  In this paper we consider the spectrum of the self-adjoint differential
operator L generated by the differential expression of order n with the m by m
periodic matrix coefficients, where n and m are respectively odd and even
integers and n>1. We prove that the number of gaps in the spectrum of L is
finite and find explicit estimation in term of coefficients for the number of
the gaps. Moreover, we find a condition on the norms of the coefficients for
which the spectrum is real axis. Besides we investigate the bands of the
spectrum and prove that most of the real axis is overlapped by m bands.
