[55] Nicomachus, Enchiridion, p. 4 ei gar tis ê dialegomenos ê apologoumenos tini ê anaginôskôn ge ekdêla metaxy kath' hekaston phthongon poiei ta megethê, diistanôn kai metaballôn tên phônên ap' allou eis allon, ouketi legein ho toioutos oude anaginôskein alla meleazein legetai.
[56] De Compositione Verborum, c. 11, p. 58 Reisk.
[57] De Comp. c. 11, p. 64 to de auto ginetai kai peri tous rhythmous; hê men gar pezê lexis oudenos oute onomatos oute rhêmatos biazetai tous chronous oude metatithêsin, all' oias pareilêphe tê physei tas syllabas, tas te makras kai tas bracheias, toiautas phylattei; hê de mousikê te kai rhythmikê metaballousin autas meiousai kai parauxousai, ôite pollakis eis tanantia metachôrein; ou gar tais syllabais apeuthynousi tous chronous, alla tois chronois tas syllabas.
[58] The metrical accent or ictus was marked in ancient notation by points placed over the accented syllable. These points have been preserved in Mr. Ramsay's musical inscription (see the Appendix, [p. 133]) and in one or two places of the fragment of the Orestes ([p. 130]). Hence Dr. Crusius has been able to restore the rhythm with tolerable certainty, and has made the interesting discovery that in both pieces the ictus falls as a rule on a short syllable. The only exceptions in the inscription are circumflexed syllables, where the long vowel or diphthong is set to two notes, the first of which is short and accented. The accents on the short first syllables of the dochmiacs of Euripides are a still more unexpected evidence of the same rhythmical tendency.
[59] Plato, Legg. p. 669.
[60] On this point I may refer to the somewhat fuller treatment in Smith's Dictionary of Antiquities, art. Musica (Vol. II, p. 199, ed. 1890-91).
[61] Plato, Legg. p. 812 d panta oun ta toiauta mê prospherein tois mellousin en trisin etesi to tês mousikês chrêsimon eklêpsesthai dia tachous.
[62] 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.
[63] The two schools distinguished by Plato seem to be those which were afterwards known as the harmonikoi or Aristoxeneans, and the mathêmatikoi, who carried on the tradition of Pythagoras. The harmonikoi regarded a musical interval as a quantity which could be measured directly by the ear, without reference to the numerical ratio upon which it might be based. They practically adopted the system of equal temperament. The mathêmatikoi sought for ratios, but by experiment 'among the consonances which are heard,' as Plato says. Hence they failed equally with those whose method never rose above the facts of sense.