Figure 3. Douthett and Steinbach’s “Cube Dance” as it appears in their original article (a) and as it is embedded in the center of three-note chord space (b).

(a) (b)

In the original representation, clockwise steps—e.g. from C to c or C to e, or e to B aug—represent descending semitonal motion, while counterclockwise steps represent ascending motion; in Figure 3(b), ascending and descending musical motion are represented by ascending and descending graphical motion. (Note that the top of this graph is glued to the bottom with a 120° twist.) The embedding can be used to show that every consistently clockwise or counterclockwise path on Douthett and Steinbach’s figure represents a particular voice leading, with the length of the path corresponding to the size of the voice leading. Furthermore, since the graph contains every “interscalar transposition” (Tymoczko 2008) between all of its chords, it necessarily contains a minimal voice leading between any two of them.