James Adam, The Republic of Plato

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τοὺς γὰρ κτλ. It is strange that in spite of οὓς ἔφαμεν νῦν δὴ κτλ. this should have been so frequently understood as referring to the school satirised by Glauco: see for example Susemihl Gen. Entw. II p. 210. Plato is of course, as Schneider pointed out, speaking about the Pythagoreans who investigated the numbers or ratios of audible consonances: see 531 A note and RP.^{7} § 56 C.

ἀλλ᾽ οὐκ κτλ. Cf. 530 B. ἀνίασιν is undoubtedly present, and not future, here: see on V 473 C.

τίνες ξύμφωνοι κτλ. As the true astronomer should study intelligible stars with the mathematical intelligence, using the visible stars only as imperfect παραδείγματα (529 C, D note), so the true ἁρμονικός must investigate intelligible, and not audible, consonances. In the words of a modern writer, he must “look, not into the tone-world here, but into the world of harmony beyond.” Plato holds that certain mathematical numbers are in themselves ξύμφωνοι, and others not: see Theo 72—75, where examples of both varieties are given. The numbers or ratios of audible consonances are only particular and imperfect embodiments or expressions of these numbers: they may serve as παραδείγματα, but nothing more. In the Timaeus Plato represents the World-soul as the grandest expression of certain ξύμφωνοι ἀριθμοί, so that it is natural enough for him to crown his προπαιδεία with the study of mathematical ξυμφωνία, and say that it is ‘useful in seeking out the beautiful and good.’ It must nevertheless be admitted that Plato's conception of Harmonics as well as of Astronomy is fundamentally different from that of modern science, in spite of the attempts which Bosanquet and others have made to prove their essential harmony. See on 530 C and App. II.

531C - 533D The pursuit of these studies, if carried far enough to reveal their mutual relationship, will contribute to the end which we desire; but after all, they are only the prelude to Dialectic. We may compare Dialectic to the prisoner's progress from looking on real animals to beholding the sun, and these preparatory studies to his release and ascent from shadows and images within the cave to shadows of real objects in the world above. Socrates declines to give an account of the method and object of Dialectic; but insists that the Good must be seen, and that Dialectic alone can reveal it, for Dialectic is the only study which ascends on the ruins of its hypotheses to the Idea of Good, leading the soul on high, and using the ‘Arts’ as handmaidens and helpers in the process of education.