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-og linear plot.Music until the 17th century was one ofthe 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.Scientific experiments have revealed the relation between note-interval and vibrational frequency produced by an instrument. We obtain an octave-higher note by doubling the sound frequency, which can be achieved by halving the length of a string. There are 12 notes in an octave in our diatonic music; i.e., the frequency difference is divided by 12 equal intervals (i) so that f /f = (2.0)1/12= 1.05946 = (15.9/15). This relation is well known among musicians, that the ratio of acoustic frequencies between successive notes, f ' and ff, is approximately 16/15. The ratio of acoustic frequencies I between any two successive music notes of an interval i is Ii= 2'/12 = (15 9/15)i, [1] where i in an integer,-ranging from 1 to 12, in the diatonic music. A semitone is represented by i = 1, a tone by i = 2, a small third by i = 3, etc. The numerical value of Ii is approximately a ratio of integers. Some notes have a ratio of small integers. A fourth (i = 5), for example, has an I5 value of 1.3382, or a ratio of about 4/3; a fifth (i = 7) has a value of 1.5036, or a ratio of about 3/2. Those used to be considered consonant tones (1). Others are represented by a ratio of larger integers. A diminished fifth (i = 6), for example, has a ratio of 1.4185 (= 10/7.05). This is not a ratio of small integers; it is not even an accurate approximation of 10/7, and this note has been traditionally considered dissonant (1).Music can be defined as an ordered arrangement of single sounds of different frequency in succession (melody), of sounds in combination (harmony), and of sounds spaced in a temporal succession (rhythm). Melody is supposedly "a series of single notes deliberately arranged in a pattern and chosen from a preexisting series that has been handed down by tradition or is accepted as a convention." Theory of harmony has taught us that the ...

Self-similarity of the "1/f noise" called music.

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Multiscale Fractal Analysis of Musical Instrument Signals With Application to Recognition

“…Su and Wu [27] applied Hurst exponent and Fourier analysis in sequences of musical notes and noted that music shares similar fractal properties with the fractional Brownian motion. Properties of self-similarity, regarding the acoustic frequency of the signals, were observed in [14], where aspects of fractal geometry were studied. Given this previous evidence of fractal properties in music, such as the fractional Brownian motion, the use of fractal and multifractal dimension for genre classification, and evidences of self-similarity properties found on musical tones, we wish to further explore whether multiscale fractal analysis could manifest supplementary facts about the structure of music signals, taking into account that such methods 1 The term 'fractal' was coined by Mandelbrot from the Latin word fractus, meaning "broken", to describe objects that are irregular (or "fragmented") to fit within the traditional geometry [16].…”

Counterpoint: The Musical Analogy, Periodicity, and Rural Urban Dynamics

“…Among her noteworthy collaborators were evolutionary biologist Stephen Jay Gould, physicist Roger Penrose, and mathematician Benoit Mandelbrot, whose fractal geometry had by then emerged as the go-to analogy for understanding principles of self-similarity in the new age of complexity. In response to a paper called The Fractal Geometry of Music (Hsu and Hsu 1990), in which a series of fractal metaphors were suggested to exist in the music of Bach, Tureck penned a detailed response nearly as long as the FIgure 12.4 Musical analogy. The musical analogy is among the most enduring of conceptual models for architecture.…”