(fl second half of the 6th century bce). Greek philosopher and religious teacher. Born on Samos, he emigrated to Croton in 531 and later settled in Metapontum in southern Italy; both moves may have been caused by political or religious persecution. He believed in the transmigration of souls, established a bios (way of life), was probably a religious leader at Croton, and emphasized the numerical underpinnings of the natural world. An extreme version of Pythagorean doctrine holds that the physical world is a material working out (representation) of numerical truth, and that this truth is immediately and easily apprehended, albeit superficially, in elementary musical consonances.
Pythagoras’s teachings, prominently publicized by Philolaus (fl second half of the 5th century bce) and further promulgated by Archytas of Tarentum (fl first half of the 4th century bce), were well known to Plato and Aristotle, by whose time Pythagoras was already a legendary figure. During the 5th century the Pythagorean community split into a scientific branch (mathēmatikoi) and a more conservative and religious one (akismatikoi). In later antiquity Pythagoras’s teachings were revived and mingled with those of Plato by Nicomachus, Iamblichus, Proclus and others. Fragments of writings on music by Philolaus and Archytas are preserved in the writings of later authors. More extended presentations of Pythagorean musical doctrine are found in the Euclidean Division of the Canon (see Euclid), the Manual of Harmonics by Nicomachus of Gerasa and the Harmonic Introduction of Gaudentius. Ancient commentaries on Pythagorean music theory, some of it critical, occurs in Ptolemy’s Harmonics (see Ptolemy, claudius) and Porphyry’s commentary on Ptolemy (see Porphyry), Theon of Smyrna’s On Mathematics Useful for the Understanding of Plato (see Theon of Smyrna) and Aristides Quintilianus’s On Music (see Aristides Quintilianus). Boethius (in De institutione musica) and other Latin writers translated Neopythagorean doctrine into Latin and transmitted it to the Western medieval world.
Certainly the most remarkable achievement attributed to Pythagoras is the discovery of the general geometric theorem that bears his name: that the square on the diagonal of a right-angle triangle equals the sum of the squares on the sides. Pythagoras’s importance for music lies in his purported establishment of the numerical basis of acoustics. On passing a blacksmith’s shop, he is said to have heard hammers of different weights striking consonant and dissonant intervals (Nicomachus, Manual of Harmonics, vi). He discovered that musical consonances were represented by the ratios that could be obtained from the musical tetractys: 1, 2, 3, 4. The ratios are relations of string lengths or frequencies. Thus 4:1 corresponds to the double octave; 3:1 to the octave plus the 5th; 2:1 (and 4:2) to the octave; 4:3 to the 4th; and 3:2 to the 5th. It follows that the interval of a whole tone – the difference between the 5th and 4th – is expressed by the ratio 9:8. A Pythagorean scale consists of 4ths subdivided into two tones plus the remainder or Limma (256:243). A 4th plus a 5th equals an octave. This scale is systematically presented in the Euclidean Division of the Canon, which may also contain a quotation from Pythagoras regarding the origin of sound. A necessary result of such a scale is that six whole-tone intervals exceed an octave by a small interval known as the Pythagorean comma (531,441:524,288).
Pythagorean music theory rests on the theory of numerical ratios presented in books 7–9 of Euclid’s Elements of Geometry and given philosophical interpretation by Nicomachus in his Introductio arithmetica. The fundamental principle of consonance for the Pythagoreans, as set forth in the Division of the Canon, holds that notes are made up of parts; pitch is raised or lowered through the addition or subtraction of percussions. Things composed of parts are related by numerical ratios, which are either multiple (i.e. the greater term contains the lesser exactly a given number of times), superparticular (the greater term contains the lesser plus one part of the lesser), superpartient (the greater contains the lesser plus more that one part of the lesser), multiple superparticular or multiple superpartient. Tacitly assuming the tetractys, the Division asserts that all consonant intervals are either multiple or superparticular.
Pythagorean doctrine also embraces theories of human harmony (between body and soul) and cosmic harmony. Book 3 of Ptolemy’s Harmonics and book 3 of Aristides Quintilianus’s On Music treat both human and cosmic harmony at length. According to the theory of the harmony of the spheres (see Music of the spheres), the distances from the earth to the visible planets and sun, as well as the speeds with which the celestial bodies circle the earth, are in the same ratios as various musical intervals, especially those of the diatonic scale. Plato’s Timaeus mentions the Pythagorean scale and contains (in chaps.xxxv–xxxvi) an early, vivid exposition of the theory of cosmic harmony, combining diatonic organization with the two fundamental celestial motions, same and other. The myth of Er, related at the end of Plato’s Republic (x, 614–18) also reflects Pythagorean influence. The pairing of Greek note names with celestial bodies varies from author to author (compare, for example, Nicomachus, Manual of Harmonics, iii, with Ptolemy, Harmonics, iii.16). Johannes Kepler made a late, complex investigation of the harmony of the spheres in Harmonices mundi (1619). See also Aristoxenus and Greece, §I, 6(i).
W. Burkert: Weisheit und Wissenschaft: Studien zu Pythagoras, Philolaos und Platon (Nuremberg, 1962; Eng. trans., rev., 1972, as Lore and Science in Ancient Pythagoreanism)
J.W. McKinnon: ‘Jubal vel Pythagoras, quis sit inventor musicae?’, MQ, lxiv (1978), 1–28
A. Barbera: ‘The Consonant Eleventh and the Expansion of the Musical Tetractys: a Study in Ancient Pythagoreanism’, JMT, xxviii (1984), 191–223
A. Barker, ed.: Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge, 1989), 28–52
A. Barbera: The Euclidean Division of the Canon: Greek and Latin Sources (Lincoln, NE, 1991)
C.A. Huffman: Philolaus of Croton: Pythagorean and Presocratic (Cambridge, 1993)
ANDRÉ BARBERA