1 That is to say, ratios of the forms n : 1 and (n + 1) : n respectively. They did not infer that all such ratios were consonant (the 9 : 8 tone was not), but that no other intervals would serve.
2 And even fewer in the Lesbian dialect of the much-admired poets Sappho and Alcaeus, some features of which were adopted by non-Lesbians for lyric verse. Contrast Hagel on ‘the not uncommon superstition of a superior ancient Greek musical ear’ (p. 207).
3 It is of course no such thing, but a sheer accident of decimal notation, which the Greeks did not use, and neither should we if humanity had six digits on its hands and feet instead of five, so that the same rational fraction of 2/3 – which has nothing to do with the octave, but is the generating ratio of the lower fifth – was written duodecimally as 0.8. On p. 216 Levin writes with like confusion of ‘real numbers, the infinite decimal fractions of which form the domain of real or continuous numbers’. It is not clear what she means by this, nor what she supposes the antonym of ‘real’ to be; certainly neither Pythagorean nor Aristoxenian music theory involves imaginary or complex numbers, which had not then been discovered.
4 Tune up a fourth from the higher note, then back down a fifth; repeat; tune down a fourth from the lower note of the original fourth, back up a fifth, and again repeat. You will have thus produced pitches a ditone above the lower note and a ditone below the higher. From the lower note of the higher ditone, tune up a fourth; from the higher note of the lower ditone, tune down a fourth; test whether the two notes thus attained make a concord. If they do, it must be greater than a fourth and less than an octave, therefore a fifth. Since the lower note of this fifth is also the lower note of a fourth, they are by definition a tone apart; since the intervals into which this tone is divided by the higher note of the original fourth are the complements of ditones to their respective fourths, they must be equal, and likewise the intervals divided by the lower note.
5 De institutione musica 3. 10, p. 285, ll. 1–2 Friedlein. In fact it is greater than the perceptible difference between the Pythagorean ditone and the major third (81 : 80, or 21.506290 cents).
6 Barker, Andrew, The Science of Harmonics in Classical Greece (Cambridge, 2007), p. 190
asserts that the experiment is offered not as a proof but as a test; that is to exalt the literal sense of Aristoxenus' words above their rhetorical force.
7 Or rather a♯–d♯ (as in Macran's exposition, p. 286, with its astonishing reference to ‘the Fourth above f namely a♯’), for Greek instrumental notation inflects the lower note by one or two sharpenings (of which vocal notation descends from the second); in the enharmonic genus modern scholars transcribe, e.g., the first sharpening of a as a↑ and the second as b♭, without intending an exact pitch.
8 This word, properly ‘fitting’ or ‘joining’, in musical contexts means variously ‘tuning-system’ (as here), ‘state of attunement’, ‘octave’ (in Pythagorean Doric), ‘enharmonic genus’, and ‘music’ in general, slipping into ‘melody’ when contrasted with ‘rhythm’.
9 Cf. Ptolemy, Harmonics 1. 10, p. 40, ll. 21–9 Düring.
10 See Harmonic Elements 1. 1, p. 5, ll. 4–11 da Rios; the brief mention of μελοποιία at 2. 48, ll. 4–10 as the final stage of the course leads to nothing in the surviving text.
11 The common translation ‘soul’ misdirects the reader towards mysticism and mumbo-jumbo, the transliteration ‘psyche’ towards emotions and away from reason.
12 ‘Intuition’ is a slippery term; Levin has elsewhere associated it with Themistocles' ability to size up a situation without prior knowledge or stopping to inform himself, οἰκείᾳ ξυνέσει (Thucydides 1. 138. 3), for which she rightly endorses ‘the gloss mother-wit or native sagacity’: see ‘Synesis in Aristoxenian Theory’, Transactions of the American Philological Society, 103 (1972), pp. 211–234
, at 211 n. 1. The question, however, is not whether ξύνεσις can bear this sense in Greek, but whether it does bear this sense in Aristoxenus. Far more relevant seems to be Aristotle, Nicomachean Ethics 10. 9, 1181a19–22: ‘those experienced in music can judge works in detail, and understand (συνιᾶσιν) by what means and how they are achieved, and what things accord with what; for those without experience it is enough if it does not escape them whether the work is well or ill made’. (Initial ξ and σ in the prefix are recognised alternatives; but although Aristoxenus used ξ, Levin transliterates s.)
13 The obvious example to prove Ptolemais' point is Pythagoras' supposed discovery of the intervallic ratios from sensory experience, and the subsequent inferences that led to the rejection of the eleventh. But though intervallic ratios are musical facts, Levin is the last person to regard them as musical meanings.
