How to soundproof a closet?
Stopping high frequencies is actually easier in a lot of circumstances than stopping low frequencies. On the other hand we’re talking about operatic soprano, which is probably the only thing more piercing than what I play (trumpet). One thing that can help is to try to eliminate echoes in your space — if you can hang drapes on the walls, or do something to kick the walls or floor/ceiling out of parallel by a little, you can keep the room resonance down and perhaps keep the level down a bit that way too. If it’s just a closet I’d honestly just start with some egg-crate foam on the walls and see if that does the trick. The shape kills reverb, the material should soak up a lot of what you’re putting out, and it’s not hideously expensive. Oh, and do what spec80 says — I’ve practiced trumpet, loudly, in apartments for 10 years and have never had a problem. I do what she says: talk with neighbor, propose hours, and give them my cell phone number in case I’m practicing when they’re home wit
Something else you might want to watch out for is what is going on behind your walls. You can soundproof your floor all you want, but if behind the existing walls are runways for plumbing or electrical lines, or any holes that are open between your floor and the downstairs for any reason, any noise or vibrations that hit your walls will find an open passageway direct to your downstairs neighbors walls. The easiest way to guard against this happening without opening up the walls is to make sure that your walls are soundproofed well enough so that no sound can get behind them.
