Recent music-theoretical research has proposed two ways of mapping the pitch-class set universe. Fourier spaces, constructed by Quinn, relate sets to one another based upon their composition from members of interval cycles, reflecting what might be called “harmonic quality.” Voice-leading spaces, generalized by Callender, Quinn, and Tymoczko, illustrate voice-leading relationships between sets. Though many researchers have noted hints of a relationship between these two types of space, their exact relation remains murky.
One way of relating these two spaces involves associating voice leading with motion through Fourier space. Voice-leading displacements of each of the six interval classes can be associated with specific changes of position in each of the six Fourier spaces for twelve-tone equal temperament. Likewise, displacement spaces, showing all of the sets that can be related by voice-leading displacements of a particular interval class, model predictable motion through each of the Fourier spaces.