1. See e.g. Macran, H. S., The harmonics of Aristoxenus (Oxford 1902) 87–8, 226Google Scholar, Lippman, E. A., Musical theory in ancient Greece (New York 1975) 145Google Scholar. I have myself rashly assumed this position in the past, explicitly in ‘Music and Mathematics’, PCPS n.s. 23 (1977) 14 n. 8Google Scholar. The present paper is in part a recantation of the view adopted in that note: the thesis I am now suggesting is stated but not argued in Michaelides, S., The music of ancient Greece: an encyclopaedia (London, 1978)Google Scholar s.v. Harmonikos.

2. Theophrastus fr. 89 (Wimmer). I have discussed a relevant part of it in the article cited in n. 1, but the beginning of the fragment, not dealt with there, is also to the point.

3. Cf. Diels, /Kranz, Die Fragmente der Vorsokratiker (1952 and later editions) 47Google Scholar, A16, A17, A19, B2 (Archytas), 44, A25, A26, B6 (Philolaus), Plato, , Timaeus 35b–36bGoogle Scholar. The procedures are fully exemplified in the Euclidean Sectio Canonis.

4. See e.g. D.K. 47, Bl (Archytas), Plato, , Tim. 67bGoogle Scholar, Aristotle, , Top. A, 107a11ff.Google Scholar, Aristoxenus, , El.Harm. 32.19–28Google Scholar, and the comments of von Jan, K., Musici scriptores Graeci (Leipzig, 1895) I. 134–41Google Scholar, Burkert, W., Lore and science in ancient Pythagoreanism (tr. Minar, E. L. Jr., Harvard, 1972) ch. 5Google Scholar, section 1.

5. Porphyry, 's Commentary on Ptolemy's Harmonics, 29.27–31.21Google Scholar (Düring). Questions about the connection between a sound's pitch and some physical velocity were also much discussed in the Peripatetic school: cf. Ar., de Anima 420a27–b5Google Scholar, de Gen.An. 786b12ff., Hist.An. 545a14ff., 581a17ff., ps.- Ar. Problems XI, 3, 6, 10, 11, 13-16, 19-21, 32, 34, 40, 47, 50, 52, 53, 56, 61, 62.

6. I discuss this matter in an article ‘Music and Perception’, JHS 98 (1978) 9–16CrossRefGoogle Scholar.

7. In the scales discussed by Aristoxenus certain notes are ‘fixed’: that is, they stand at set intervals from one another. Between them lie notes which are moveable, and with certain systematic variations in the intervals between them and the fixed notes we get what Aristoxenus calls change of γένος. There are three genera, enharmonic, chromatic, and diatonic. The enharmonic, which is referred to here, is characterised primarily by two very small intervals (quarter-tones) in the lower part of each tetrachord, followed by a jump of a ditone to the next fixed note, a fourth above the original pitch. For the details of his analysis of the interval structures of the genera, see 22-7 and 46-52.

8. D.K. 47 A16 (from Ptolemy), A17 (from Porphyry). See also the passages cited in n.3 on Philolaus.

9. Tim. 35b-36b. See the discussion of the Timaeus scale with reference to the supposed fragments of Philolaus in Burkert op.cit., ch.5, section 2.

10. El.Harm. 56.13ff. Eucl., ContrastSect. Can. 15Google Scholar.

11. El.Harm. 50.15ff. and passim. Eucl., ContrastSect. Can. 16Google Scholar (the argument depends on the proof given in 3).

12. Explained in Ptol., Harm. 11.18-12.7Google Scholar (Düring). The classification of ratios from which the problem arises can be found in Eucl., Sect.Can. 1Google Scholar.

13. Porphyry op. cit. 23.24-31, 25.3-28.27.

14. Ibid. 25.10-14.

15. Ibid. 25.14-16.

16. Ibid. 26.6-15.

17. A σύστημα is a scale, in the sense of a particular order of intervals. A τόπος, is a range of pitch. Aristoxenus is concerned, in the passage cited, with the question whether the different συστήματα are to be arranged at any particular pitches relatively to one another. See Macran op. cit. 229-32, 262-6.

18. Macran op. cit. 170.

19. Theophrastus ap. Porphyry op. cit. 61.24-62.3.

20. The procedure is never fully described by Aristoxenus, but can be fairly confidently reconstructed in the way I have indicated on the basis of his scattered remarks about its uses and misuses. See the passages referred to in the Index Verborum of the edition of R. da Rios (Rome, 1954) s.vv. καταπύκνωσις and διάγραμμα.

21. Macran op. cit. 251-2, and see my paper cited in n.6 above.

22. See e.g. Winnington-Ingram, R. P., Mode in ancient Greek music (Cambridge, 1936), especially 81 ff.Google Scholar

23. The subject of Greek notation is a delicate one, and this matter in particular is far more ambiguous and controversial than my bald statement suggests. I cannot give it here the space which it deserves, but it may be worth stating in a crude and unargued way what I think the situation probably was. I believe (a) that the system of notation in common practical use was an early version of something like that of Alypius, based on δυνάμεις, and (b) that an alternative, quantitative system had perhaps been developed by the ἁρμονικοί, based on the calculations whose results were set out in the καταπύκνωσις diagrams. There is no evidence that any such system was in use among practical musicians, but the remarks of Aristoxenus in the passages at present under discussion suggest that one had been constructed as a tool of analysis by those whose views he is attacking. See Winnington-Ingram, R. P., ‘The First Notational Diagram of Aristides Quintilianus’, Philologus 117 (1973) 244 n. 2CrossRefGoogle Scholar. In general on notation, see Michaelides op.cit. svv parasemantike and onomasia, with the references given there, and Potiron, H., ‘La notation grecque au temps d'Aristoxene’, Revue de Musicologie 50 (1964) 222–5CrossRefGoogle Scholar.

