Simple. You either have a cracked/loose washer in the faucet or debris from those 80 year old pipes have made their way into the faucet or the shower head. Either will cause humming, and since it started suddenly either could be the most likely cause. You don't experience it at any other faucet in the house (yet), so it's not a pressure issue. A washer can be slightly cracked and flutter, enough so that it creates the resonance, but still seals enough to prevent leaks (for now anyway).
Grab your mechanics stethoscope and find out where it's the loudest-either at the faucet or the shower head.
If it's the faucet, open it up, flush it out and rebuild or replace the cartridge (assuming it's a cartridge model) and replace the washers. If it's the shower head pull it off, take it apart, soak it in CLR, replace any washers, and then put it together and back onto the stem.
The harmonics can resonate through the pipes, but it's going to be at the faucet or shower head-most likely chunks of debris from those 80 year old pipes lodged at the faucet.