James Adam, The Republic of Plato

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τὰς γὰρ ἀκουομένας κτλ. The intervals reckoned as consonant (σύμφωνα) were such as the octave, double octave, fifth and fourth: see on IV 430 E. These the Pythagoreans ‘measure by’ (or ‘against’) ‘one another,’ by comparing the lengths of vibrating strings of the same material, thickness and tension. It is thus found that the octave is 2 : 1, the double octave 4 : 1, the fifth 3 : 2, and the fourth 4 : 3. See Dict. of Ant. II p. 193 with Theo Smyrn. pp. 48—51, 56—61 Hiller, and Aristox. Harm. 20 ff. Marquard. Richards proposes <ἐν> ἀλλήλοις, but the dative is strictly accurate: cf. Tim. 39 D τῷ τοῦ ταὐτοῦ καὶ ὁμοίως ἰόντος ἀναμετ ρηθέντα κύκλῳ.

ὥσπερ οἱ ἀστρονόμοι. The parallel is exact: as the astronomers studied visible, so the Pythagoreans investigated audible φοραί (Theo l.c.). To Plato, on the other hand, ἁρμονίη ἀφανὴς φανερῆς κρείσσων (Heracl. Fr. 47 Bywater). ‘Heard harmonies are sweet, but those unheard are sweeter.’ See above on 530 C.

νὴ τοὺς θεοὺς κτλ. There were two rival schools of musical theory in Greece, viz. “(1) the Pythagorean or mathematical, who identified each interval with a ratio, (2) the ‘musical’ (μουσικοί), who measured all intervals as multiples or fractions of the Tone” (Monro in Dict. Ant. II p. 193). Cf. Modes of Anc. Gk. Mus. p. 124. Plato's criticism was intended to apply to the first school; but Glauco erroneously understands it of the second.

πυκνώματα κτλ . ἄττα (nescio quae) and ὀνομάζοντες shew that πυκνώματα is a technical term. The word πύκνωμα does not appear to occur elsewhere in this sense, but πυκνόν was a favourite word with writers of the ‘musical’ school, as may be seen from its constant employment by Aristoxenus. πυκνόν is thus defined: τὸ ἐκ δύο διαστημάτων συνεστηκὸς ἃ συντεθέντα ἔλαττον διάστημα περιέξει τοῦ λειπομένου διαστήματος ἐν τῷ διὰ τεσσάρων (Aristox. Harm. 24. 10 ff. Marquard) i.e. any combination of two intervals which are together less than the interval remaining in the Fourth when the πυκνόν is subtracted from the Fourth, e.g. two quarter tone intervals, or even two semitone intervals (but not more): see Aristox. l.c. 50. 15 ff. The definition in Bacchius Isag. 20 von Jan τὸ ἐκ δύο διαστημάτων ἐλαχίστων συγκείμενον ἐν ἑκάστῳ γένει is less exact, but not, so far as it goes, inconsistent with that of Aristoxenus. Plato's πυκνώματα must be “haec ipsa πυκνά vel alia parva et tamen composita intervalla,” so called “propter sonorum in angusto spatio quasi confertorum frequentiam” (Schneider). Cf. πυκνότης in Laws 812 D, καταπυκνοῦσθαι, καταπύκνωσις etc. in Theo 91 and often in Aristoxenus, and see generally Westphal and Rossbach Gr. Harm. etc. pp. 105 ff. It is possible that the musical application of these terms was originally a metaphor borrowed from the art of weaving: for “vestes spatha textae, ob densitatem, quam inde consequebantur, πυκνώματα dictae ap. Aesch. Suppl. 235 πέπλοισι βαρβάροισι, καὶ πυκνώμασι” (Stephanus-Hase s.v. πύκνωμα, where reference is made also to Hesych. s.v. σπάθημα and a Scholiast on Ar. Ach. 180). I agree with Schneider in doubting whether Gellius' “frequentamenta” (I 11. 12, V 1. 1) are the same as Plato's πυκνώματα.

οἷον ἐκ γειτόνων κτλ.: ‘as if they were trying to catch a sound in the neighbourhood.’ Cf. Heliod. I 17 πίνει δὲ ἐνταῦθα ἐκ γειτόνων and Blaydes on Ar. Plut. 435 or Stephanus-Hase Thes. s.v. γείτων, where numerous examples of this highly idiomatic phrase are quoted. J. and C.'s translation “from a neighbour's house” is incorrect and pointless: still worse is Westphal's “als ob sie die Intervallgrösse dem Nachbarton ablauschen wollen.” The idiom was understood by Ficinus, who translates it by “viciniore loco.”

οἱ μέν φασιν κτλ. Some will have it that they overhear a note between (let us say) B and C, and that this is the smallest interval, and should be the unit of measurement: others say ‘No! it is not different from B.’ Plato (who is all for simplicity in music Laws 812 C) here satirises the μουσικοί, who made the quartertone or δίεσις their unit: see Theo 55 δίεσιν δὲ καλοῦσιν ἐλαχίστην οἱ περὶ Ἀριστόξενον τὸ τεταρτημόριον τοῦ τόνου, ἥμισυ δὲ ἡμιτονίου, ὡς ἐλάχιστον μελῳδητὸν διάστημα, and on the ἐναρμόνιον γένος generally, which Plato strongly disliked (Theo 56; cf. also Procl. in Tim. 191 E), and in which the δίεσις played a large part, Dict. of Ant. l.c. and Westphal and Rossbach l.c.

ἀμφισβητοῦντες. We should expect ἀμφισβητοῦσιν (so Theo 6) or else φάσκοντες instead of φασιν above. Cobet would emend, but the anacoluthon is not difficult in a writer like Plato: see on VI 488 C, D and supra 519 A note

φθεγγομένων: sc. τῶν χορδῶν, omitted as in ἡ διὰ πασῶν.

ὦτα κτλ. This bitter epigram was applied by Adrastus to Aristoxenus (Procl. in Tim. 192 B). The cap fits admirably; for Aristoxenus was afterwards the leader of the μουσικοί whose principle is here ridiculed. With the expression itself cf. Pliny Epp. VII 27, 8 sed offirmare animum auribusque praetendere.