  Virtually all of the analysis of quantum contextuality is restricted to the
case where events are represented by rank-one projectors. This restriction is
arbitrary and not motivated by physical considerations. We show here that
loosening the rank constraint opens a new realm of quantum contextuality and we
demonstrate that state-independent contextuality can even require projectors of
nonunit rank. This enables the possibility of state-independent contextuality
with less than 13 projectors, which is the established minimum for the case of
rank one. We prove that for any rank, at least 9 projectors are required.
Furthermore, in an exhaustive numerical search we find that 13 projectors are
also minimal for the cases where all projectors are uniformly of rank two or
uniformly of rank three.

Quantum Physics