James Adam, The Republic of Plato

τρἴ ἄττα εἴδη. Arist. Quint. I 34 ed. Meibom γένη τοίνυν ἐστὶ ῥυθμικὰ τρία: τὸ ἴσον (2/2), τὸ ήμιόλιον (3/2), τὸ διπλάσιον (2/1). To the first belong dactyls, spondees, anapaests: the second includes paeons, cretics, and bacchei: under the third fall trcchees, iambics, ionics. See Gleditsch l.c. p. 694.

ὥσπερ -- ἁρμονίαι. What are the τέτταρα εἴδη? The following answers (among others) have been given: 1. the intervals of the fourth, fifth, octave, and double octave (Ast): 2. the four notes of the tetrachord, which was probably the historical and at all events the ‘theoretical unit of the scale’ (Stallbaum, Jowett and Campbell): 3. “the four ratios which give the primary musical intervals—viz. the ratios 2:1, 3:2, 4:3 and 9:8, which give the octave, fifth, fourth, and tone” (Monro l.c. p. 106 note; cf. also Dict. Ant. II p. 193): 4. the four ἁρμονίαι Φρυγιστί, Λυδιστί, Δωριστί, Λοκριστί (Westphal Rhythmik p. 238). Ast's view cannot be right, unless we suppose that ἁρμονίαι here includes scales of double compass, which is most unlikely. Westphal's explanation is improbable, for Plato has said nothing of Λοκριστί, and (though perhaps no great stress should be laid on this) it is awkward to derive the ἁρμονίαι (ὅθεν αἱ πᾶσαι ἁρμονίαι) from themselves. If the principle of Westphal's interpretation is right, I should be inclined to substitute Ἰαστί for Λοκριστί, having regard to 398 E, where see note Cf. Cl. Rev. X p. 379. (I have since found that Prantl also took this view: see note 116 in his translation.) I do not think that Stallbaum has hit the truth, for Plato's language is not suggestive of any allusion to the origin of the octave from the combination of two tetrachords, and a single tetrachord cannot produce a ἁρμονία (ὅθεν αἱ πᾶσαι ἁρμονίαι). Possibly the τέτταρα εἴδη ἐν τοῖς φθόγγοις denote simply the keynote, its octave, and the intervals of a tone and a semitone: for these are as it were the threads out of which all modes ‘are woven’ (πλέκονται should be repeated with ἁρμονίαι), the difference between the modes depending on the difference in position of the tones and semitones. But Euclid lays the greatest stress upon the ratios 3:2 and 4:3 as the component elements of the octave: see for example Sect. Can. 6 τὸ διπλάσιον διάστημα ἐκ δύο τῶν μεγίστων ἐπιμορίων συνέστηκεν, ἔκ τε τοῦ ἡμιολίου καὶ ἐκ τοῦ ἐπιτρίτου and cf. ib. 8, 12, and for this reason I now believe that Monro's view has most in its favour.

ποῖα δ᾽ ὁποίου κτλ. On ὁποίου see I 348 B note, and for the error in Paris A Introd. § 5.