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An Oxyrhynchus Fragment on Harmonic Theory
Published online by Cambridge University Press: 11 February 2009
Extract
The tattered remains of a few paragraphs of a work on harmonic theory were published in 1986 as P. Oxy. LIII.3706, with a careful commentary by M. W. Haslam. There are six fragments. Four of them (frr. 3–6) are too small for any substantial sense to be recovered; and while fr. 2 and the second column of fr. 1 allow us to pick out significant words and phrases here and there, the remnants of these columns are very narrow, and the line of reasoning seems inaccessible. Musicological analysis must focus on the first column of fr. 1. There is little enough even of that, and in attempting a relatively detailed interpretation I shall have to be rather less cautious than Haslam quite properly was. But I think that something can be made of it without stretching speculation too far, and if I am right the piece is of some genuine interest. Here are the two versions of the text that Haslam prints. The first records what is decipherable on the papyrus itself, while the second represents a partial reconstruction, restoring word-divisions and some of the missing letters.
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References
1 The Oxyrhynchus Papyri, vol. LIII (1986), ed. M. W. Haslam, pp. 49–55.Google Scholar
2 See his prefatory comments (p. 49) and note on line 5 (p. 52).
3 Haslam's statements in his note on this line (p. 52) that μελοποι- is ‘very probable’ and that μεικτ μελοποια ‘would well suit the remains’ seem to me to underplay his hand. I would reckon the case almost conclusive, despite the different spelling μικτή in fr. 2.12.
4 Aristoxenus uses the terms μίξις, μικτός, etc. in this kind of context at El. Harm. 7.3, 7.6, 17.26, 44.26; compare e.g. Cleonides, Eisagoge 6 (189.11, 15 Jan), Ptol. Harm. I.16 (especially 38.33–39.14, 40.8–10 Düring), II.6 (56.1–4). In these Ptolemaic passages references to μίξις and to μεταβολή are regularly linked. For other discussions of μεταβολή in these authors see Aristox. El. Harm. 7.10–8.3, 38.6–17, cf. 34.8–11, 40.12–24, Cleonides, Eisagoge 11 and 13, Ptol. Harm. II.6–11, and see also especially Aristides Quintilianus, De mus. 22.11–26 with context.
5 For a convenient summary see West, M. L., Ancient Greek Music (Oxford, 1992), pp. 195–6Google Scholar. More detail is offered in his notes on individual compositions (ch. 10): for fuller discussion see Pöhlmann, E., Denkmäler Altgriechischer Musik (Nürnberg, 1970Google Scholar). The brief remarks introducing each composition in Chailley, J., La musique grecque antique (Paris, 1979Google Scholar), ch. 8 are especially helpful for present purposes, since the presence or absence of modulation is a feature to which the author regularly draws attention.
6 For lists and brief characterisations see Cleonides, Eisagoge 13, Bacchius, Eisagoge 50–8, Anon. Bell. II.27, III.65.
7 In his note to line 5, p. 52. According to the Greek theorists, the basis of all musical systems (scales) is the division of each octave into two stretches each spanning a perfect fourth, together with the interval of a tone (in the paradigmatic form of the octave this is placed between the two fourths), thus: E–A, B–E′. Each of the fourths is then subdivided by the insertion of two further notes, which with the notes bounding the fourth constitute a ‘tetrachord’. While the positions of the notes bounding the tetrachord are fixed relative to one another and to other notes in the initial framework, those of the notes inside each tetrachord are not. The three ‘genera’ and their subspecies are distinguished from one another by the positions of the two inner notes of the tetrachord, especially the higher of them, in relation to the tetrachord's boundaries. Very roughly speaking, if the higher moveable note is placed very low in the tetrachord, the genus is enharmonic: if it is placed at or above the mid-point in this span the genus is diatonic: between these extremes the genus is chromatic.
