We show that the Mahler measure of the Jones polynomial and of the colored Jones polynomials converges under twisting in any link diagram. In this respect, the Jones polynomial, like the Alexander polynomial, behaves like hyperbolic volume under Dehn surgery. We discuss related results for pretzel knots, torus knots, and the simplest hyperbolic knots. The main proofs combine the representation theory of braid groups with linear skein theory. This is joint work with Abhijit Champanerkar.