24. Implied apparently by Macran op. cit. 277.

25. The evidence is conveniently summarised in Lippman op. cit. ch. 1, and in Burkert op. cit. ch. 5 section 1.

26. Ar., Metaph. 1016b18, 1052b20, 1083b33Google Scholar. Cf. 1053b32 ff. and An. Post. 84b ad fin.

27. There is evidence that Aristotle was aware of some of the relevant contrasts and complications. See Metaph. 1053a14-16.

28. , El.Harm. 24.11-14. The πυκνόν is, effectively, the lower portion of an enharmonic or chromatic tetrachord. For an account of its nature and status, see 48.9-33.

29. As previously explained, it is impossible for them to be mathematically equal according to the Pythagorean system of ratios. See Eucl., Sect.Can. 16Google Scholar, and cf. the passages concerning Archytas and Philolaus cited at n.3 above.

30. See e.g. El.Harm. 19.23-9, Anon. Bell. 26.30, Aristox. ap. Plut., De Mus. 1134FGoogle Scholar.

31. The Hibeh Papyri ed. Grenfell, B. P. and Hunt, A. S. (London, 1906) part 1, no. 13Google Scholar. The papyrus itself dates from the mid-third century. The commonly accepted dating of its contents is based on Cronert, W., ‘Die Hibehrede uber die Musik’, Hermes 44 (1909) 503–21Google Scholar. One of his main criteria is that of style, which he identifies as Isocratean: this seems to me doubtful. The other lies in the author's reference to the use of ἁρμονία (i.e. the enharmonic genus or style) by tragedians (col. 1,16, col.2,3-5). This is taken to prove a pre-Aristoxenean date, on the basis of El.Harm. 23.12-23. But that passage does not show, as generally supposed, that the use of the enharmonic had disappeared: it shows that what Aristoxenus regarded as the proper intervals of the enharmonic were in his time played inaccurately by most musicians. There is nothing here which entails that contemporary tragedians did not think of themselves as still employing the old forms: the new tendency is simply that of performers to raise the λίχανος when playing enharmonic music, thus pushing it in practice towards the form of the chromatic. A further piece of evidence for an early date is extracted from the reference to (col.2,13) by Anderson, W. D., Ethos and education in Greek music (Harvard, 1966) 149–50CrossRefGoogle Scholar and Appendix C. He takes this as a reference to the plank benches of the theatre in Athens: since the stone theatre of Dionysus was built in the mid-fourth century, and the seats were then no longer wooden, the fragment must pre-date the new building. Since the fragment makes no reference to Athens, let alone to any particular theatre, and since this ingenious interpretation of the phrase in question is scarcely proved, this suggestion can only charitably be described as speculative.

32. Col.2,1-6 plainly indicate that the author makes no distinction between διάτονος μουσική and χρῶμα.

33. See especially Rep. 398c-400c.

34. E.g. Anderson op.cit. ch.1, Michaelides op.cit.s.v. ethos.

35. Aristides Quintilianus 21-2 (Meibom) = 18-20 (Winnington-Ingram, Teubner). See Winnington-Ingram, , Mode in ancient Greek music 22–30Google Scholar.

36. Cf. Winnington-Ingram loc.cit., with his references, and Anderson op.cit. 18-19.

37. E.g. Hibeh Papyrus 1, 13 col.2,3-4. But cf. Plut., De Mus. 1137EGoogle Scholar, Quaest. conviv. Bk.III, I, 11-12.

38. A certain indefiniteness about the distinctions between genera is suggested by the passage cited at n.32 above.

39. In the paper cited at n.6 above.

40. This seems to be an implication of El.Harm. 23.3-24, and is perhaps part of the position against which the author of the Hibeh Papyrus on music is arguing. See also Ar.Quint. 19 (Meibom) = 16.13 ff (Winnington-Ingram), and 133-4 (Meibom) = 110.28-111.27 (Winnington-Ingram). Cf. Vogel, M., Die Enharmonik der Griechen (Dusseldorf, 1963)Google Scholar with the discussion by Winnington-Ingram, in Die Musikforschung 18 (1965) 60–4Google Scholar.

41. There is some ground for suggesting that much of the groundwork for the systematisation is that done by the Pythagoreans, drawn on surreptitiously by Aristoxenus. This question is too large to go into here, but with the more detailed mathematical sections of El.Harm. cf. Burkert op.cit. ch.5 section 2, and Winnington-Ingram, R. P., ‘Aristoxenus and the Intervals of Greek Music’, CQ 26 (1932) 195–208CrossRefGoogle Scholar.

42. See the account of Macran op.cit. 262-266.

43. In theorists from Aristoxenus onwards, the term μέλος may refer either to an actual melody, or to the legitimate series of notes which form a scale. The latter is in a sense an abstraction, constructed by identifying the note-types into which the individual sounds of the actual melody fall, and arranging them in an ordered sequence. For present purposes the ambiguity is unimportant: in fact whichever meaning we adopt the sense of the passage comes to much the same thing. The quarter-tone series cannot form the basis for any actual melody; but this means no more than that it cannot form the scale produced by setting down in order the note-types, actual or implied, which make up the legitimate set from instances of whose members given melodies are constructed. I am grateful to Professor Winnington-Ingram for pointing out to me the ambiguity in the term, and also for his helpful comments on other aspects of this paper, which he was kind enough to read in draft. I am encouraged by his general approval of the main lines of my argument, but errors of fact and judgement which remain are of course my own.