From watching other "singing Tesla coil" videos, my understanding is that the music is actually played by the Tesla coil. Think of how what appears to be a single lightning bolt typically has ~20 strokes back and forth between the ground and the sky. In similar fashion, they've set up the coil so that it's actually discharging hundreds of times per second but to us (and to the video recording equipment) appears to be on continuously. Each discharge makes a little snapping sound and in aggregate you get something like a square wave playing music.