8 The reference to chromatic is inferred rather than observed, but is really in little doubt. Haslam's note to line 13 (p. 54) presupposes that where we have a phrase of the form ‘x and enharmonic’, ‘x’ must be either ‘chromatic’ or ‘diatonic’. At first blush this seems a pretty sweeping assumption. I have not run a detailed check on it. In fact, however, I would be astonished if it turned out to be wrong in more than a tiny minority of cases.
9 El. Harm. 21.31–22.21 (cf. 46.19–24). Aristoxenus explains that his choice of this tetrachord to exemplify the changes by which shifts of genus are produced is only a matter of convenience: of the various groups of notes that could have been chosen by way of example, this one happens to be σχεδν λνωριμωτάτη τοῖς άπτομένοις μουσικς.
10 Tetrachords are said to be in conjunction (συνημένα, κατ συναϕήν) when they are so linked in a series that the top note of one tetrachord serves also as the bottom note of the next. They are in disjunction (διεζευγμένα, κατ διάζευξιν) when they are separated by an interval that stands outside either tetrachord, and this interval is always a tone. (Thus the series of white notes on a keyboard from B up to A constitutes a pair of tetrachords in conjunction, sharing the note E, while the series from E to E′ is a pair of tetrachords in disjunction, E–A and B–E, separated by the tone A–B.) See especially Aristox. El. Harm. 58.14–60.9.
11 Here the subject is broached initially in a discussion of forms of melodic progression, rather than merely of scale structure, and what is defined is γωγ περιϕερς ‘circular progression’: it is said to proceed upwards through notes in conjunction, downwards through notes in disjunction. But Aristides adds that the topic is one that is also studied in the context of modulation.
12 The emendation is small but crucial, involving the insertion of a negative. See Zanoncelli, L., La manualistica musicale greca (Milan, 1990), p. 295 n. 37.Google Scholar
13 Haslam hints at some of the points on which this interpretation is based at the end of his note on line 6 (pp. 52–3), but seems not to think them worth pursuing.
14 The word τριτοειδής is not met elsewhere, but its formation is parallel to terms such as λιχανοειδής, ὑπατοειδής, and so on, which are quite common in harmonic texts. Close analogues for the usage we seem to have here are at e.g. Aristides Quintilianus, De mus. 9.21, 25, 26. The sense of such terms can vary with context: thus e.g. Aristides' use of ὑπατοειδής and μεσοειδής at 81.21–3 is substantially different, as is that of Anon. Bell, at III.63–4.
15 There are a good many other possibilities. An author with Platonist leanings, like Aristides Quintilianus for instance, might have written μοιοται or one of its compounds: cf. e.g. De mus. 76.21, 79.25.
16 References to changes in ἦθος might suggest that the kind of μεταβολή being discussed is after all not μεταβολ κατ σύστημα as Cleonides defines it, but what Cleonides calls μεταβολ κατ μελοπιαν and defines by reference to change of ἦθος (Eisag. 13, 206.3–18), while Bacchius straightforwardly calls it μεταΒολή κατ ἦθος (Eisag. 50 and 54): cf. also Aristides Quint. De mus. 30.1–17. The association of the terms μελοποιία and ἦθος in Cleonides (also in Aristides) might seem a strong pointer. But I think this is unlikely. None of the Aristoxenian sources gives any sign of believing that such ‘modulation of ethos’ could be given a closely technical analysis: none of the structuring concepts of harmonic theory, such as those used to characterize other forms of modulation, appear in connection with this one, which is defined only in impressionistic terms related to aesthetic response. The Oxyrhynchus author, by contrast, is plainly offering a technical analysis grounded in Aristoxenian theory. It is however worth noting the unusual definition given for μεταβολ κατ ἦθος at Anon. Bell. II.27: it occurs ὅταν ν αὐτοῖς τοῖς τετραχόρδοις τ ἢθη τν ϕθόγγων τν μετάπτωσιν λαμβάνῃ. The kind of modulation envisaged is certainly different from that given the same name by Bacchius. But what does the definition mean? It cannot be alluding to those movements of notes in the tetrachord that create modulations of genus, since that sort of modulation is explicitly distinguished from μεταβολ) κατ ἦθος, and has been defined in the preceding lines. Most probably, I think, ν αὐτοῖς τοῖς τετραχόρδοις is to be taken closely with τ ἢθη. What changes is a note's character in a tetrachord: that is, it acquires a different role in the tetrachord, for instance by becoming a tetrachord's bounding note when previously it was not. This is just the sort of ‘change of character’ that is generated by Cleonides' μεταβολ κατ σύστημα (a kind of modulation not mentioned by this name in the list given by Anon. Bell.).
