The only thing I don't get about the orbifold representation of chords is that chord inversions do actually effect the transition probabilities from chord to chord in most musical styles, because many musical pieces are composed voice by voice, and there is even correlation over time that gives rise to such techniques as suspensions and anticipations. (D-G-Bb) ---> (D-F#-A) are very near when inversion is taken into account, and it is easy to see that the pedal tone on D is justified by way of the progression G minor (2nd inversion) ---> D major (root)