The gyrotropic eigenmodes in magnetic vortex crystals are studied by means of broadband ferromagnetic-resonance measurements. An artificial crystal structure is realized with two staggered ellipses as building block. Making use of the anisotropic field dependence of the vortex resonance frequency in a ferromagnetic ellipse, distinct resonance modes can be observed that are easily tunable via a small magnetic bias field. The gap between the modes can be continuously adjusted. In the presence of dipolar interaction, a bending and shifting of the modes is observed. A rigid particle model is adopted to analytically describe the experimental results. Calculations for the crystal basis reveal an anticrossing of the resonance modes depending on the vortex-core polarization configuration.