
Pythagoras Greek philosopher, mathematician, and founder of the Pythagorean brotherhood that, although religious in nature, formulated principles that influenced the thought of Plato and Aristotle and contributed to the development of mathematics and Western rational philosophy (Pythagoreanism). Pythagoras migrated to southern Italy about 532 BC, apparently to escape Samos' tyrannical rule, and established his ethicopolitical academy at Croton (now Crotona). It is difficult to distinguish
Pythagoras' teachings from those of his disciples. None of his writings
has survived, and Pythagoreans invariably supported their doctrines by indiscriminately
citing their master's authority. Pythagoras, however, is generally
credited with the theory of the functional significance of numbers
in the objective world and in music. Other discoveries often attributed
to him (e.g., the incommensurability of the side and diagonal of a square,
and the Pythagorean theorem for right triangles) were probably developed
only later by the Pythagorean school. More probably the bulk of the
intellectual tradition originating with Pythagoras himself belongs to mystical
wisdom rather than to scientific scholarship. Harmony of the Spheres The astronomy of the Pythagoreans
marked an important advance in ancient scientific thought, for they were
the first to consider the earth as a globe revolving with the other planets
around a central fire. They explained the harmonious arrangement of
things as that of bodies in a single, allinclusive sphere of reality, moving
according to a numerical scheme. Because the Pythagoreans thought that
the heavenly bodies are separated from one another by intervals corresponding
to the harmonic lengths of strings, they held that the movement of the spheres
gives rise to a musical soundthe "harmony of the spheres." 
The sacred decad in particular has a cosmic significance in Pythagoreanism: its mystical name, tetraktys (meaning approximately "fourness"), implies 1 + 2 + 3 + 4 = 10; but it can also be thought of as a "perfect triangle," as in the Figure. Speculation on number and proportion led to an intuitive feeling of the harmonia ("fitting together") of the kosmos ("the beautiful order of things"); and the application of the tetraktys to the theory of music revealed a hidden order in the range of sound. Pythagoras may have referred, vaguely, to the "music of the heavens," which he alone seemed able to hear; and later Pythagoreans seem to have assumed that the distances of the heavenly bodies from the Earth somehow correspond to musical intervalsa theory that, under the influence of Platonic conceptions, resulted in the famous idea of the "harmony of the spheres." Though number to the early Pythagoreans was still a kind of cosmic matter, like the water or air proposed by the Ionians, their stress upon numerical proportions, harmony, and order comprised a decisive step toward a metaphysic in which form is the basic reality. In reviewing the accounts of
music that have characterized musical and intellectual history, it is clear
that the Pythagoreans are reborn from age to age. The German astronomer
Johannes Kepler (15711630) perpetuated, in effect, the idea of the harmony
of the spheres, attempting to relate music to planetary movement. René
Descartes (15961650), too, saw the basis of music as mathematical. He was
a faithful Platonist in his prescription of temperate rhythms and simple
melodies so that music would not produce imaginative, exciting, and hence
immoral, effects. For another philosophermathematician, the German
Gottfried von Leibniz (16461716), music reflected a universal rhythm
and mirrored a reality that was fundamentally mathematical, to be experienced
in the mind as a subconscious apprehension of numerical relationships.
