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：我们从组合学视角探讨与和声相关的音乐理论，包括音阶、和弦构成与即兴演奏。目标是通过少数假设推导基本结构，而非罗列需死记硬背的和弦/音阶列表，从而为该学科奠定基础。我们的方法通过引入约束条件限定可考虑的音阶范围。例如，可设定约束：两个声部不能仅相隔半音，因其过于不协和。进而研究不含半音相邻音符的音阶。更精细的约束通过研究无三音仅以半音分隔的音阶来避免三声部碰撞。此外，我们要求音阶具有完备性——大致指其是满足这些约束的最大音高集合。事实证明，应用于简单二声部/三声部约束的完备性特征，恰好刻画了音乐创作中常用的音阶类型。令人意外的是，受二声部约束的音阶与受三声部约束的音阶之间存在对应关系。我们将该对应关系表述为对偶命题，提供了一种通过一种约束条件下的音阶理解另一种约束条件下的音阶的途径。最终，我们将这些约束思想结合，实现对和弦的分类。