In this talk, a generic class of metamaterials is introduced and is shown to exhibit frequency dependent double negative effective properties.  Explicit power series are developed for the dispersion relation and associated Bloch wave solutions on some frequency intervals. These intervals are explicitly determined by two distinct spectra: the Dirichlet spectrum of the Laplacian and the generalized electrostatic spectrum. Direct numerical simulations  show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial.