"The Pythagoreans... were fascinated by certain specific ratios, and especially by those which relate what we call today the arithmetic [A = \frac{a + b}{2}], geometric [G = \sqrt{ab}], and harmonic [H = \frac{2ab}{a + b}] means. ... The particular ratios between these means in which the Pythagoreans were interested were...A:G = G:HThe other relationship is expressed bya:A = H:bThe Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the ns by Pythagoras himself after having been taken prisoner in Egypt. s lay at the heart of the Pythagorean theory of music. If a string is divided into 12 parts, the ratio 12:6, or 2:1, gives us the octave. If the arithmetic and harmonic means of 12 and 6 are now taken, we haveA = \frac{6 + 12}{2} = 9andH = \frac{2 \times 6 \times 12}{6 + 12} = 8The [perfect proportions] 9:6 and 12:8 both equal to 3:2, correspond to the fifth in the theory of music. Similarly... 8:6 and 12:9, both equal to 4:3, corresponding to the fourth. In this way, certain intervals, crucial in the theory of music, were all obtained by ratios involving the numbers 1, 2, 3, and 4, which came to be of mystical significance for they also represented the perfect triangle, yielding the 10, the sum of 1, 2, 3, and 4. These ratios of the musical fifth and fourth were used by the Pythagoreans to obtain the whole tone of the ,\frac{3}{2} : \frac{4}{3} = 9:8and the semi-tone,\frac{4}{3} : (\frac{9}{8})^2 = 256:243."