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The word "fractal" means "broken" and originally referred to fractional dimensions and irregular surfaces. Today the word is heavily associated with computer generated images that attempt to display a representation of the complex relationships of fractal geometry. Fractal geometry is a branch of what is popularly called chaos theory - the science of nonlinear relationships. The human consciousness tends to revolve around the objects and relationships that it can readily understand - the earth to most people is a sphere. In reality, however, nature is infinitely more complex than this. Most equations that describe nature do not plot onto a straight line (linear equations) but are nonlinear and essentially have no solution. If what the study of nonlinear equations and fractal geometry is telling us is to be believed, the earth is not a sphere but an irregular shape of infinite complexity with coastlines of infinite length. Perspective plays an important part in understanding the significance of this. While a photo of the earth from space makes it look like a near perfect sphere or ellipsoid, the closer you get to it the less like a sphere it becomes. Its coastlines are infinite for the same reason - they are only as long as the unit that you use to measure them. Do it again with a smaller unit and you will arrive at a higher measurement. As there is no smallest possible unit the length is therefore infinite.
So how does any of this apply to music? Anyone who has listened to music while watching fractals cycle will surely have been struck by how well the two go together. Both shift and change with a just imperceptible inner purpose. Why does a particular piece of music reach into your soul? Why are fractals aesthetically pleasing? It is possible that the human senses find more pleasure and interest in the fractal rather than the geometric? Trees are commonly considered more pleasing than buildings and the murmur of the sea makes more enjoyable listening than the repetitive hum of machinery.
"Why is geometry often described as cold and dry?
One reason lies in its inability to describe the shape of a cloud, a mountain,
a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines
are not circles, and bark is not smooth, nor does lightning travel in a straight
line...Nature exhibits not simply a higher degree but an altogether different
level of complexity."
Complexity is another link between fractals and music. A mandelbrot fractal can be viewed at any magnification and the same complex patterns can be seen although they are never the same. Nature is similar - the smaller the scale you examine it with,the more elaborate and complex it seems to get. Music is no different. If the rules for creating pleasing, original music were simple or fixed we would not need composers - a computer could do it for us. Even where computers are used in music it is often necessary to humanise the result (by introducing small timing errors or processing the sound to add higher harmonics, reverb etc.) so that it does not sound too mechanical.
There are two ways to approach music and fractals. The first is to look at the fractal nature that already exists in music and the second is to look at how fractals can be used to create new music.
Fractals In Existing Music
Although the study of fractals in relation to music is comparatively new the relationship itself has always existed. The research that has been done suggests that the self-smilarity and complexity of nonlinear mathematics is at the very heart of musical form.
Creating New Music With Fractals
Clearly the relationship between fractals and music is not a simple one. If it was we could simply generate music on a computer in the same way that we create images of fractals. Experiments at doing this have created some interesting sounds but none are what the majority of people would identify with music. A few musicians and listeners would probably disagree strongly with this and indeed there has been a tendency in recent years towards ambient music whose soundscapes are often almost fractal in nature. The background music for this page was generated using fractals so you can make your own mind up. There are also a number of links to fractal music on the links page.
Creating Order From Chaos
Often art and the creative process is about imposing some kind of order onto the chaos of nature and this is perhaps a good way to approach fractal generated music. Just as mandelbrot fractal images and phase space portraits are expressions or representations of natural relationships, so music generated from equations is not chaos itself but rather a pattern imposed on it in order to see and make sense of it. The colours in a fractal image represent different cutoff points for various equations and similarly if we wish to make music from chaos we must apply some more or less arbitrary rules.
Just as we must pick a starting colour for generating a fractal image, so we must pick starting notes and scales if we want to have some kind of order in a musical composition. To what extent we do this is up to the individual. A single starting pitch may suffice but including some kind of key element will produce more pleasing results.
Instrumentation is another area that may need to be pinned down. If we are being purists we might confine the piece to a single instrument or let some aspect of the equations change the instrument. If we are looking for more mainstream results however we may cheat and edit the tracks of data in sequencer, picking instruments that suit the results we have obtained.
The final thing to consider is timing. The structure of most human music is really fairly ordered and even rigid. Allowing too much non-linearity to influence the timing of a piece may leave it sounding a little too disturbed for the average human ear. This, of course, may be just what you had intended.
Creating something beautiful and meaningful from the raw chaos that nature provides is not a new pursuit. I would recommend that that the use of fractal formulae in the creation of music should be approached in the same way as using any other tool in an artistic endeavour. It has it's uses but it will seldom do all the work for you - it is skillful use of tools that results in masterpieces. If you are partial to abstract, experimental music then stretch the tool to its full potential but if you are looking to create more widely accepted compositions then you will need to set some boundaries. Using nonlinear results to pick linear values will often prove more rewarding (i.e. let the result of the formula pick the length or pitch of the next note rather than actually be the length or pitch of the next note.