I'm a composer who makes a lot of algorithmic music, and I wanted to make a Pi Day video for my YouTube channel about sonifying pi. However, I've seen a lot of nonsense where people make music out of the digits of pi, and that feels pretty much like making music out of a random sequence. So my thought, instead, was to make music out of a sequence that converges to pi in an interesting way.
I think it would be particularly interesting to find a sequence that doesn't just get closer and closer monotonically but has some irregularity or strange deviations. I wouldn't want it to converge particularly efficiently either, since I'd like to be able to listen to it for a while before it converges.
Does anyone have any ideas for strange sequences that converge to pi, that might be interesting to listen to?
The Leibnitz series converges very slowly, and oscillates between values above and below. There are ways to speed it up.
You could consider Archimedes' method (inscribed and circumscribed regular polygons with an increasing number of sides). The continued fraction approximation converges very quickly. You can find both those methods described on pi's wikipedia page.
You might be able to do something interesting with the algorithm that calculates the $n$th hexadecimal digit of $\pi$ without having to know all the previous digits.