Fractal Patterns in Herbie Hancock's Solo on "Orbits"
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Chapter 1: Introduction to Fractal Music
Fractal music exemplifies how mathematical principles can influence artistic expression. Renowned jazz musician Herbie Hancock once remarked, "It pulled me like a magnet, jazz did, because it was a way that I could express myself." One striking illustration of this concept is found in Hancock's solo on Wayne Shorter's composition "Orbits," which is featured on the iconic Miles Davis album "Miles Smiles." The track boasts an impressive ensemble, including:
- Miles Davis - trumpet
- Wayne Shorter - tenor sax
- Herbie Hancock - piano
- Ron Carter - bass
- Tony Williams - drums
In this analysis, we will delve into the intricacies of Hancock's solo. Its fluidity and improvisational style reveal a fractal quality in how intervals evolve, particularly showcasing a power-law relationship in the distribution of intervallic distances.
Chapter 2: Understanding Power-Laws in Music
Power-laws are integral to the study of fractal geometry, capturing relationships between two variables where one is proportional to the power of the other. A classic example in biology is the allometry of metabolic rates: metabolic rate = k⋅(weight)^(3/4). These relationships pervade various phenomena in both natural and social sciences, from biodiversity to the distribution of wealth.
Power-laws also manifest in music through elements such as pitch, duration, and rhythmic variation. They can create fractal structures known as motivic scaling, where motifs are repeated across different time scales.
Section 2.1: Fractal Music Defined
A search for "fractal music" often yields diverse interpretations. Many refer to compositions generated by mapping fractal images to musical elements like pitches and dynamics. However, true fractal music must possess measurable scaling properties. Benoit Mandelbrot suggested that music could be fractal in a quantifiable sense, distinguishing it from mere pixel-to-sound translations.
Fractal music has historical roots, with motivic scaling evident in the works of Flemish composers like Johannes Ciconia and Josquin des Prez, marking some of the earliest examples of this phenomenon.
Section 2.2: Detecting Power-Law Characteristics
To identify a power-law in a series of melodic intervals, we can count the number of intervals at each size. If we plot these counts on a log-log scale, a linear trend indicates melodic interval scaling. The slope of this line corresponds to the dimension of the set of elements, establishing a direct connection between melody and intervallic distribution.
Chapter 3: Analyzing Hancock's Solo
For Hancock's piano solo, I utilized pitch data from The Jazzomat Research Project. The intervals were calculated using the formula 2^(k/12), where k represents the number of semitones.
Binning techniques can enhance our analysis by grouping intervals of similar sizes, thus revealing underlying power-law relationships. A linear regression on this binned data yields an excellent fit, suggesting a fractal structure.
Chapter 4: Conclusion and Reflection
In studying music through this lens, we acknowledge the limitations of our datasets. Nonetheless, a strong relationship exists between melodic character and its intervallic distribution. As we observe the self-similarity in fractal music, it becomes apparent that the greatest artists resonate with the fractal nature of the world.
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