For an elliptic operator with high-contrast periodic matrix-inclusion coefficients in \(\mathbb {R}^n\) , \(n\ge 2\) , we obtain improved results on uniform approximations of typical Floquet-Bloch eigenvalues in terms of those of an explicit two-scale limit operator. As a result, we obtain not only improved rates for convergence of the spectra to the limit spectrum, that contains band gaps, but also improved uniform estimates for an explicit asymptotics for the integrated density of states.