Self Similarity

Theories, Musings & Observations

"When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar."
- Mandelbrot

How Long is the Coastline of Britain?

Mandelbrot's definition of self similarity means that each piece is similar to the whole - not necessarily identical to the whole. His classic example of self similarity is the coastline of Britain. From a satellite view, it looks jagged. If you zoom in to 10,000 feet, it looks, well, pretty similarly jagged. If you zoom in to 5,000 feet, the coastline looks similarly jagged yet again. And so on and so forth. The concept of self similarity doesn't require that the coastline looked exactly the same from each altitude, only that it was similar in its texture, irregularity, or coarseness. How jagged it is. One way of looking at the fractal dimension, D, is that it measure of how coarse a self-similar shape is.

Self-similarity in a coastline. One of Mandelbrot's many examples of self-similarity found in nature.

Examples of Self-similar Fractal Shapes

In the following examples, the concept of self-similarity is used to produce extremely complex shapes from very simple shapes. This parallels the approach the Fractal Sequencer uses to produce complex music from simple phrases, which is covered in detail on the Approach page.

The first five stages of Mandelbrot's Plate 141.

Various stages of Mandelbrot's magnificent Plate 68.

Western Music from 30,000 Feet...

Throughout this site, and throughout many other sites devoted to fractal music, it is asserted that music is self-similar. Let me attempt to explain this...

Consider what tonal music is, in its broadest sense. If you zoom in and look at its smallest component, the note, you see that the note's main attribute is its tone. If you zoom out a bit, you will see short 'phrases' are constructed of notes. The notes in these phrases are usually connected by tonality - typically, they are derived from the same chord.

If you zoom out yet further, you will notice that the phrases form 'sentences'. In each sentence, the tonality varries from phrase to phrase until the sentence ultimately resolves itself. Sentences, in turn, vary in tonality and construction to produce a movement. Movements vary in tonality and construction to produce a song. Although the terminology is different, you see a similar pattern in classical, rock, jazz - any musical style. Western music itself, from 30,000 feet, is a collection of varied tonal constructions of greater and lesser dimensions.

The view from 30,000 feet is not the exact same as the view up close. From 30,000 feet, you will notice that there are common song structures underlying most Western music. Up close, you see what someone has done with the song structure, what unique attributes they have imposed. Historically, these song structures have made songs easy to follow. These common underlying patterns help listeners and performers anticipate what is coming.

Song Structure - 12 Bar Blues

Probably the most common song structure in existence, 12-bar blues exhibits self similarity in and of itself. Envision an initiator, A, and a rule, B-A. Apply the rule to itself (flipping the second substitution, F->F, -F), you would get something that looks like CBAC. If you string together the first three iterations, the pattern looks something like A-BA-CBAC. If you stretch the first two iterations to 4 measures to obtain three sequences of equal length, you get AAAA-BBAA-CBAC, which looks remarkably like 12-bar blues: EEEE-AAEE-BAEB. An intriguing fact is that a fourth is an inverted fifth - one 5 half-steps up, the other 5 half-steps down. Another intriguing fact is that the dominant 7th is 2 iterations x 5 = 10 half steps up.

Song Structure - Sonata Form

Sonata form is another song structure that is frequently used. Although sonata form varies significantly from usage to usage, it is usually described as an A-B-A song structure. Within each component of the song, variations are introduced before you are led back home, e.g., A-A'-A--B-B'-B--A-A'-A. Although this is a gross oversimplification of sonata form, there is a kernel of truth here that ties sonata form to self-similarity, allowing many layers of tonal resolution in a fairly compact song structure.

Bach's Canons & Fugues

Probably the most blatant self-similarity in Western music is found throughout Bach's Canons and Fugues. From Hofstadter's classic, Godel, Escher, Bach: "A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards."

Notice a trend here?

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