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：我们从组合数学的视角发展音乐理论中与和声相关的若干方面，如音阶、和弦构成及即兴创作。本研究的目的是通过从若干基本假设推导出核心结构，为该领域奠定理论基础，而非仅罗列大量需要机械记忆的和弦/音阶列表。我们的研究方法引入约束条件以限定可考虑的音阶范围。例如，可施加"两个声部不得仅相距半音"的约束，因为这种音程过于不协和。随后可研究不包含半音相邻音符的音阶体系。更精细的约束通过研究不存在三个音符均以半音相邻的音阶，来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。事实证明，针对这些简单双声部/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桥梁。最后，我们整合这些约束思想，提出系统的和弦分类方法。