Proto-tonal theory: Tapping into 9th-century insights

The 9th-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.