Homogenization of a spectral problem in a bounded domain with a high
contrast in both stiffness and density is considered. For a special
critical scaling, two-scale asymptotic expansions for eigenvalues
and eigenfunctions are constructed. Two-scale limit equations are
derived and relate to certain non-standard self-adjoint operators.
In particular they explicitly display the first two terms in the
asymptotic expansion for the eigenvalues, with a surprising bound
for the error of order $\varepsilon^{5/4}$ proved.