Proto-tonal Theory: Tapping into Ninth-Century Insights

The ninth-century treatise Scolica enchiriadis (SE) offers two notions of “interval,” namely ratio (proportion) and step distance. The latter notion entails a “generic” distance (cf. “fifth”); however, suggestive diagrams clarify that a “specific” distance is assumed as well (cf. “perfect fifth”). SE raises the question, how to pair step distances such as perfect octave (diapason), perfect fifth (diapente), and perfect fourth (diatessaron), with ratios such as 2:1, 3:2, and 4:3, respectively. In answer, SE departs from the Boethian tradition whereby the distinction between say, duple (2:1) and diapason, is merely terminological. Moreover, SE points out that multiplication of ratios corresponds to addition of step distances, in a manner to which a modern-day mathematician would apply the term homomorphism. Even though the “daseian” tone system proposed in SE (and the “sister” treatise Musica enchiriadis) was discarded already in the middle ages, the SE insights into “proto-tonal” theory, the background system of tones prior to the selection of a central tone or “final,” are still relevant.