In this paper, we propose a new generalization of the well-known Cahn-Hilliard two-phase model for the modelling of n-phase mixtures. The model is derived using the consistency principle: we require that our n-phase model exactly coincide with the classical two-phase model when only two phases are present in the system. We give conditions for the model to be well-posed. We also present numerical results (including simulations obtained when coupling the Cahn-Hilliard system with the Navier-Stokes so as to obtain a phase-field model for multiphase flows) to illustre the capability of such modelling.