¹ Cf. NA 18.10 and my remarks at Aulus Gellius (London, 1988), 224 and n. 50Google Scholar.

² Cf. Theol. Arith. 55.4–5 De Falcod ξ' λγεται τς πρώτης συμφωνας ριθμς εἶναι τς δι δ' δ^γ [4:3].

³ See now von Staden, Heinrich, Herophilus: The Art of Medicine in Early Alexandria (Cambridge, 1989), 276–84, 346–61, 391–3Google Scholar.

⁴ Both Roscher, W. H., Die Hebdomadenlehre der griechischen Philosophen und Ärzte (Abh. K. Sächs. Ges. Wiss., ph.-hist. Kl. 24/6 [1906]), 139Google Scholar and von Staden 279 n. 136 refer the text to rhythm without more ado. To be sure extant Greek authors offer no support and Herophilus likened the healthy adult pulse to the spondee, in even rhythm (δι ἵσου): ‘Ruf. Eph.’ Syn. puls. 4.5–6, p. 225.2–6 Daremberg-Ruelle—cf. Aulus Gellius 234 n. 4, where for ‘2/2’ read ‘even minims in 4/4’ and correct the Ar. Quint, reference to 2.14 (p. 82.25–8 W.-I.). But it is no objection that Aristoxenus did not recognize 4:3 as rhythmical (El. Rhythm. 2.35), since medical writers were not bound by his pronouncements; he equally rejected 5:2, which Galen allowed (see below). The extent of Herophilus' debt to Aristoxenus is unclear (von Staden 278–9, but cf. 391).

⁵ See West, M. L., Ancient Greek Music (Oxford, 1992), 244 with nn. 64–5Google Scholar.

⁶ Bk. 1, fann (‘fen’) 2, ta'līm (‘doctrina’) 3, jumla (‘summa’) 1, fasl (‘capui’) 1; ed. Institute of History of Medicine and Medical Research (New Delhi, 1982-), i.202–3; Gerard, e.g. Padua, 1476 edn., sigs. e4^v–e5^r. French trans, by Shiloah, Amnon, ‘“Ên-Kol”—Commentaire hébraïque de Šem Tov ibn Šaprût sur le Canon d'Avicenne’, Yuval: Studies of the Jewish Music Research Centre, 3 (1974), 267–87 at 272–3Google Scholar; for this reference and that in n. 12 I am indebted to Dr Bonnie Blackburn, and for helpful comments to Dr Charles Burnett and the anonymous referee for this journal.

⁷ muttafiq = ‘consonant’, but is also used more loosely; here applied to rhythms of even periodicity (Shiloah, 279–80, §16 n. a). The antonym is mutanāfir; Avicenna's privative corresponds to σμφωνα. Dr Burnett points out that elsewhere dissonance may be ihtilāf, which in this text denotes the variation of pulse (category 8), itself either orderly or disordered.

⁸ al-madkūra; but Gerard of Cremona renders ‘…nominatarum aut non nominatarum. nominatarum uero…’.

⁹ Lit. ‘the relation of the whole and five’, a blend of nisbat al-kull wa-hums (‘relation of the whole and a fifth [part]’ = 6:5) and al-nisba ‘llati bi-’l-kull wa-'l-hamsa, ‘the relation that is through the whole and five’.

¹⁰ This proportion generates the major third of just intonation, of little account in ancient music theory; ‘que proportio non est proprie musicalis’, de Abano, Peter, Conciliator differentiarum philosophorum et precipue medicorum (Pavia, 1490)Google Scholar, sig. [r6]^ra (note that ‘Ga. in de pulsuum compendio’, ibid., sig. r4^ra, is ‘Ruf. Eph.’ Syn. puls.), but see Ptol, . Harm. 1.13, p. 31.1Google Scholar Düring; 1.15, pp. 35.3–4, 36.31–2 (10:9 × 9:8); 2.13, p. 68.20–22.

¹¹ tumma lā yahuss = postea non sentitur (Gerard), cf. μετ τατα similarly used at Ptol, . Harm. 1.15 (p. 34.3 During)Google Scholar. Ibn Šaprût, in his commentary though not his translation, declares that 5:4 cannot be perceived ‘à cause de sa grande subtilité’ (larôb daqqôtô).

¹² Siraisi, Nancy G., ‘The Music of Pulse in the Writings of Italian Academic Physicians (Fourteenth and Fifteenth Centuries)’, Speculum, 50 (1975), 689–710, esp. 693–4, 699–700CrossRefGoogle Scholar.

¹³ It is uncertain whether we should measure dilation against contraction, dilation and pause against contraction and pause, or dilation against contraction and both pauses: see Galen, , περ διαγνώσεως σφυγμν 3.3 (viii.911–13)Google Scholar, but cf. below, n. 19.

¹⁴ Nicomachus, Ench. 4; Boeth. Inst. mus. 1.3; the former also considers how the sound of wind-instruments is affected by intensity of breath and the dimensions of bore and holes.

