  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.
