So I tweeted a picture of a hyperbolic geometry earlier, as a joke.

And now I'm wondering how you'd implement hyperbolic geometry in music.

Obviously you'd define a tonal center at the locus. Superimpose it over a complex plane. The further you go from that tonal center, the note changes in smaller and smaller intervals until it is one cent from the next.

Define infinity as say, two octaves.

For each point in the complex plane, you'd have a real component, and an imaginary component.

Maybe declare the real component as hertz +- the tonal center, the imaginary component the timbre

Then, you step through a polar function around the center, weaving through the hyperbolic geometry in an intricate spirographic dance.

The note would shift tones, timbres, smoothly and continuously.

Now take snapshots of this trajectory at the nyquist rate and suddenly it becomes a discrete series of notes.

And bam: you'd have an extremely complex musical pattern, arising from simple functions on a hyperbolic plane.

Minimal complexity music, like Schmidhuber's minimal complexity art?

It'd also work with normal euclidean geometry on a complex plane, but it'd be a lot more boring.