Listen to this, as the frequency goes up, splits into multiple tones, and then turns into chaos, briefly reintegrates, and then turns back into chaos. You might also like this version, where I've simplified to pure semi tones (i.e. the keys on a piano).

[UPDATE: New personal favourite - in C major - much more dramatic.]

The logistic map is probably the simplest and most celebrated math lab example of chaos.

It's a pretty simple function *f*. There's a control parameter *r*. When you take a number, say 0.5 and compute f(0.5) and then f(f(0.5)) and so on, interesting things happen. When the parameter *r* is low, you quickly end up at a fixed value, some point p where f(p) = p, so the iteration just stays there. When you increase *r* however, a lot of stuff happens - first a split, so the iteration flip flops between two values, and then that happens again into four values and so on. Above a certain value of *r* you reach chaos. This famous image shows the fixed points and chaos of the iteration for values of r.

The image however is static - you don't get a feel for how the dynamics of the iteration hops around on the image.

I was curious how that sounds, so I made this Pure Data patch and took a slow slide up the chaos scale. The result is above.