Music Theory Spectrum, 35:1, 103–110

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 homomor phism. 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.