Human ear & Mathematics

The musical octave is universally divided into twelve semitones. As the frequency doubles in each octave, the semitones form a geometric progression with the common ratio being the twelfth root of two or approximately 1.06. (In a geometric  progression, each number is multiplied by a common ratio to get the next number in the progression.)

M. Pietsch & others in their paper "Spiral forms of the human Cochlea", write "Inner ear geometry is compared to shells of mollusks. Due to the apparent similarity to the Cochlea, the Nautilus shell has become a symbol of hearing. The Nautilus shell is a perfect example of a logarithmic Fibonacci spiral." (A logarithm is the exponent to which the base number should be raised to yield a given number. A Fibonacci sequence is one, in which each number is the sum of the two preceding ones.)

Samuel Arbesman writes in his "Fractal Musical rhythms" that "White"noise is the static we hear between radio stations. The other extreme is "Brown" music, which is simply a random walk up & down the musical scale. In between is 1/f noise, called the "Pink" noise. Most music that we listen to is 1/f noise. It has the right combination of pattern & unexpectedness, & is pleasing to the human ear. The shape of the curve described by 1/f music, has a fractal shape!

(Euclidian Geometry deals with straight lines. Analytic Geometry deals with curves. Fractal Geometry deals with irregular surfaces.)