Greek Music Theory and Archytas’s Theorem

Athenaeus, Deipnosophistae 3.60 quotes the comic writer Damoxenus, Syntrophoi [14, p. 215]: What are akin By octaves, fifths, or fourths I weave all in At the proper ‘intervals’ and suitably To their ‘resolution.’ octave: διὰ πασῶν, “through all”, “through the whole”, e.g. 2 Corinthians 8:18. fifth: διὰ πεντε, fourth: διὰ τεττάρον. intervals: διαστήματα. A διάστημα is an interval, extent, extension, distance, line segment, space between things, gap; see LXX Ezek. 41:8, Acts 5:7, Asclepiodotus, Tactics iv.1. Plato, Timaeus 35b–36b [4, pp. 59–60], about the Demiurge cutting off pieces from a long strip: This is how he began to divide. First he took away one part from the whole, then another, double the size of the first, then a third, hemiolic with respect to the second and triple the first, then a fourth, double the second, then a fifth, three times the third, then a sixth, eight times the first, then a seventh, twenty-seven times the first. Next he filled out the double and triple intervals, once again cutting off parts from the material and placing them in the intervening gaps, so that in each interval there were two means, the one exceeding [one extreme] and exceeded [by the other extreme] by the same part of the extremes themselves, the other exceeding [one extreme] and exceeded [by the other] by an equal number. From these links within the previous intervals there arose hemiolic, epitritic and epogdoic intervals; and he filled up all the epitritics with the epogdoic kind of interval, leaving a part of each of them, where the interval of the remaining part had as its boundaries, number to number, 256 to 243. And in this way he had now used up all the mixture from which he cut these portions. intervals: διαστήματα. hemiolic: ἡμιολίαν, “half as much again”, epitritic: ἐπιτρίτον, “a third as much again”, epogdoic: ἐπόγδοος, “an eighth as much again”. The words ἐπιτρίτον and ἐπόγδοος are used in many extant writings to talk about lending money, e.g. Demonsthenes, Against Polycles 50.17. Aristotle, Rheotric 3.10.7, 1411a [27, p. 220]: And Moerocles said he was no more wicked than–(naming someone of the upper class); for that person was wicked “at thirty-three and a third percent interest” he himself “at ten.”