17 τόπος is Aristoxenus' regular term for the range within which a moveable note can move. For the expression τόπον διαβαίνειν, though in a slightly different context, see El. Harm. 9.16. Haslam (p. 50) reports traces as of a π in the gap before -ονδιαβ.
18 The concept of a note's δύναμις is of central importance in Book II of Aristox. El. Harm. It is nowhere defined, but seems to be constituted by the relations in which a note stands, and may legitimately stand, to others and to the system of which it is a part. The doctrine that a note's identity lies in its δύναμις is contrasted with the view (stated in Book I, 15.15–16) that a note is merely ‘the incidence of the voice on a single pitch’: see 36.6–14. For other important occurrences of the term see 33.6–9, 33.34–34.5, 36.4–6, 40.4–24, 47.29–48.7, 49.2–7; cf. also my discussion in ‘Aristoxenus' Theorems and the Foundations of Harmonic Science’, Ancient Philosophy 4 (1984), pp. 23–64Google Scholar, particularly Section VII. No author uses the word ἦθος as a direct equivalent for the Aristoxenian δύναμις (though the usage at Anon. Bell. II.27, n. 16 above, seems to be close), and I do not suggest that the present author does so. We do however find the view that individual notes possess ‘character’, and that this is closely linked to their positions in the tetrachord and other ‘dynamic’ properties: the view is elaborated most fully at Aristides Quintilianus, De mus. 77.17–82.3.
19 I should emphasize that the theoretical apparatus required is specifically Aristoxenian, not shared by theorists in general. Though many of the conceptions involved are indeed common to writers of all schools, the quantifications given below, on which the detailed interpretation of the passage depends, are specific to Aristoxenus and his followers.
20 See especially the long discussion at El. Harm. 22.24–27.14.
21 These divisions of the tetrachord are ξαίρτοί τε κα γνώριμοι, El. Harm. 50.19–22.
22 It is identical with the range of parhypatē, the corresponding note in the tetrachord mesōn: this is specified at El. Harm. 23.25–6.
23 Diatonic tetrachords include all those in which the two lowest intervals are jointly equal to or greater than the highest interval. Where this is not the case the pair of intervals at the bottom constitutes what is called a πυκνόν, which is characteristic of chromatic and enharmonic divisions. See e.g. El. Harm. 24.11–14, 25.6–9: the distinction is consistently maintained in Aristoxenus and all subsequent theorists.
24 Compare what is said of the note lichanos at El. Harm. 26.13–18.
25 The interval from mesē to tritē diezeugmenōn varies between 1½ and 1¼ tones, as we have seen. Paranētē synēmmenōn, however, can be a mere ½ tone above mesē in enharmonic and ⅔ tone in chromatic. Even the highest chromatic version of the note must inevitably be less than 1¼ tones above mesē, since at that point the highest interval in the tetrachord becomes equal to the combination of the two lower intervals, and the division must be diatonic (see n. 23 above).
26 For both aspects of the matter see Ptol. Harm. II.6: cf. Cleonides, Eisagoge 205.5–206.2.
27 See El. Harm. 38.17–18 with 34.34. Many later authors include μελοποιία in their lists of the ‘parts’ of harmonics. But contrast El. Harm. 1.22–2.7.