A body of recent work by Richard Cohn, Julian Hook, Brian Hyer, Henry Klumpenhouwer, David Lewin, and others has employed groups of transformations traceable to Hugo Riemann's Schritte (dualistic transpositions) and Wechsels (contextual inversions). Much of this work involves circles of major and minor triads such as Cohn's hexatonic systems. The paper asks: what is the space of all such circles--that can be formed by alternating major an minor triads (or other inversionally related types) in a consistent manner? It then proceeds to enumerate and characterize the 168 circles, called uniform flip-flop circles (UFFCs), that comprise this space, and to show how they relate to certain subgroups of the Schritte-Wechsel (S/W) group defined by Klumpenhouwer.
Each UFFC may be viewed as an overlay of two constituent circles, one generated by transposing major triads by a constant interval and the other by transposing minor triads by the same interval. The enumeration proceeds by counting the possible constituent circles and their approriate pairings, and accounting for rotations. Alternatively, but with the same result, each UFFC may be seen as arising from a pattern of two distinct Wechsels; however, a pattern may give rise to more than one UFFC. Navigation among the triads of any UFFC is accomplished by means of the appropriate simply transitive subgroup of the S/W group.
The paper connects with Hook's work proposing a group of 288 transformations, of which Hyers group of order 144 is a subgroup, and touches on extensions to the arbitrary asymmetrical trichord, to dichotomies other than inversionally related chords, and to universes of other than 12 pcs.