  A minor change to the standard epistemic logical language, replacing $K_{i}$
with $K_{\node{i,t}}$ where $t$ is a time instance, gives rise to a generalized
and more expressive form of knowledge and common knowledge operators. We
investigate the communication structures that are necessary for such
generalized epistemic states to arise, and the inter-agent coordination tasks
that require such knowledge. Previous work has established a relation between
linear event ordering and nested knowledge, and between simultaneous event
occurrences and common knowledge. In the new, extended, formalism, epistemic
necessity is decoupled from temporal necessity. Nested knowledge and event
ordering are shown to be related even when the nesting order does not match the
temporal order of occurrence. The generalized form of common knowledge does
{\em not} correspond to simultaneity. Rather, it corresponds to a notion of
tight coordination, of which simultaneity is an instance.
