How could I make a playable LEGO piano?
Donny Chen made a LEGO piano that I believe is just large enough to play. Details are available on LEGO Ideas. You can watch him show it off in a video from Brickvention 2019. He has it motorized in that video.

From your comments, it sounds like you are interested in not simply having a piano with keys that can move, but one that actually emits proper sound. I think it will be quite difficult to do that effectively using LEGO elements (string, rubber bands, etc). The best that you can probably do to deliver sound using LEGO elements would probably be using EV3 or BOOST. This would be similar to the guitar model from the BOOST set: