Fractal Music

Common Errors

Google the term "fractal music" and you will find a vast array of misconceptions concerning the subject.  There are three common errors:

• Perhaps the most popular misunderstanding comes from assuming that if an image is fractal, then any music derived from it in a methodical manner must also be fractal.  Take for example, the interesting, if sometimes disconcerting, succession of notes that can be generated from images of the Sierpinski Gasket, Koch curve, and Mandelbrot set.  This type of "fractal music" is often neither and is probably more aptly described as “fractal inspired, musical sound.”  While it is possible that compositions generated in this fashion possess some inherent power-law relation, this is by no means guaranteed.

• There is also a common error of logic whereby it is assumed that because fractal patterns often emerge from iterative processes, iteration must always result in some sort of fractal structure. As the logistic map illustrates, this is clearly not the case.

• Another fundamental misunderstanding arises from the notion that self-similarity is synonymous with fractality.  Self-similarity is a necessary but insufficient condition for claiming that a structure is, indeed, fractal.  To be clear, onions, spirals, and Russian dolls are not fractal; they do not contain a minimum of two matching or similar regions in which the arrangement of elements either mirrors or imitates the structure of the object as a whole.

Finally, there is a less common mistake that occurs when one attempts to obtain a box-counting dimension based on dissimilar units of measurement such as pitch and time. While the graphs of such plots may suggest the roughness of mountains or noise-like structure, the relative scale of the axes is entirely arbitrary. Changing the relative scale would change the dimension obtained. Box-counting only works when the units of measurement are identical.

Clearly, iterative algorithms can generate interesting and, in the hands of a talented musician, pleasing compositions and musical textures. However, the burden of demonstrating specific fractal characteristics falls to those making the claim.