In this paper, several methods of analysing vibrational properties of crystal lattice with defects are developed. Integral expressions for additive functions of normal mode frequencies are derived following the work of Montroll and his collaborators. It is shown that the Helmholtz free energy can be evaluated at high and low temperatures without performing the integrations. The methods presented are valid for lattices of all odd dimensions, although specific results are presented here for one-dimensional monatomic and diatomic lattices. Using a method similar to that develpoed by Lifshitz, it is shown that the properties of a lattice with defects can be expanded in a series of powers of the concentration of defects. The coefficient of the nth power depends on the properties of a lattice with n defects. Examples of such expansions are given. An exact expression for the frequency distribution function of a monatomic linear chain with an isotope defect is given.