If you arrange the musical notes C, C#, D, D#, E, F, F#, G, G#, A, A# around a circle (in that order), like this:
The resulting structure has some interesting properties. Firstly, each interval becomes a directed chord (in the geometric sense) on this circle, and transposition is just rotation.
The trouble is that because of octaves, this picture doesn't tell the whole story. If you go all the way around the circle you'll find yourself an octave higher (or lower) than when you started. Not all the intervals will sound alike as some of them will include an extra octave factor.
Unless you use Shepard tones.
Shepard tones have frequency components of the same pitch class in all octaves (tapering off at low and high frequencies according to a bell curve in log frequency space), like this:
They sound kind of like notes on a church organ.
The Shepard tones sound like musical notes but because the peak of the bell curve is always the same frequency, once you have progressed through a full scale you're back where you started instead of being up an octave. This can be used to create an auditory illusion of a pitch which sounds like it is continually getting higher (or lower), but is in fact looping.
It would be interesting to compose some music using Shepard tones. Another interesting feature of them is that all chords are identical to their own inversions. So in a number of ways Shepard tones drastically simplify music theory.
Another fun thing to do with Shepard tones is to leave the pitch class alone and vary the frequency that is at the peak of the bell curve. That way you can make a sound which continuously increases in frequency but which never seems to change pitch.