2023/09/18 by Maksim Lipyanskiy, Lipyanskiy, Maksim
Arts and Humanities · Computer Science · Neuroscience · #Audio and Speech Processing (eess.AS) #FOS: Computer and information sciences #FOS: Electrical engineering #Music Technology and Sound Studies #Musicology and Musical Analysis #Neuroscience and Music Perception #Sound (cs.SD) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2309.10719
openalex publication_date 2023/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.