This paper studies the eigenvalues problem associated with the vibrations of a composite medium assumed to have a periodic microstructure. The investigation concentrates on the first-order correction to the homogenized eigenvalues. A major difficulty arises due to the interaction of the periodic microstructure with the boundary of the medium. The way in which the boundary intersects the underlying periodic structure strongly influences the value of the correction. A rigorous derivation is given of the eigenvalue correction formula. The properties of the eigenvalue correction are studied in one- and two-dimensional numerical calculations.