Iterated maps: self-similarity and fractals

One practical way of generating new structures is by means of iterating a procedure. We illustrate this with the construction of a famous fractal. Many fractals can be obtained by performing one transformation on an object, and then applying it again to the resulting sub-units, and so on.

Take, for instance, an equilateral triangle and divide each side into three segments of equal length. Then for each side draw an equilateral triangle in the middle segment, as in the figure:


The contour of the outer edge forms a star, for which the same procedure can be applied to each of the six resulting triangles. Continuing this an infinite number of times gives rise to the fractal known as the ‘Koch snowflake’.

There are several ways in which this procedure can be applied to music, for instance if the melodic contour is thought of as a curve. As a matter of fact, using fractals to generate music has become quite popular in recent years.

One simple example would be to take the three motif sequence of notes:

cefg, edcf, gagc

where each fragment corresponds to a side of the triangle above. The next level of iteration would then be

ce cefg fg, ed edef cf, ga gagc gc.

Another approach would be to assign a musical equivalent value of some kind to the fundamental triangle shape; this value could reduce in size or amplitude as the shape expands outwards in six directions.

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