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Methods and aims in the Euclidean Sectio Canonis

Published online by Cambridge University Press:  11 October 2013

Andrew Barker
Andrew Barker
Affiliation:
University of Warwick
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Extract

For the writers of the imperial period, music theory presented a sharp dichotomy. One might be an Aristoxenus, or one might be a follower of the mathematikoi or Pythagoreans. The methods and doctrines of the two schools were thought of as radically incompatible with one another; and it is true that from the fourth century B.C., when their respective doctrines were first formally articulated, members of each tradition held views which their opposite numbers denied. But their disagreements are easily misunderstood. I have argued elsewhere that Aristoxenus' attitude to the Pythagoreans has sometimes been distorted. Here, in discussing the Sectio, which appears to be the earliest continuous treatise surviving from the other side of the fence, I shall try to show some of the ways in which its objectives differ from Aristoxenus', and thus the sense in which certain major disagreements between them reflect an oblique rather than a direct confrontation. I hope also to show that further study of this rather arid looking work for its own sake would repay both musicologists and philosophers.

Information

Type
Research Article
Information
The Journal of Hellenic Studies , Volume 101 , November 1981, pp. 1 - 16
Copyright
Copyright © The Society for the Promotion of Hellenic Studies 1981

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References

1 See e.g. Porphyry's Commentary on Ptolemy's Harmonics, 23.24–31, 25.3–28.27 (Düring).

2 ‘οἱ καλοúμενοι άρμονικοί: the predecessors of Aristoxenus’, PCPS cciv (1978) 1–21Google Scholar.

3 For a brief account of the controversies about authorship and date, see Burkert, W., Lore and Science in Ancient Pythagoreanism (Harvard 1972) 375Google Scholar n. 22, and the references given there.

4 von Jan, K. (Carolus Janus), Musici Scriptores Graeci (Leipzig 1895) i 117Google Scholar.

5 Cf. Burkert (n. 3) 379–83, esp. 380 n. 47; and some intriguing speculations on the development of the Pythagorean conception of an interval in Szabó, A., The Beginnings of Greek Mathematics (Dordrecht 1978) 107–34CrossRefGoogle Scholar.

6 Τούτων δὲ οἱ μὲν πολλαπλάσιοι καὶ ἐπιμόπιοι ἑϝὶ ὀνόματι λέγονται πρὸζ ἀλλήλουζ p. 149, II. 14–16 in von Jan (n. 4).

7 Burkert (n. 3) 383–4 interprets the passage as I do, but without explanation or discussion. On the reasons behind the Pythagorean acceptance of the principle which assigns concords to these two classes of ratio, see also Ptolemy Harmonics 11.8–20 (Düring).

8 That it was already in the fourth century a definitive mark of a distinct school of thought is clear from e.g. Aristoxenus El.Harm. 32.24 and Theophrastus fr. 89 (Wimmer) = Porphyry op. cit. 62.1 ff. (Düring). The Aristoxenus passage also explicitly associates the use of the language of ratios with a special attitude to the deliverances of reason. The fullest discussion of the wider implications of the two positions is Ptolemy Harm. 19.16–21.20.

9 That is, if two terms, B and C, are in the ratio n + 1: n, there is no term D, between B and C, such that B:D = D:C, and no series of terms D, E, F,…, N, such that B:D = D:E = E:F =…= N:C. The proof given is of some historical interest, being substantially identical to that ascribed by Boethius to Archytas. See Heath, T. L., A Manual of Greek Mathematics (Oxford 1931Google Scholar) republ. as Greek Mathematics (Dover N.Y. 1963) 136–7;Google Scholar and Burkert (n. 3) 442–7.

10 Von Jan seems to have been unable to make sense of the argument as it stands in the MSS. His version can be reconstructed from the emendations recorded in the three following notes. See Musici Scriptores Graeci 163.

