The ATtiny13 chips finally arrived, and I can test the code from the "tiny13" branch on real hardware (find it at https://bitbucket.org/thesheep/nyan/src/tiny13). After some struggling with getting my USBASP programmer to work with the chip (I needed a newer firmware and a -B option for avrdude to slow the clock), I got it flashed and playing:

It's a bit too fast, but changing the clock definition in the Makefile to 1200000 fixed that. The tone is probably wrong too, but I'm tone-deaf, so I don't care.

The difference from the regular Nyan Board is that it uses Timer0 (because there is no Timer1 on ATtiny13), and a different pin for the speaker. If you look at the six cats on the PCB, you will see that one is slightly different -- that's the ATtiny13 version.

I still have one more chip to try for the Nyan Board, but that is going to be a surprise.