The Vial/Weber chordal space depicts two different sorts of relationships: shared common tones and efficient voice leading.
The close correspondence between these properties -- shared common tones, efficient voice leading, and the two-dimensional pitch lattices -- is in some sense a lucky accident.
As Richard Cohn (1997) explained, analogous constructions depicting relationships among other types of chords do not have these properties.
Interest in common-tones and voice leading gradually led music theorists to modify Heinichen's original proposal.
In the circular arrangement F - d - C - a ..., the chords F and d share two common tones, and can be linked by efficient voice leading.