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Published online by Cambridge University Press: 11 October 2013
The construction of Pythagorean musical theory rests philosophically on the foundation provided by Sectio Canonis. Indeed, the treatise may have performed this role historically too. Andrew Barker has recently contributed to this journal a discussion of the methods and aims of the Sectio—JHS ci (1981) 1—16. In so doing he has pinpointed lapses in the theoretical reckoning of the treatise, especially in the case of proposition 11 (P11). I should like to reply to Barker's article. My remarks concern the authorship and date of the treatise, the introduction, a few propositions, and ultimately the historical and philosophical settings for the Sectio.
1 Porphyrios Kommentar zur Harmonielehre des Ptolemaios, ed. Düring, I. (1932; repr. N.Y. 1980) 99–103Google Scholar.25.
2 Boethius, , De Institutione Arithmetics libri duo. De Institutione Musica libri quinque, ed. Friedlein, G. (1867; repr. Frankfurt 1966) 301.6–308.15.Google Scholar
3 For instance, in his definition of consonant notes as a blend, Boethius inserts the phrase ‘struck at the same time’, simul pulsae, referring to the individual notes that make up a consonance. The Greek equivalent, ἅμα κρούω, appears in most Pythagorean definitions but not in Sectio Canonis. See Bower's, Calvin M. discussion of this matter, ‘Boethius’ The Principles of Music, An Introduction, Translation, and Commentary’ (Ph.D. thesis, George Peabody College for Teachers 1967) 213, 440–3Google Scholar. Unlike the Sectio, furthermore, Boethius does not make the important connection between a ‘single name’ or ‘one term’ for multiple and superparticular ratios and the single blend of sound formed by two consonant notes (see below and also Aristotle, de Sensu 447a12 ff.).
4 See Bower, C. M., ‘Boethius and Nicomachus: an essay concerning the sources of De Institutione Musica’, Vivarium xvi (1978) 1–45CrossRefGoogle Scholar, and Pizzani, U., ‘Studi sulle fonti del De Institutione Musica di Boezio’, Sacris erudiri xvi (1965) 5–164.CrossRefGoogle Scholar
5 Proclus, , Prodi Diadochi in Primum Euclidis Elementarum librum Commentarii, ed. Friedlein, G. (Leipzig 1873) 69.3Google Scholar; Marinus, , Commentarius in Euclidis Data, ed. Menge, H. (Leipzig 1896) 254.20–7Google Scholar, vol. vi of Euclid, , Opera omnia, ed. Heiberg, I. L. and Menge, H..Google Scholar
6 von Jan, K., Musici Scriptores Graeci (Leipzig 1895; repr. Hildesheim 1962) 115–20Google Scholar. A new edition of Sectio Canonis is in order.
7 ‘Inauthenticité de la “Division du canon” attribuée à Euclide’, CRAI iv (1904) 439–45Google Scholar = Mémoires scientifiques, ed. Heiberg, J. L. and Zeuthen, H. G. (Paris 1915) iii 213–19.Google Scholar
8 ‘Republic 530c–531c: another look at Plato and the Pythagoreans’, AJP cii (1981) 395–410.Google Scholar
9 Mathiesen, T. J., ‘An annotated translation of Euclid's division of a monochord’, J. Music Theory xix (1975) n. 34.Google Scholar
10 Boethius devotes much of the third book of his de Musica to a repudiation of Aristoxenian theory. See also my ‘Interpreting an arithmetical error in Boethius's De Institutione Musica (iii 14–16)’Google Scholar, Archives Internationales d'histoire des sciences xxxi (1981) 26–41.Google Scholar
11 To some extent, this argument depends on the modern critical editions of the three versions, one of which—Porphyry's commentary—may be in need of considerable revision.
12 Nicomachus, , Enchiridion, in Jan, Mus. Script. Gr. 235–65Google Scholar. Ptolemy, , Die Harmonielehre des Klaudios Ptolemaios, ed. Düring, I. (1930; repr. N.Y. 1980) 13–14.Google Scholar
13 ‘Die Harmonielehre des Pythagoreer’, Hermes lxxviii (1943) 175.Google Scholar
14 In this respect, see Burkert, W., Lore and Science in Ancient Pythagoreanism, trans. Minar, E. L. Jr, (Cambridge, Mass. 1972).Google Scholar
15 For a detailed presentation of the Pythagorean order of ratios, see: Nicomachus, , Introductionis Arithmeticae ed. Hoche, R. (Leipzig 1886) 44.8–72 and 119.9–144.19Google Scholar, and Theon of Smyrna, Expositio rerum mathematicorum ad legendum Platonem utilium, ed. Hiller, E. (Leipzig 1878)Google Scholar. See also Barbera (n. 8) 406 n. 29.
16 Lippman, E. A., Musical Thought in Ancient Greece (New York 1965) 154.Google Scholar
17 Mathiesen (n. 9) n. 12.
18 Burkert (n. 14) 444–5.
19 See e.g. Nicomachus, , Enchiridion, ed. Jan, 255–65Google Scholar; Gaudentius, , Introduction to Harmonics, ed. Jan, 343–5Google Scholar; and Boethius, , De Musica iv 3–13Google Scholar, ed. Friedlein 308–37.
20 In late antiquity, for instance, see Theon of Smyrna, Expositio 58.13HillerGoogle Scholar. For the Pythagorean oaths involving the tetractys, see: Aëtius, , Placita i 3.8Google Scholar, in Diels, H., Doxographi graeci4 (1879) 181Google Scholar; Iamblicus, , De Vita Pythagorica, ed. Deubner, L. (Leipzig 1937) 47.15–16, 85.4–5Google Scholar; Sextus Empiricus, Adversus Mathematicos iv 2, ed. Mau, J. (Leipzig 1954) iii 133.16–17Google Scholar; and Theon, , Expositio 94.6–7Google Scholar. See also: Delatte, A., Études sur la littérature pythagoricienne (1915; repr. Geneva 1974) 253 ff.Google Scholar; Kucharski, P., Étude sur la doctrine pythagoricienne de la létrade (Paris 1952) 75–7.Google Scholar
21 Boethius, , De Musica ii 27Google Scholar, and Plutarch, , On the E(psilon) at Delphi, Mor. 389d–e.Google Scholar
22 See my ‘The consonant eleventh and the expansion of the musical tetractys: a study in ancient Pythagoreanism’, y. Musk Theory (forthcoming).
