  We put a cochain complex structure ${CH}^*(\mathcal Z_K)$ on the cohomology
of a moment-angle complex $\mathcal Z_K$ and call the resulting cohomology the
double cohomology, ${HH}^*(\mathcal Z_K)$. We give three equivalent definitions
for the differential, and compute ${HH}^*(\mathcal Z_K)$ for a family of
simplicial complexes containing clique complexes of chordal graphs.
