Sounds that bikes make are difficult to describe. "Fingernails on a chalk board", so sort of a high pitched dry scraping noise, not a squeal, squeak or scream?
That sounds like something ceramic, stone or metal rubbing against the metal rim to me. The last loud noise where it then goes quiet could be whatever it is getting turned just before getting entirely dislodged. Does that sound about right?
I'm not an expert, but my best guesses:
- Poorly made brake pads made with chunks of stuff in them.
- You're picking up some sort of grit that's getting stuck in the brake pad.
The consistent pattern points much more to the pads being manufactured that way. But I've had similar kinds of sounds when I picked up a bit of sand on the rims.
Brake pads aren't solid rubber/poly-whatever, the're actually a compound, typically with a sort of ceramic or metallic material embedded in the rubber-like material. Poorly made brake pads can have large chunks of that material instead of consistent fine particles.
Either way, when the noise starts, examine the brake pads. You'll probably see something embedded in the pad that you can pry out with something pointy (screwdriver, tip of a knife) or scrape out with a razorblade. It's quite possible it's something the size of a grain of sand, not a huge chunk.
And if it definitely looks like it's something that the pad was manufactured with in it, consider replacing the pads with a different brand; It's likely that your rims are getting worn faster.