We propose a "phase diagram" for particulate systems with purely repulsive contact forces, such as granular media and colloids. We characterize two classes of behavior as a function of the input kinetic energy per degree of freedom T-0 and packing fraction deviation from jamming onset Delta phi = phi - phi(J) using simulations of frictionless disks. Isocoordinated solids (ICS) exist above jamming; they possess an average contact number equal to the isostatic value z(iso). ICS display "strict" harmonic response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of sharp peaks at eigenfrequencies omega(d)(k) of the dynamical matrix. In contrast, hypocoordinated solids (HCS) occur above and below jamming and possess fluctuating networks of interparticle contacts but do not undergo cage-breaking particle rearrangements. The density of vibrational frequencies for the HCS is not a collection of sharp peaks at omega(d)(k), but it does possess a common form over a range of Lambda phi and T-0.