Fractal geometry of music

Abstract

 

Music critics have compared Bach’s music to the precision of mathematics. What “mathematics” and what “precision” are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot’s fractal geometry and the precision is the deviation from a log-log linear plot.

 

Content

 

Music until the 17th century was one of the four mathematical disciplines of the quadrivium beside arithmetic, geometry, and astronomy. The cause of consonance, in terms of Aristotelian analysis, was stated to be numerous sonorus, or harmonic number. That the ratio 2:1 produces the octave, and 3:2 produces the fifth, was known since the time of Pythagoras. Numerologists of the Middle Ages speculated on the mythical significance of numbers in music. Vincenzo Galilei, father of Galileo, was the first to make an attempt to demythify the numerology of music (1). He pointed out that the octave can be obtained through different ratios of 2″:1. It is 2:1 in terms of string length, 4:1 in terms of weights attached to the strings, which is inversely related to the cross-section of the string, and 8:1 in terms of volume of sound-producing bodies, such as organ pipes.

http://www.pnas.org/content/87/3/938.full.pdf

PDF – PNAS-1990-Hs-938-41

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