Note 62 — § 37. Epilogue—Speech and Song.

In Euclid's Sectio Canonis the Pythagorean division is assumed, and there is no hint of any other ratio than those which Pythagoras discovered. Prop. xvii shows how to find the Enharmonic Lichanos and Paranêtê by means of the Fourth and Fifth. Prop. xviii proves against Aristoxenus (of course without naming him), that a pyknon cannot be divided into two equal intervals; but there is no attempt to explain the nature of the Enharmonic diesis. It is worth notice that in these propositions the Lichanos and Paranêtê of the Enharmonic scale are called lichanos and paranêtê simply, as though the Enharmonic were the only genus—a usage which agrees with that of the Aristotelian Problems (supra, p. 33).

According to Ptolemy (i. 13) the Pythagorean philosopher Archytas was the author of a new division of the tetrachord for each of the three genera. In it the natural Major Third (5: 4) was given for the large interval of the Enharmonic, in place of the Pythagorean ditone (81: 64); and the Diatonic was the same as the Middle Soft Diatonic of Ptolemy. But, as Westphal long ago pointed out (Harmonik und Melopöie, p. 230, ed. 1863), this scheme is probably the work of the later Pythagorean school. It seems to be unknown to Plato and Aristoxenus,—the latter wrote a life of Archytas—and also to Euclid, as we have seen. The next scheme of musical ratios is that of Eratosthenes, who makes no use of the natural Major Third.