14 When in 1970 China launched a satellite that played the Cultural Revolution anthem The East is Red, the sound, to my untutored ear, suggested a pipes' lament. This was presumably not the ethos intended.
15 §6, in Jan, C., Musici scriptores Graeci (Leipzig, 1895), p. 415
, ll. 3–4.
16 Science of Harmonics, pp. 424–9.
17 Barbera, André, The Euclidean Division of the Canon (Greek and Latin Music Theory; Lincoln, Nebr., and London, 1991)
18 Demetrius, son of Demetrius the Besieger and Ptolemais daughter of Ptolemy I king of Egypt, was betrothed to the future Berenice II but fell victim to domestic murder instigated by her when he turned his attentions to her mother Apame. Since he married neither woman, and Berenice passed for a virgin when rebetrothed to the future Ptolemy III of Egypt, Ptolemais would have been Apame's child, though our only source for the bad business says nothing of a pregnancy, let alone of a parturition.
19 This would also contrast with the simple ὁμοίως said of Aristoxenus and be taken up by the opposite inequality of the following group.
20 The proofs were not corrected with sufficient care: quotations in prose are set out as verse, at p. 119, n. 58 the fourth line should be the seventh, on p. 202 the lower intervals of the hemiolic chromatic are made ¾ tone instead of ⅜, and on p. 245 the diagram intended to illustrate conjunction of tetrachords represents only one tetrachord instead of two (though here the text too is at fault). In the bibliography Andrew Barker's works have been added to André Barbera's and Sir James Jeans is disguised as ‘James, Sir J.’ Nor has Levin been well served by the indexer, who registers Lejeune Dirichlet under the second element of his surname, distinguishes ‘Philodemus’ from ‘Philodemus of Gadara’, and, because at p. 93, n. 7 Athenaeus is misprinted ‘Athenaus’, lists the latter separately, complete with book number, as ‘Athenaus XIV’, like the Roi Soleil or mad King Erik of Sweden.
21 Or rather, the second sharpening of A; cf. above, n. 7.
22 The internal structure was (ascending) tone, two conjunct tetrachords, tone, two conjunct tetrachords; in the diatonic genus this yielded the surface pattern of a modern minor key; tone, semitone, tone, tone, semitone, tone, tone, tone, semitone, tone, tone, semitone, tone, tone.
23 Likewise, the diatonic Lydian species corresponds to our major scale, our Lydian with B♮ matches the Greek diatonic Hypolydian.
24 This equation, implicit in the use of synemmenon for b♭, is taken for granted in Ciconia's Nova Musica; that he wrongly supposed it to be Phrygian need not concern us.
25 I prefer ‘positional’ and ‘functional’, with their corresponding adverbs, to ‘thetic’ and ‘dynamic’, less because the one baffles and the other misleads the Greekless (Hagel, p. 103), but because the former will invite the reader with Greek to think of the opposition ϕὐσει ‘by nature’ ∼ θέσει ‘by positive determination’, which indeed Gaudentius uses in this very context, so that his ϕύσει corresponds to Ptolemy's κατὰ θέσιν.
26 And likewise that Lydian mese was b or a little lower, hence that its conventional transcription ought to be d′.
27 The most frequently attested note, as Hagel observes (p. 453), in our musical remains.
28 Besides, the modern meaning is closer to the original metaphor of colouring (p. 448, n. 7).
29 A citharist (κιθαριστής, feminine κιθαρίστρια) simply plays the cithara, whereas a citharode (κιθαρῳδός) also sings to it.
30 The ninth string of the enneachord, a tone below hypate; in the double octave called ὑπάτων διάτονος.
31 Aristoxenus had denied that even human beings and instruments taken together could reach so far; but music must have advanced since his day, for Greek notation extends from E to g′.
32 A similar conception of a double octave within a triple may inform Aristeides Quintilian's statement (1. 10, p. 21, ll. 13–16 Winnington-Ingram) that the Dorian τόνος is sung entire because the voice serves all twelve tones (meaning the double octave), and because its proslambanomenos (B♭) lies a fourth above its Hypodorian counterpart (F); although it is two tones lower than Lydian, Hagel is not unduly perturbed by the contradiction ‘[e]specially if the pitch of the system was not perfectly fixed’ (p. 72), but demonstrates that while most attested notes fall within the Lydian double octave, nearly all do within the Dorian.
