# Preprint: group theory and chords

A while ago I wrote a paper called A Group-theoretical Classification of Three-tone and Four-tone Harmonic Chords (arXiv link). The main contribution is an analysis of four-tone harmonic chords in music theory. These are: major-major, minor-major, augmented-major, major-minor, diminished-minor, minor-minor, and diminished-diminished. X-Y in this scheme means X is the basic triad (so major, minor, augmented, or diminished), and Y is the quality of the seventh factor. These four-tone chords can be represented as a quadruple $(0,a,b,c)$ where the numbers $a,b,c$ are the distance from the root $0$ in semitones. I define three fundamental operations on these chords: inversion (usual inversion on a keyboard instrument) denoted $i$, major-minor duality denoted $d$, and augmented-diminished duality denoted $a$. These three operators are elements of the symmetric group on four letters.

I show how all the harmonic chords are related under these operators, and each of these relations reveals the harmonic relations between these chords as we hear them in music. The highlight of the paper is this diagram, which shows all these relations (M=major, m=minor, A=augmented, d=diminished):

Unfortunately, I recently found out that it was rejected :( It was suggested that I could resubmit on the condition that I significantly expand its relation to the rest of the literature and make the notation more consistent with some other parts of the literature. I am going to have to figure out what to do about that because I think too much notational change might actually make this paper more confusing and hard to read. So, I am sharing it with you so you can see this cool diagram. I am not sure if I can really satisfy the requirements of the field of mathematics of music so I might have to leave this one as a preprint, especially since I would like to keep working on ecology.