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：我们从组合学视角系统发展了音乐理论中与和声相关的若干方面，包括音阶、和弦构成及即兴创作。其目标是通过少量假设推导出基本结构，而非罗列需要机械记忆的大量和弦/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来限制可考量的音阶范围。例如，可设定两个声部不得相距半音这一约束，因为该音程过于不协和。由此可研究不含半音间隔音符的音阶。更精细的约束通过避免三个声部碰撞，研究不含仅相隔半音的三音组合的音阶。此外我们要求音阶具有完备性，大致指其是满足这些约束条件的最大音集。研究发现，将完备性应用于这些简单的二声部/三声部约束时，恰好能刻画音乐创作中常用的音阶类型。令人惊讶的是，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。我们将此对应形式化为对偶性陈述，从而提供了一种通过一类约束音阶理解另一类约束音阶的途径。最后，我们融合这些约束思想对和弦进行分类。