 One of the most startling discoveries of early Greek science was that the fundamental intervals of music, the octave, the fourth, and the fifth, corresponded to whole number ratios of string length. Thus, if we pluck a string of length x and then a string of length 2x, we will hear the interval of an octave between the two sounds. If the two string lengths are in the ratio 4 : 3, we will hear a fourth, and, if the ratio is 3 : 2, we will hear a fifth. This discovery that the phenomena of musical sound are governed by whole number ratios must have played a central role in the Pythagorean conception, first expressed by Philolaus, that all things are known through number (DK 44 B4). The next step in harmonic theory was to describe an entire octave length scale in terms of mathematical ratios. The earliest such description of a scale is found in Philolaus fr. B6. Philolaus recognizes that, if we go up the interval of a fourth from any given note, and then up the interval of a fifth, the final note will be an octave above the first note. Thus, the octave is made up of a fourth and a fifth. In mathematical terms, the ratios that govern the fifth (3 : 2) and fourth (4 : 3) are added by multiplying the terms and thus produce an octave (3 : 2 × 4 : 3 = 12 : 6 = 2 : 1). The interval between the note that is a fourth up from the starting note and the note that is a fifth up was regarded as the basic unit of the scale, the whole tone, which corresponded to the ratio of 9 : 8 (subtraction of ratios is carried out by dividing the terms, or cross multiplying: 3 : 2 / 4 : 3 = 9 : 8). The fifth was thus regarded as a fourth plus a whole tone, and the octave can be regarded as two fourths plus a whole tone. The fourth consists of two whole tones with a remainder, which has the unlovely ratio of 256 : 243 (4 : 3 / 9 : 8 = 32 : 27 / 9 : 8 = 256 : 243). Philolaus’ scale thus consisted of the following intervals: 9 : 8, 9 : 8, 256 : 243 [these three intervals take us up a fourth], 9 : 8, 9 : 8, 9 : 8, 256 : 243 [these four intervals make up a fifth and complete the octave from our staring note]. This scale is known as the Pythagorean diatonic and is the scale that Plato adopted in the construction of the world soul in the Timaeus (36a-b).