Harmony

Pythagorean–Platonic musical mathematics was transmitted to scholars during the Latin Middle Ages principally through the following texts: Calcidius's late-fourth-century Latin translation of most of the Timaeus (up to Timaeus 53c) and his accompanying commentary on that text; Macrobius's fifth-century commentary on Cicero's Somnium Scipionis (Scipio's dream); Martianus Capella's De nuptiis philologiae et Mercurii, quoted above; book 2 of Cassiodorus's sixth-century Institutiones divinarum et humanum litterarum (Institutions); book 3 of Isidore of Seville's sixth-century Etymologiae (Etymologies); and finally the extremely important work of Boethius (c. 480–c. 524), De institutione musica (The fundaments of music), a learned statement of the Neopythagorean and Neoplatonic traditions in music probably based on the work of the Greek writers Nicomachus of Gerasa (fl. c. 100 C.E.) and Ptolemy (2nd century C.E.). After affirming the correctness of Plato's assertion that the soul of the universe is joined according to musical concord (Boethius, book 1, chapter 1), Boethius formulates an evocative triadic version of the tradition (Boethius, book 1, chapter 2): musica mun-dana (cosmic music), the principle of the universe controlling planetary motion, seasons, and elemental combinations; musica humana (human music), the integrating force between body and soul; and musica instrumentalis (instrumental music), the music produced by string, wind, and percussion instruments.

By about the middle of the ninth century, during the Carolingian era, writers on the subject of music began to produce treatises that contained large excerpts from the writers cited above, but that attempted to place the Pythagorean–Platonic harmonic tradition within a Christian framework and adapt it to the current need to codify and regulate the performance of liturgical chant within the mass and the holy office. For example, Aurelian of Réôme in his Musica disciplina (The discipline of music), written around the middle of the ninth century, justifies the classification of liturgical chant into eight different modes by referring to the seven planetary orbits plus the outer starry sphere (now referred to as the Zodiac), exactly as laid out in Plato. Referring to his Latin authorities (Boethius, Cassiodorus, Isidore), he writes that "the whole theory of the art of music consists of numbers … Music has to do with numbers that are abstract, mobile, and in proportion" (Aurelian, Musica disciplina, chapter 8, pp. 22–23).

The last truly original and complete statement of Pythagorean-Platonic speculative harmony was given in the Renaissance by the German astronomer and writer on music Johannes Kepler (1571–1630), who in his treatise Harmonice mundi (The harmony of the world) posits a world created by God in accordance with the archetypal harmonies represented in the principal musical consonances. The tradition, however, is clearly implicit in subsequent works, such as Die Welt als Wille und Vorstellung (The world as will and representation) by Arthur Schopenhauer (1788–1860 ), and explicit in the work of the twentieth-century Swiss aesthetician Hans Kayser (1891–1964). Kayser's principal contribution was a harmonic theory, informed by a close reading of Kepler's Harmonice mundi, according to which the measurement of the intervallic properties of sound could also serve as an exact measurement of feeling. Finally, late-twentieth-century "superstring theory," a cosmic physical theory that, in principle, is capable of describing all physical phenomena, is clearly Pythagorean in essence and scope:

Music has long since provided the metaphors of choice for those puzzling over questions of cosmic concern. From the ancient Pythagorean "music of the spheres" to the "harmonies of nature" that have guided inquiry through the ages we have collectively sought the song of nature in the gentle wanderings of celestial bodies and the riotous fulminations of subatomic particles. With the discovery of superstring theory, musical metaphors take on a startling reality, for the theory suggests that the microscopic landscape is suffused with tiny strings whose vibrational patterns orchestrate the evolution of the cosmos. The winds of change, according to superstring theory, gust through an aeolian universe. (Greene, p. 135)