Mathematical approaches to defining the semitone in antiquity

Abstract

Connections between mathematics and music have been recognized since the days of Ancient Greece. The Pythagoreans' association of musical intervals with integer ratios is so well known that it occludes the great variety of approaches to the music-mathematical relationship in Ancient Greece and Rome. The present article uncovers this diversity by examining how authors from Antiquity used one mathematical element – the number series 16, 17, 18 – to serve different ends. The number 17 provides the simplest way to divide the 9:8 whole tone into two parts, but the status of those two parts was debated, as was their relationship to the standard 256:243 semitone of Greek theory. I account for this diversity by appealing to the contexts in which the authors wrote – music treatises vs. commentaries on Plato's Timaeus – and to the importance placed on mathematics in Neoplatonist curricula. The article concludes by examining how confusion regarding the 16, 17, 18 series lingered even in the medieval period due to ambiguities in Boethius's De institutione musica, and how medieval authors eventually superseded the debate through a Euclid-inspired geometric division of the 9:8 ratio.