Getting Fancy

Just as images of chaos often look random to the casual observer, fractal music often sounds just as random, regardless of how complicated the underlying mathematics is. Something that sounds like a three-year old plunking on the piano would seem destined for limited release in CD stores. We can get visually stunning pictures out of the Mandelbrot Set, but is it possible to get the aural equivalent?

The first thing to point out is that it is quite possible to get pictures from the Mandelbrot Set that are not visually stunning. Consider the image to the left, which is certainly interesting, but fairly garish in it's colour scheme. Contrast it with any number of more beautiful images to be found online. The image to the right is another portion of the Mandelbrot set, calculated in exactly the same way as figure 1. The difference is the system used for assigning colour values to pixels and the detail with which each image was rendered. In the case of figure two, the result is smooth gradients and a pleasing colour scheme.

Similarly, better results can be achieved with fractal music using better algorithms for assigning calculated values to pitches. Just as there are rules for producing visually pleasing images, like "solid reds and pinks don't work well together" there are (arguably more) rules for producing pleasing music: notes should be restricted to those of a particular key, small hops between notes are far better than large ones, runs of sequential notes are usually good. There are also numerous other rules about phrasing, what sort of intervals (cadences) phrases should end with, etc. Creating music with more than one note played at a time introduces even more rules - which intervals produce harmonies, which produce dissonance, how chords should be resolved, what chords should follow other chords, etc.

Software has been produced that allow compositions that take these rules into consideration. One such program, is available from http://www.fin.ne.jp/~yokubota/mandele.shtml. The software allows a user to provide a melody and chord progression, and then fits the values taken from the Mandelbrot set into the provided framework. Examples of more "normal" sounding music produced using this technique include:

• rendition of Auld Lang Syne
• an English dance tune, Shepherd's Hey
• a piece entitled "Ragtimer", though to me it sounds less like rag-time and more like the finale of a Tchaikovsky piano concerto with all of the cadences removed.

Other fractal musicians try to improve their sound by varying the tempos of different lines, or using a variety of different instruments for different musical parts. Phil Jackson uses these techniques, which are demonstrated in a few samples at think.com.