Modal logics often have the finite model property. However, for many applications the state space is not only finite, but of fixed size n for a positive number n. An example is reasoning about a social network where the relations can vary but the set of agents (corresponding to states) is fixed. In this paper we investigate the modal logic of frames with n states, for any positive integer n, in the basic modal language. This defines a family of modal logics parameterized by n. We give complete axiomatizations of many of these logics, and characterize the computational complexity of the basic logic.