  We show that the Koszul calculus of a preprojective algebra, whose graph is
distinct from A$\_1$ and A$\_2$, vanishes in any (co)homological degree $p>2$.
Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to
its (higher) homological calculus, by exchanging degrees $p$ and $2-p$, and we
prove a generalised version of the 2-Calabi-Yau property. For the ADE Dynkin
graphs, the preprojective algebras are not Koszul and they are not Calabi-Yau
in the sense of Ginzburg's definition, but they satisfy our generalised
Calabi-Yau property and we say that they are Koszul complex Calabi-Yau
(Kc-Calabi-Yau) of dimension $2$. For Kc-Calabi-Yau (quadratic) algebras of any
dimension, defined in terms of derived categories, we prove a Poincar\'e Van
den Bergh duality theorem. We compute explicitly the Koszul calculus of
preprojective algebras for the ADE Dynkin graphs.
