Group Theory in Music: 8. Permutation group of order 4

This video examines the symmetrical properties of the group containing the first 4 integers under addition, which will be applied in the next video to a specific chord.

[dyllanusher]

I was just having a nice conversation with a friend about this. I watched the series and it was great!

I have so many avenues I want to further explore:
- Alternate tuning systems?
- What happens if we think of Chords as elements of a group?
- Chord movements as elements of a Group where the elements are "functions" mapping the previous groups?
-- Do the chord movements that "sound good" follow a particular symmetry in this group?

Also, if music = harmony + rhythm, what else might we discover applying rhythm to groups?

Also, there's an intimate connection between rhythm and harmony, for instance, a 2/3 polyrhythm sped up becomes a perfect fifth (at least if we're in just intonation)
So I wonder if there's a group theoretic connection to be made somewhere there between some group of rhythms and some group of harmony?

Oh man so much to explore so little time. Posting in case someone has inspiration to follow down this path.