Resonances can be tuned in dielectric resonators in order to construct single-resonator, negative-index metamaterials. For example, high-contrast inclusions in the form of metallic dipoles can be used to shift the first electric resonance down (in frequency) to the first magnetic resonance, or alternatively, air splits can be used to shift the first magnetic resonance up (in frequency) near the first electric resonance. Degenerate dielectric designs become especially useful in infrared- or visible-frequency applications where the resonator sizes associated with the lack of high-permittivity materials can become of sufficient size to enable propagation of higher-order lattice modes in the resulting medium.