(Note: An expanded version of this article has now been published in Rain Pro. If you’ve already read that, there won’t be much new in the following words. However, you will find uncompressed .wav files for all of the sounds mentioned, which you are free to use under the Creative Commons license detailed below.)
Following are samples of the most interesting examples of the fractals I rendered as audio files as part of my thesis research. The first question most people have asked me about my thesis is “What is a fractal?” The term was coined by Benoit Mandelbrot, and there are lots of ways of thinking about and defining fractals, but I generally conceptualize them as a structure which has an infinite level of detail, but can be described with a finite amount of information. A good example of this is the Sierpiński triangle:
Although this image obviously has a finite number of pixels, it is easy to see how an infinitely detailed shape is implied, with smaller and smaller triangles appearing as one looks closer. Also notice that the triangles at the top, lower left, and lower right corners each have the same structures as the whole shape. This is another common characteristic of fractals, self-similarity with respect to scale.
I was curious what it would sound like if a fractal structure was rendered into a sound file instead of an image, so I wrote a series of small C programs which created sound files sample-by-sample according to various (usually recursive) algorithms which described fractal structures. The first program was based on the Cantor set, a “dust” of points created by removing the middle third of a line segment, then removing the middle thirds of the two remaining segments, then the middle thirds of the four remaining segments, and so on. Here’s what the first few steps look like:
This is what my rendering sounds like:
Because fractals are infinitely detailed, it’s possible to make the sounds as long as you like. But for the moment I’ve kept to fairly short renderings. What I’m mainly interested in here is the interaction between rhythm, pitch, and timbre, which in sense are all the same thing occurring at different scales. It’s like when you put a baseball card on a bike frame so that the spokes of the wheel slap against it. When you move the wheel slowly you can hear each individual slap, but once you get going they blend into a continuous buzzing sound. These fractals are kind of like baseball cards stuck to dozens of different wheels all spinning at different speeds, so that some of them you hear as individual slaps, some you hear as buzzes, and a lot are right around the cusp. Here’s another couple fractals I made that perhaps illustrates this a bit more clearly:
The moniker “opposite-average” comes from the algorithm, which isn’t that complicated but is harder to describe conceptually, so I won’t bother with the details. I find it a bit more interesting than the Cantor Set because the sound is continuous. Still monaural (as opposed to stereo) though. Deciding how to implement a stereo algorithm was a bit trickier, but I did come up with one that turned out rather well:
By themselves these are just curiousities, but I think they have a lot of potential for use in electronic music and for creating sound effects.