Exploring Fractal Dimensions in John Coltrane's Giant Steps
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Chapter 1: The Fractal Nature of Music
The "Nature of Fractal Music" series delves into the captivating world of John Coltrane's famed solo in "Giant Steps." This piece is a prime illustration of how power-laws operate within music.
The exploration of Coltrane's music began during my time at the Berklee College of Music, where I transitioned from rock to jazz. Influenced by artists like Chick Corea and Wes Montgomery, I found Coltrane's artistry particularly enchanting. His improvisational skills, especially in "Giant Steps," recorded in 1959, are unparalleled and remain a cornerstone of jazz.
The composition features 26 chord changes over a 16-bar structure, with the tonal center shifting among three keys (B, G, and Eb) a total of ten times. The rapid tempo of around 291 beats per minute sets a high bar for jazz improvisation.
Coltrane himself described the song's title as a reflection of its bass line's unique movement, which deviates from traditional patterns.
Section 1.1: Understanding Power-Laws
Years ago, while attending the Yale Fractal Geometry Workshops, I pondered whether the fractal structures I'd noted in Bach's music could also be found in Coltrane's work. I discovered that his solo not only flows melodically but also embodies power-law characteristics in its interval distribution.
Power-laws are essential in fractal geometry, establishing a relationship where one variable scales with a power of another. They are prevalent in various natural and social phenomena, from galaxy distributions to wealth inequality.
Fractal structures in music can manifest in various ways, such as pitch, interval size, duration, and rhythmic variations, often appearing as motivic scaling—where melodic themes repeat across different time frames.
The Visualization of Coltrane's Solo
I was intrigued to visualize the inherent power-law within Coltrane's solo through animation. The resulting video captures the entire band's performance, allowing viewers to see the solo's structure unfold dynamically. Each instrument's data is represented as a time series, where the height indicates note value and the length signifies duration.
This visualization provides a clear view of the distribution of the orange saxophone notes, revealing a pattern of smaller intervals punctuated by larger leaps. The relationship between the count of intervals and their size can be effectively described using a power-law.
Chapter 2: Measuring Fractal Dimensions
To analyze the melodic intervals in Coltrane's solo, we count the number of intervals of varying sizes. A log-log plot of these counts against their respective sizes should yield a straight line if the sequence demonstrates melodic interval scaling. This line's slope can help us infer the dimension of the musical elements.
In Equation 1, where y = Log(N) and x = Log(1/s), the linear equation y = mx + b can be interpreted, with the slope representing the dimension of the examined set.
Benoit Mandelbrot believed in the measurable fractal nature of music, suggesting that music exhibiting quantifiable power-laws could be classified as "fractal music." However, this term should be reserved for music with clear scaling properties rather than simply mapping pixels to sound.
Fractal characteristics have been present in music for centuries, with early examples emerging from Flemish composers like Johannes Ciconia and Josquin des Prez. Coltrane's "Giant Steps" reflects this self-similar distribution, demonstrating how fractal structures can be observed in music.
Section 2.1: Analyzing Coltrane's Melodic Moments
For my analysis of the saxophone solo, I utilized pitch data from The Jazzomat Research Project. The intervals were calculated based on equal temperament, revealing a fractal dimension of approximately 8.3.
The data can be further smoothed through binning, where we group intervals into equal ranges. This technique aids in recognizing trends within the data, which can reveal power-law characteristics.
A binned interval distribution displayed a fractal dimension of around 8, with an improved fit in terms of R² value, indicating a respectable correlation.
Closing Thoughts
As we analyze music through this lens, we must remember that our datasets are relatively small, leading us to hypothesize a strong link between melody and intervallic distribution. Great music, like that of Bach or the Beatles, often reveals a timeless quality, and Coltrane's work embodies this principle. His music reflects the fractal nature of the world, as he experienced it.
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