并非罗列大量需机械记忆的和弦/音阶列表，而是通过若干基本假设推导出该学科的基础结构，从而建立理论框架。我们的研究方法通过引入约束条件来限定可考虑的音阶范围。例如，可施加约束禁止两个声部仅相差半音，因其不协和度过高；进而可研究不含半音相邻音符的音阶体系。更精细的约束通过考察不存在三个音符均以半音相邻的音阶，以避免三个声部的冲突。此外，我们要求音阶具有完备性，这大致意味着它们是满足这些约束的最大音符集合。研究结果表明，针对简单双声部/三声部约束的完备性条件，恰好刻画了音乐创作中常用的音阶类型。令人惊讶的是，满足双声部约束的音阶与满足三声部约束的音阶之间存在对应关系。我们将这种对应关系表述为对偶定理，从而建立通过一类约束下的音阶理解另一类约束下音阶的理论桥梁。最后，综合这些约束思想，我们提出了和弦的系统分类方法。