The self-similarity of a real fern is limited. But the fact that a fern has self-similar characteristics means that we can create a fern-like object with a few mathematical rules, such as the Barnsley fern.
Start with a simple line drawing:
Replace each of the lines with the entire drawing, at the scale of that line:
And eventually you will have an object which is self-similar at many scales. These objects tend to have a certain naturalness or harmony about them.
If we use a fractal that's in one dimension, such as the Cantor Set, then we can read off the points of the fractal as rhythmic elements, as if it were a score over time. By varying the ratios of this fractal, we can create more interesting rhythms. This is the core of Fractal Music Machine's algorithm.
The video below shows Fractal Music Machine creating a fractal. It takes pattern of notes (Recursive depth 1) and, for each note, repeats the whole pattern (recursive depth 1->2), and does this repeatedly a few times to get a much deeper pattern. The pattern of notes includes frequencies, which are multiplied by each other to obtain new frequencies as the pattern evolves.
Here is a small selection of some other interesting links to fractal music experiments and research: