Fractal Music

What makes something Fractal?

Often fractals are characterized as objects exhibiting some type of scaling symmetry.
However, as the filled-in circle shows (area scales as radius2), scaling alone does not imply fractality.

The simplest fractals are self-similar with respect to scale, by which we mean that any such object satisfies the following three requirements:

  1. It is constructed of a collection of different-size elements whose size distribution satisfies a power-law relationship spanning at least three scales:

  1. It must comprise at least two similar regions in which the arrangement of elements either mirrors or imitates the structure of the object as a whole.

  1. Its features must possess sufficient detail that the overall structure cannot be more easily explained in Euclidean terms.

Evenly subdivided square Sierpinski Carpet
Not fractal Fractal

Although the blue and red outlines in the left-hand figure mark smaller copies of the large square, the overall shape can be adequately described in terms of an evenly subdivided square; specifically, it is not necessary to invoke the words "scaled" or "recursive" to explain, in a concise manner, how it is constructed. The structure of the pattern on the right clearly requires a more complicated description.

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