We apply the idea of choosing new variables that are nonlinear functions of the old in order to simplify calculations of irrotational, surface gravity waves. The usual variables consist of the surface elevation and the surface potential, and the transformation to the new variables is a canonical (in Hamilton's sense) one so as to maintain the Hamiltonian structure of the theory. We further consider the approximation of linear dynamics in these new variables. This approximation scheme exactly reproduces the effects of the lowest-order nonlinearities in the usual variables, does well at higher orders, and also captures important features of short waves interacting with longer waves. We provide a physical interpretation of this transformation which is correct in the one-dimensional case, and approximately so in the two-dimensional case.