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
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The fractal geometry of music is a fascinating concept that explores the mathematical principles and structures underlying musical compositions. Fractals are complex geometric shapes or patterns that exhibit self-similarity at different scales. In the context of music, fractal geometry can be applied to analyze various aspects of musical composition, performance, and perception. Here’s how fractal geometry relates to music:
1. **Self-Similarity**: One of the key characteristics of fractals is self-similarity, meaning that the same pattern or structure repeats at different scales within the overall composition. In music, self-similar patterns can be observed in melodies, rhythms, harmonies, and musical forms. For example, a musical motif or theme may appear in different variations throughout a composition, creating a sense of coherence and unity.
2. **Fractal Dimension**: Fractal dimension is a measure of the complexity or irregularity of a fractal pattern. In music, the fractal dimension can be used to quantify the degree of complexity or randomness in a piece of music. Compositions with higher fractal dimensions may exhibit greater diversity and unpredictability in their musical elements, while those with lower fractal dimensions may have more repetitive or structured patterns.
3. **Fractal Analysis**: Fractal analysis techniques can be applied to analyze musical compositions and performances, providing insights into their underlying structures and patterns. Researchers have used fractal analysis to study various aspects of music, including rhythm, pitch, dynamics, and timbre. By quantifying the fractal properties of music, researchers can gain a deeper understanding of how musical elements are organized and perceived by listeners.
4. **Fractal Music Generation**: Fractal geometry can also be used as a creative tool for generating new musical compositions. Fractal algorithms and mathematical models can generate musical patterns and sequences based on iterative processes and recursive functions. These generated musical structures can exhibit fractal properties such as self-similarity and complexity, resulting in unique and intriguing compositions.
5. **Perception and Aesthetics**: The fractal geometry of music can influence how listeners perceive and appreciate musical compositions. Some researchers suggest that fractal patterns in music may enhance listeners’ engagement and aesthetic experience by creating a sense of depth, richness, and coherence. Fractal structures in music can evoke feelings of wonder, curiosity, and aesthetic pleasure, contributing to the emotional impact and artistic value of the composition.
Overall, the exploration of fractal geometry in music offers a fascinating interdisciplinary perspective that bridges the realms of mathematics, science, and art. By uncovering the fractal patterns and structures inherent in musical compositions, researchers and musicians can gain new insights into the nature of music and its profound connections to the principles of fractal geometry.