¹⁵ Iacobi Foroliuiensis medici singularis expositio et quaestiones in primum Canonem Avicennae (Venice, 1547)Google Scholar, fo. 132^Va, a misconstruction of Gerard's ‘proportio suorum temporum in uelocitate et spissitudine est sicut proportio sonorum eius’.

¹⁶ Arist. De anima 420^a29–^b4; for other views Archytas 47 B 1 DK sub fin.; PI. Tim. 80A; Thphr, . Sens. 85Google Scholar; Ps.-Arist. Probl. 899^a27–8, 900^b7–14; Nicomachus and Boethius locc. citt. Cf. Towey, Alan, ‘Aristotle and Alexander on Hearing and Instantaneous Change: A Dilemma in Aristotle's Account of Hearing’, in Burnett, Charles, Fend, Michael, and Gouk, Penelope (edd.), The Second Sense: Studies in Hearing and Musical Judgement from Antiquity to the Seventeenth Century (Warburg Institute Surveys and Texts, XXII [London, 1991]), 7–18Google Scholar; Burnett, ‘Sound and its Perception in the Middle Ages’, ibid. 43–69 at 64–5.

¹⁷ Whence Isid, . Etym. 3.17.3Google Scholar; cf. Siraisi 702. Whereas Aug, . De musica 6.3.4Google Scholar speaks only of rhythm, the speaker of Mart. Cap. 9.926 is called Harmonia (cf. Serv./DS Aen. 12.394, 397) rather than Musica to show her cosmic significance (cf. Cristante, edn. of bk. 9, pp. 13–18).

¹⁸ Alex. Eph, . SH 21Google Scholar, Plin, . NH 2.84Google Scholar, Cens. 13.3–5, Fav. Eul. 25.1–2, Mart. Cap. 2. 169–98. The Romans follow Varro, who apparently assigned the doctrine to Pythagoras; the additive conception of intervals is not the only ground for disbelief: Heath, T. L., Aristarchus of Samos (Oxford, 1913), 113–15Google Scholar.

¹⁹ Applied to the pulse, this would entail comparing the time from the beginning of one contraction to the end of the next with that from the beginning of the first to that of the second.

²⁰ Retaining αὐτος against Jahn's αὑτος, cf. τν μιταων οἰ τοτοις παρακεμενοι, with Mathiesen, Thomas G., Aristides Quintilianus: On Music, in Three Books (New Haven, CT, 1983), 65, 172Google Scholar, and Barker, Andrew, Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge, 1989), 506Google Scholar.

²¹ In Ptolemy's syntonic diatonic, though not in Pythagorean tuning nor in Didymus', a major sixth (3:2 × 10:9).

²² Respectively Johannes de Sarto, in the canonic inscription to the tenor of his motet Romanorum rex (composed for the death of Albrecht II in 1439, and previously thought to be by Johannes Brassart), and Johannes Tinctoris, in his Terminorum musicae diffinilorium (written c. 1473, published at Treviso, c. 1495; facs. Leipzig, 1983), sigs. [A5]^r, [A5]^v, [A6]^v. See now Blackburn, Bonnie J., Lowinsky, Edward E., and Miller, Clement A., A Correspondence of Renaissance Musicians (Oxford, 1991), 658 with nn. 19–20, 663–4Google Scholar, and for the resulting ambiguity ibid. 695 with n. 28. (The proportions refer to the overall time-relation, e.g. ‘three breves in the time of four/two’; hence 3:4 makes the music slower, 3:2 faster.)

²³ Cf. Jacopo of Forlì, fo. 132^vb: ‘in musica proportio dupla uocatur dyapeson [sic], sexquialtera uero diapente, sexquitertia uero diatessaron.’

²⁴ Cf. Aulus Gellius, 169 n. 26; Guillaumin, J.-Y., ‘La terminologie latine de la série des épimores’, RPh³ 53 (1989), 105–9Google Scholar; see too Aug, . De mus. 2.10.18Google Scholar. Evidently Gellius had not read Cic, . Tim. 22–3Google Scholar; for subsesquitertia = ὑπεπτριτος see Boeth, . Inst. arith. 1.24 (p. 49.24 Friedlein)Google Scholar.

²⁵ Of the examples given, 15:10, 30:20; 12:9, 40:30, only 12:9 is of musical interest, being part of the intervallic formula 12:9:8:6. Some commentators refer 18.15.2 to a division of the octave into 5 + 7 semitones, representing respectively the 4:3 and 3:2 relations of the preceding chapter; but even if the coarse blend of Aristoxenean and Pythagorean conceptions were no objection, cf. n. 18, ‘ratione quadam geometrica’ suits better with the 3, 4, 5 triangle of Aug, . De mus. 5.12.26Google Scholar. See my ‘Parva Gelliana’, CQ (forthcoming).

²⁶ Cf. Aulus Gellius, 233–4.