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.