TLDR: Fractals are repeating patterns that are found in Nature. They can be described mathematically. I wondered if this could be applied to music, so decided to investigate. Conclusion: interesting, but not sure.
Table of Contents
Introduction: your sonic accompaniment:
While you read, I invite you to listen to this gem of an album: Charanjit Sing’s “Ten Ragas to a Disco Beat”. Absolutely mind-blowing fractal aesthetic chills.
Fractals
The word “fractal” was coined in the year 1978. Robert Wolfe Brooks and J. Peter Matelski studied photographs of fractals found in nature and derived a mathematical equation. Two years later, Polish-born mathematician, Benoit Mandelbrot, used IBM computers to visualise the output of the algorithm. This is known as the “Mandelbrot Set”.
The central concept is that fractals “iterate” upon themselves. That is, the output of one “turn” of the algorithm becomes input for the next “turn.”
Four facts about fractals
- They are everywhere
These fascinating repeating motifs are found almost everywhere in nature; they are infinitely reproducing and iterating patterns. Anyone who has ever used psychedelic substances will surely have seen the patterns. Mountain ranges and succulent plants, conch shells and vegetables (see left)…all allow us to marvel at the intricacy of their design and in so doing ponder whether there really is some higher rational order to this hot mess of the human Universe.
2. They are never-ending patterns
Fractals are self-similar. What that means is they are alike within themselves. What that means is that you can take a tiny part of a fractal, and it will be the same as a large part of the same fractal. Therefore, they are never-ending arrangements, no matter how big or how small they get.
To explain the mathematics simply, let’s consider compound interest: You invest €100 and earn €1. The next time interest is calculated, it’s based on the new balance of €101. So it increments a little more, because the initial input is now €101, not €100. Fractals work a bit like this.
3. They are algorithms that iterate.
If you use an algorithm to get an output “x,” and then use “x” as the starting value for the same algorithm, you have iterated. Let’s say the result is “y”. Another iteration using “y” as the starting point will give you the next result. Plug these results into something that can model them (like a computer program) and you may see the fractal pattern.
4. They are present in music.
Patterns of all sorts are present in music. Composers routinely create melodic motifs that repeat in different voices (instruments), velocities, and intensities. The leitmotif (guiding theme) is a recurring musical phrase or theme in a piece of music. The leitmotif iterates upon itself to build up a musical piece. This is especially noticeable in loop-based electronic music, and DJing, where repeating patterns are layered and filtered to create a track.
In the case of visual fractal patterns, the tools are 2-D or 3-D space (dimension) and color. In the case of music, the tools are sequences, rhythms, pitches and velocities.
Mandelbrot suggested that there exists a type of sound that remains unchanged in quality regardless of variations in playback speed. He referred to this sound as “scaling noise.”
Math, Music, Mind…
The mathematical basis of music is well established. When tuning to equal temperament, music frequencies inherently exhibit a binary nature. On a piano, if middle A is tuned at 440 Hz, then one octave down, the frequency of A is 220. One octave up, it’s 880 Hz. And so on. There is no deviation from this rule. Another frequency is either dissonant, or another note.
Except, there is a deviation. If you tune by octaves, you double or divide the note frequency by two to get the octave. But if you tune by fifths, the pattern is not so clear. The Pythagorean idea of “Music of the Spheres” , or Musica Universalis, says that the notes on a musical scale increase each time they go up an octave. So, in fact, there is a kind of mathematical iteration here.
Fractal Patterns in Music – Exploration and Experiment
This video gives a very complete description of how fractal patterns can be used in Ableton Live, or any DAW. Fractal theory perfectly complements electronic music. The magic of electronic music is in the precision of editing, both sounds and patterns. Iterations are only limited by your patience and the RAM in your workstation! However, artistic value and aesthetic cannot, and must not, be sacrificed for the sake of some mathematical concept!
This fascinating paper about fractals and music
By applying fractal tools in the study of music, researchers were surprised to discover that the self-similarity property, which is ubiquitous in nature, also exists in music. Such an observation may be regarded as the first step toward a further understanding of what music is and explaining how music simulates the harmony of nature.
https://www.researchgate.net/publication/222531523_Multifractal_analyses_of_music_sequences
Fractals and Neuromusic
My question is: how does this relate to Neuromusic?
In earlier posts about Neuromusic I have explored the many and various powers of music. There is something mysterious about music, and about our relationship to music. The fact that there isn’t even an agreed definition of music is mind-blowing.
Music affects the brain in so many different ways. Music, and sound, are everywhere. Fractal patterns are everywhere. Gentle music with repeating, overlapping patterns could be used to entrain the brain for meditation, a neuromusical application par excellence.
Many genres of electronic music (especially psy-trance and trance) use repetitive iterating motifs. Dancing and music are known to change mood very quickly; this is another neuromusical application of this concept.
Binaural beats can be accompanied by fractal imagery to create surreal, seemingly deep and complex tools to focus the mind. The objective: cognitive enhancement.
Conclusion
It is interesting to research topics like this, and imagine how they might affect us in real life. I am not sure if fractals and neuromusic have a particularly tight relationship. But, there is surely some overlap. What do you think?