  We obtain the generic complete eigenstructures of complex Hermitian $n\times
n$ matrix pencils with rank at most $r$ (with $r\leq n$). To do this, we prove
that the set of such pencils is the union of a finite number of bundle
closures, where each bundle is the set of complex Hermitian $n\times n$ pencils
with the same complete eigenstructure (up to the specific values of the finite
eigenvalues). We also obtain the explicit number of such bundles and their
codimension. The cases $r=n$, corresponding to general Hermitian pencils, and
$r<n$ exhibit surprising differences, since for $r<n$ the generic complete
eigenstructures can contain only real eigenvalues, while for $r=n$ they can
contain real and non-real eigenvalues. Moreover, we will see that the sign
characteristic of the real eigenvalues plays a relevant role for determining
the generic eigenstructures of Hermitian pencils.