11 Von Jan emends CE to BE.

12 Von Jan emends FC to FE and DE to DB.

13 Von Jan emends to ‘DB is equal to FE’.

14 The best description of the Pythagorean canon, or monochord, is at Ptol. Harm. 17.27–19.15 (Düring).

15 Cf. Ptolemy loc. cit., Nicomachus Enchiridion 246.22–247.4 (von Jan).

16 DK 47, A16, A17.

17 Cf. the discussion of Aristotle's use of the notion of a phainomenon in Owen, G. E. L., ‘Tithenai ta phainomena’ in Aristote et les problèmes de la méthode (Symposium Aristotelicum: Louvain 1961) 83–103,Google Scholar repr. in Moravcsik, J. M. E. (ed.), Aristotle: a collection of critical essays (Macmillan 1968) 167–90Google Scholar.

18 Aristoxenus El. Harm. 20.17–22, 45.20–34. Cf. Ptol. Harm. 13.1–23 (Düring).

19 Repub. 531C1–4.

20 Cf. my paper ‘σúμϕωνοι ἀριθμοί: a note on Republic 531c 1–4’, CPh lxxiii (1978)Google Scholar, esp. 339–40.

21 Cf. Ptolemy's remarks at Harm. 13.23–14.2 (Düring).

22 Harm. 13.1–23.

23 See e.g. Nicomachus Enchiridion section 3, pp. 241–2 (von Jan).

24 E.g. Aristoxenus El. Harm. 19.20–29. This genus was also central to certain fundamental constructions of theory. Cf. e.g. Chailley, J., La musique grècque antique (Paris 1979)Google Scholar chs 5–7.

25 Timaeus 35b–36b.

26 E.g. Ps.-Plutarch De Musica 1134f–1135b.

27 On the Χρóαι ‘shades’, see Aristoxenus El. Harm. 47–52.

28 Op. cit. 2.6–11, 351–13.

29 Cf. Repub. 531a7, Arist. Metaph. 1016b18, 1052b20, 1083b33.

30 See my paper cited at n. 2, 10–12.

31 Aristoxenus El.Harm. 19.26–9.

32 Division of the tone, including reference to the enharmonic diesis, El.Harm. 21.20–31, 46.2–7, and elsewhere: the sizes of the fourth and the fifth, assumed elsewhere (e.g. 46.1–2) are derived at 56.13–58.6. The crux of the derivation lies, I think, in Aristoxenus' implied claim that the ear ‘accepts’ the difference between the fourth and the fifth (the tone) as identical with the difference between the fourth and the ditone taken twice.

33 The passages cited at n. 16, as well as the scale in the Timaeus, are evidence of earlier attempts at such an analysis within the ‘Pythagorean’ tradition.

34 The arguments against the authenticity of Propositions 19–20 have hinged on the supposition that Propositions 17–18 indicate the original author's primary concern with the analysis of a scale in the enharmonic genus. If I am right in suggesting that they indicate nothing of the kind, but are by intention largely polemical in the way I have outlined, these arguments can have no weight. If the early date of P19–P20 is admitted, van der Waerden's arguments, cited by Burkert, for the later invention of the monochord must also fail. See Burkert (n. 3) 375 n. 22, and Szabó (n. 5) 118–19.

35 Burkert (n. 3) 384 states that the argument presupposes only two ‘empirical observations’—that the octave consists of a fourth and a fifth, and that while a double octave is a concord, a double fourth or a double fifth is not. As will become clear, the empirical status of some of the presuppositions I shall mention is in doubt: but the discussion which follows will suggest that the system of presuppositions is rather more complex than Burkert implies.

36 El.Harm. 55.3–10.

37 El.Harm. 21.20–3, 45.35–46.1.

38 El.Harm. 55.10–56.12.

39 Cf. Repub. 531a–c.

40 Best explained at El.Harm. 53.34–54.21, which, ith its continuation to 55.2, strongly suggests the ‘less ambitious’ form of the project.

41 The procedure described in the passage cited at n. 36 is plainly at least capable of practical application.

42 See the passages cited in Burkert (n. 3) 383 n. 62.

43 Cf. especially the discussion of the criteria of musical judgement at El.Harm. 33.1–34.34.

44 El.Harm. 55.3–7.

45 I should like to thank David Fowler for his comments on an earlier draft of this paper. He has also pointed out to me recently sources for the principles not proved, but assumed from elsewhere, in P2 and P3. See Euclid Elements viii, Propositions 6–9.