(fl ?2nd half of 1st century bce). Greek music theorist. Fragments of his work survive in quotation by Porphyry and Ptolemy. Most musicological studies have hitherto tacitly assumed him to be identical with the Alexandrian grammarian and lexicographer Didymus, nicknamed ‘Chalkenteros’ (or ‘Chalcenterus’; fl second half of 1st century bce), who, according to Quintilian (i.8, 10) and Athenaeus (iv.139c), produced more than 3500 books on literary and antiquarian subjects; these included compilations of Hellenistic philology, much drawn on by later authors, although only a few fragments now survive. The qualification ho mousikos (‘the musician’), almost invariably added to the name by Ptolemy and Porphyry, suggests, however, that this identification is incorrect. Classical scholars have proposed that Didymus was a younger man of the same name, a grammarian and musician at Rome in the time of Nero, probably the Didymus who wrote a work, now lost, cited by Clement of Alexandria (Strōmateis, i.26) as Concerning Pythagorean Philosophy. The latter work may have served Ptolemy as a source in the final chapters of his Harmonics (iii.3–13).
In the preface to his commentary on Ptolemy (ed. Düring, 3.13), Porphyry cited Didymus as a primary authority. He quotes a fragment (26.6–28.6) which, he said, was from Didymus's treatise Concerning the Difference between the Pythagorean and Aristoxenian Theories of Music (5.11ff; 25.4ff). This fragment, like that from a certain Ptolemaïs of Cyrene quoted immediately before (22.25–26.6), criticizes music theory according to the criteria of reason (logos) and perception through the senses (aisthēsis).
Ptolemy (ii.13–14) discussed Didymus's doctrines of the division of the monochord and the divisions of the tetrachord. He sought to correct Didymus's theory of intervals and genera (ii.13), criticizing it as contrary to the findings of empirical observation. He tabulated the calculations of the divisions of the tetrachord made by Didymus and others, together with his own (ii.14–15). Those of Didymus are as follows: diatonic tetrachord – 9:8, 10:9, 16:15; chromatic tetrachord – 6:5, 25:24, 16:15; enharmonic tetrachord – 5:4, 31:30, 32:31.
Unlike his predecessor Eratosthenes, who had divided the diatonic tetrachord into two equal whole tones (each 9:8) and a limma (256:243), Didymus introduced a distinction in the diatonic tetrachord between a major and minor whole tone (respectively 9:8 and 10:9). The major and minor whole tone together constitute a major 3rd (5:4), previously found only in the enharmonic tetrachord of Archytas; and in including a major 3rd, the diatonic tetrachord of Didymus resembles the upper or lower tetrachord of the modern major scale (e.g. C–D–E–F, or G–A–B–c; see the table in MGG1, iii, 435–6). This tetrachord was adopted by Ptolemy, but with the positions of the major and minor whole tones reversed, as his ‘tense’ diatonic tetrachord. The difference between the major and minor tones (9:8 × 9:10 = 81:80) is known as the ‘syntonic comma’, or ‘comma of Didymus’; this is also the difference between the Pythagorean major 3rd (81:64) and the pure major 3rd (5:4).
The chromatic tetrachord of Didymus, besides a harmonic minor 3rd (6:5) and the semitone of the diatonic tetrachord (16:15), contains another, rather small semitone (25:24) that was adopted by no other Greek theorist. His enharmonic tetrachord again includes a pure major 3rd (5:4), with the remaining diatonic semitone (16:15) divided into two quarter-tones that are almost equal (32:31, 31:30). In his tunings Didymus was thus able to achieve pure major and minor 3rds while adhering strictly to the principle of superparticularity (for an explanation of the latter concept see Ptolemy).
M. Schmidt, ed.: Didymi Chalcenteri grammatici Alexandrini: fragmenta quae supersunt omnia (Leipzig, 1854/R)
H. Diels, ed.: Doxographi graeci (Berlin, 1879/R), 79–80
L. Cohn: ‘Didymos (8), (11)’, Paulys Real-Encyclopädie der classischen Alterthumswissenschaft, v (Stuttgart, 1905), 445–72, 473–4
S. Wantzloeben: Das Monochord als Instrument und als System entwicklungsgeschichtlich dargestellt (Halle, 1911)
L. Schönberger: Studien zum 1. Buch der Harmonik des Claudius Ptolemäus (Augsburg, 1914)
P. Tannéry: Mémoires scientifiques, iii (Toulouse and Paris, 1915), 68ff
R.P. Winnington-Ingram: ‘Aristoxenus and the Intervals of Greek Music’, Classical Quarterly, xxvi (1932), 195–208
I. Düring, ed. and trans.: Ptolemaios und Porphyrios über die Musik (Göteborg, 1934/R), 139, 145ff, 155ff, 267
W. Dupont: Geschichte der musikalischen Temperatur (Kassel, 1935)
J.M. Barbour: Tuning and Temperament: a Historical Survey (East Lansing, MI, 1951/R, 2/1953)
H. Husmann: Grundlagen der antiken und orientalischen Musikkultur (Berlin, 1961), esp. 15, 27, 33ff
L. Richter: Zur Wissenschaftslehre von der Musik bei Platon und Aristoteles (Berlin, 1961), 178–9
M. Vogel: Die Enharmonik der Griechen (Düsseldorf, 1963), i, esp. 33–4, 46
R. Pfeiffer: Geschichte der klassischen Philologie (Hamburg, 1970), 331ff
A. Barbera: ‘Arithmetic and Geometric Divisions of the Tetrachord’, JMT, xxii (1977), 294–323
A. Barker, ed.: Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge, 1989), 229–30, 242ff, 342ff
LUKAS RICHTER