(fl first half of the 4th century bce). Mathematician, music theorist and inventor. A friend of Plato, he may have been taught by Philolaus, the first man known to have publicized Pythagorean discoveries widely. Although no extended writing by Archytas survives, fragments attributed to him are contained or summarized in the works of others. He may have been the first author to establish the subjects of the Quadrivium (geometry, arithmetic, astronomy and music). He also expounded a theory of acoustics that associated pitch with the speed of sound as it passed through the air, noting that sounds arriving swiftly and strongly appear high-pitched, whereas those arriving slowly and weakly appear low-pitched (Diels, 47b1).
Archytas presents the three mathematical means of music (Diels, 47b2): arithmetic [(a+b)÷2], geometric [√ab] and subcontrary or harmonic [2ab÷(a+b)]. The geometric mean divides a musical interval exactly in half. Accordingly, Pythagorean music theory uses it to characterize the octave as the mean interval between the double octave and the unison. The arithmetic and harmonic means, since they always produce rational numbers provided that the original terms are rational, have the potential for wider application in music theory. Within an octave, the arithmetic mean determines the frequency ratio of the ascending 5th and the harmonic mean determines that of the ascending 4th.
Boethius (De institutione musica, iii.11) attributes to Archytas a proof that no mean falls proportionately between the terms of a superparticular ratio. This fundamental precept of Pythagorean music theory is also demonstrated in the third proposition of the Euclidean Division of the Canon. Both proofs rely on propositions established in the numerical books of Euclid’s Elements of Geometry. Ptolemy (Harmonics, i.13) reports and discusses Archytas’s divisions of the tetrachord into three genera. Descending from mesē to hypatē, the intervals are enharmonic (5:4, 36:35, 28:27), chromatic (32:27, 243:224, 28:27) and diatonic (9:8, 8:7, 28:27).
Archytas’s other achievements included, apparently, a solution to the Delian problem of doubling the cube, and the construction of both a mechanical wooden dove that could fly and a child’s rattle.
H. Diels, ed.: Die Fragmente der Vorsokratiker (Berlin, 1903, rev. 6/1951–2/R by W. Kranz; Eng. trans., 1948, 2/1959)
A. Barker, ed.: Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge, 1989), 39–52
C.M. Bower, trans.: Boethius: Fundamentals of Music (New Haven, CT, 1989), 103–5
A. Barbera: The Euclidean Division of the Canon: Greek and Latin Sources (Lincoln, NE, 1991), 58–60, 124–7
A. Barker: ‘Ptolemy’s Pythagoreans, Archytas, and Plato’s Conception of Mathematics’, Phronesis, xxxix (1994), 113–35
ANDRÉ BARBERA