Given a multimodal interval map f, I will show the existence and uniqueness of the `natural equilibrium states', equilibrium states for the `natural potential' -t\log|Df|, for the largest possible range of t. This also gives the smoothness of the pressure function t--> P(-t\log|Df|), a graph which provides a good deal of information about the dynamics. One byproduct of this can be put in terms of a converse to Ledrappier's theorem on SRB measures: if the pressure function is not smooth at t=1 then there is an SRB measure. This is joint work with Godofredo Iommi.