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.

Case study: Laurence Glazier’s Golden Spiral Method

Composer and mathematician Laurence Glazier has experimented extensively with a fractal compositional technique which he calls the ‘Golden Spiral Method’. As Laurence explains:

“The method grows a musical composition by an iterative expanding process, starting from a single bar. A succession of compositions is made. Each consists of two parts, one longer than the other by the Fibonacci ratio, approximately 1.62, and is a template for the smaller part of the next, longer composition.

These compositions all have structural integrity and are written out as chorales. The final stage has counterpoint, colour and texture added. This is what is heard in performance, but while the listener expects music, like experience, to unfold from a beginning, tell a story, and then end, music produced by this method is not sequential in origin, but a result of scaleable fractal production.

In practice this method has been used to compose several works, most of which have been performed. It is, however, extremely labour-intensive, as to create the performed music, several transitional compositions are built along the way.”

Learn more about Glazier’s work here:

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