33 He then contradicts himself by adding: ‘then descending the Hypolydian tetrachord and next likewise ascending the Hyperlydian’. But Hagel's and our concern is not with this man's muddled mind, but with the facts that have been confused.
34 Ed. Blackburn, Bonnie J. and Lowinsky, Edward E., ‘Luigi Zenobi and his Letter on the Perfect Musician’, Studi Musicali, 22 (1993), pp. 61–114
, at 82, §10, translated ‘of the same timbre throughout’ (p. 99); cf. commentary, p. 92.
35 At p. 90, ll. 16, 23, f and c, which pertain to the octave below middle C, ought to have been F and C.
36 So the MS, as clearly visible in Hagel's photograph (p. 123), though scholars generally accent ὁρμασία in accordance with the apparent etymology; the word is attested nowhere else in Greek. Did the scribe know better, or did he, contemplating his table, think of αἱμασιά, ‘dry-stone wall’?
37 Although he acknowledged that the wider πυκνόν preferred by some was still agreed to be enharmonic, in his own classification the division of the tetrachord into ⅜, ⅜ and 1¾ tone was chromatic (specifically the hemiolic ‘shade’ or χρόα).
38 Harmonic Elements 1. 20, p. 26, ll. 5–7 da Rios, understood as the maximum range for any one person or instrument; Ctesibius had not yet invented the hydraulis. The Pythagoreans allowed the music of the spheres a range of four octaves and a major sixth, or 27 : 1, considerably less than that of modern keyboard instruments.
39 To be sure it is not superpartient but multiply superparticular, being of the form (an + 1) : n, specifically διπλασιημιόλιος or duplex sesquialtera, (2 × 2 + 1) : 2.
40 Hagel, in accordance with his frequent practice, transcribes these notes a fourth lower, contradicted by the Greek notation (which he himself has added, since Ptolemy never uses it).
41 i.e. nete synemmenon considered in a disjunctive context, forming an octave with hyperhypate.
42 At p. 409, n. 102 Hagel observes that the inclusion of the tritone precludes a reference in Gaudentius to minor resonance in the third; but the specification of this third demonstrates that it was not like other ditones, such as that between parhypate meson and mese.
43 The date of the play's first performance, when, left unfinished by Euripides, it was patched up by his like-named son; there are signs of a major revision for a probably fourth-century revival. The Oxford editor, James Diggle, regards vv. 1500–9 (the first extract on the papyrus) as ‘perhaps not Euripidean’, vv. 784–94 as ‘hardly Euripidean’, without distinguishing between Euripides junior and a later poet as the likely author; the Loeb editor, David Kovacs, rejects vv. 1500–9 and also 784, but accepts vv. 785–94 as the work of one or both Euripidai, treating the original production as their joint achievement.
44 In principle, new music might have been composed for an existing text; but the Orestes papyrus suggests otherwise.
45 When inferring from Pseudo-Plutarch's report of Lamprocles' reanalysis that this ἁρμονία must have existed before the last quarter of the fifth century (p. 375), Hagel might have cited the genuine Plutarch's anecdote (On Listening to Lectures 15, 46 b) that Euripides, at a rehearsal, rebuked a member of the chorus for sniggering when he sang μιξολυδιστί.
46 We are surprised to read on p. 418: ‘For those who are not convinced by such organological considerations, here are some general thoughts’ (sc. on why the three-quartertone enharmonic should be older than the semitonal). Plausible as the general thoughts are, they remain aprioristic, overturnable by good hard fact; but the borings of such pre-Pronomian auloi as we have, and the unhandiness of the seven-stringed lyre for tuning to Aristoxenus' enharmonic, are harder to dismiss.
47 At p. 402 ϕαίνεται δ' Ὄλυμπος αὐξήσας μουσικήν is mistranslated ‘It seem s that Olympus furthered music’ (= ϕαίνεται … αὐξῆσαι) instead of ‘It is manifest that’.
48 The Pythagorean Thrasyllus conceived the enharmonic in this manner (pp. 159–60); in the sixteenth century Vicentino too left out the quartertones.
49 Hagel's English retains a few infelicities and a few Germanisms: the worst respective instances occur on p. 406, where in two consecutive sentences ‘supplanted’ appears to be used for ‘substituted’ and ‘must not’ for ‘need not’. At p. 66, l. 4 up, ‘Hyper’- should be ‘Hypo-’; p. 105, l. 3: clearer would have been ‘the lowest string becomes the functional mésē’. There are indexes of ancient sources and of personal names (none later than Boethius), but unaccountably and inexcusably not of subjects.