In my precalculus class, I have spent the last month learning about fractals. Fractals are non-regular geometric shapes that have the same degree of non-regularity on all scales. In other words, they are never-ending patterns, demonstrating self-similarity on infinitely small scales. No matter how far you zoom in or out on a fractal, it will always appear the exact same as it did originally.
To me, one of the most intriguing aspects of fractals is their ubiquitous nature. The natural world is full of fractals, from jagged coastlines to vasculature to heads of Romanesco broccoli. Fractal theory is used to produce striking computer graphics, analyze seismic patterns, diagnose diseases, and much more. I, however, am most interested in how fractals can also appear in music.
Fractal music can be thought of as a single phrase – whether a melodic pattern or a rhythmic one – combining to form larger groups of phrases, which themselves combine to form still larger phrases. This process can be repeated – or iterated – a few times or infinitely to produce a piece of fractal music. Each group or phrase reflects the form of the whole in terms of pitch, rhythm, or speed. This is known as self-similarity.
A fascinating example of fractal music was observed by mathematician Harlan Brothers in Bourrée Part I from Johann Sebastian Bach's Cello Suite No. 3. The piece starts off with two eighth notes and a quarter note (m1), repeats that pattern (m2), then continues with a phrase (m3) that is twice as long. The same pattern of short, short, long (s1) is repeated (s2), followed by a longer sequence (s3).
The first eight measures are then repeated, producing two "short" sections that are followed by a 20-measure "long" section.
The structure of this piece resembles that of a classic type of fractal known as a Cantor set. In this type of structure, you start with a line segment and remove its middle third. Then you remove the middle third again. You continue iterating this process by removing the middle third from the prior lines.. The result is called a "Cantor comb."
Harlan concludes, "The fact that Bach was born almost three centuries before the formal concept of fractals came into existence may well indicate an intuitive affinity for fractal structure.”
This raises a fascinating question: why are humans innately drawn to fractal structures in music? Richard Taylor, a physicist at the University of Oregon, might have uncovered the answer. Since we know that nature is full of fractals, Taylor believes that when we look at fractal images or listen to fractal music, our brains recognize its kinship to the natural world, producing profoundly soothing and mood-boosting effects.
To test this hypothesis, Taylor used electroencephalogram (EEG) to study participants’ brain waves while viewing geometric fractal images. Indeed, he discovered that their frontal lobes produced the brainwaves of a relaxed state even after looking at the images for just one minute. Further, using functional MRI studies, he identified that fractals activate the ventrolateral cortex (associated with high-level visual processing) and the dorsolateral cortex (responsible for encoding spatial long-term memory). Notably, fractals also engage the parahippocampus which is involved with regulating emotions; this same structure is also highly active while listening to music.
I’ll close with links to a few clips of fractal music for you to enjoy… and maybe you’ll begin to notice self-similarity in your own favorite music!
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