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Golden Mean, Irrational Numbers & Fibonacci Series

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...have argued, lies in art's expression of mathematical ratios, such as the golden mean, which is also the limiting ratio of the Fibonacci series (Kak). This ratio can be found in architecture, painting and even in music. Donald E. Knuth relates the golden mean to the "art of computer programming" (Kak). Leonardo Fibonacci developed his series in 1202 and it consists of a series of numbers in which each number is the sum of the two numbers preceding it (Sabine). Therefore, the Fibonacci series begins 1, 1, 2, 3, 5, 8, 13, 21, etc. There are many parallels between the Fibonacci series and the way in which the octave is constructed in Western music.
First of all, the Western octave is composed of twelve tones plus one, which is the octave itself, which makes 13 tones, which is part of the series (Sabine). Pentatonic scales have 5 tones; diatonic scales have 8 tones; and the first, third and fifth tones of a diatonic scale...

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