Introduction

In neo-Riemannian theory, an essential construct is the cycle (or circle) of triads, alternately major and minor. Figure 2.1 shows four such cycles, called “hexatonic systems” by Cohn, which partition the set class (sc) of consonant triads into four subclasses of six triads each. The four systems are transpositions of each other, and they embody relations traceable to Riemann: Parallel (P) (equivalent to Riemann's Quintwechsel ) and Leittonwechsel (L) apply alternately as we make our way around any of the circles. The meanings of these terms, based on double common-tone retention, are evident from the context.

Cohn also describes three octatonic systems—cycles of eight triads each, which also partition the sc of major/minor triads. These are shown in figure 2.2. Like their hexatonic cousins, the octatonic cycles, again all transpositions of one another, arise from alternation of two Wechsel, in this case P and Relative (R) (equivalent to Riemann's Terzwechsel ). The remaining pair from P, L, and R (L and R), when applied alternately, generates the circle of figure 2.3—a circle running through the full set class of 24 consonant triads, put forth by Werckmeister in 1698, and placed in historical context in the work of Joel Lester. P, L, and R show parsimonious voice leading: they are the only transformations that change one consonant triad into another by replacing a single pitch class. They are also exchange operations: they transform a major or minor triad to a triad of the opposite mode. There are 12 Wechsel in the Riemannian group, including P, L, and R.

It seems natural to ask: what is the space of all circles formed by alternating a pair of Wechsel ? It is this question that first motivated the present investigation. In the next few sections of the paper, I will enumerate and characterize the circles that comprise the space and show how they relate to subgroups of the Schritt/ Wechsel group. In the final section of the paper I will expand the field of inquiry to other kinds of circles, including diatonic sequences, and study other groups that support the circles in question. The objective is to extend neo-Riemannian theory to address a broad range of circular musical objects.