  We investigate the spectral properties of the maximal operator $A$ associated
with a differential expression $\frac 1 w(-\frac d {dx}(p\frac d {dx}) + q)$
with real-valued periodic coefficients $w$, $p$ and $q$ where $w$ changes sign.
It turns out that the non-real spectrum of $A$ is bounded, symmetric with
respect to the real axis and consists of a finite number of analytic curves.
The real spectrum is band-shaped and neither bounded from above nor from below.
We characterize the finite spectral singularities of $A$ and prove that there
is only a finite number of them. Finally, we provide a condition on the
coefficients which ensures that $\infty$ is not a spectral singularity of $A$.
