  We semiclassicalise the standard notion of differential calculus in
noncommutative geometry on algebras and quantum groups. We show in the
symplectic case that the infinitesimal data for a differential calculus is a
symplectic connection, and interpret its curvature as lowest order
nonassociativity of the exterior algebra. Semiclassicalisation of the
noncommutative torus provides an example with zero curvature. In the
Poisson-Lie group case we study left-covariant infinitesimal data in terms of
partially defined preconnections. We show that the moduli space of bicovariant
infinitesimal data for quasitriangular Poisson-Lie groups has a canonical
reference point which is flat in the triangular case. Using a theorem of
Kostant, we completely determine the moduli space when the Lie algebra is
simple: the canonical preconnection is the unique point for other than sl_n,
n>2, when the moduli space is 1-dimensional. We relate the canonical
preconnection to Drinfeld twists and thereby quantise it to a super
coquasi-Hopf exterior algebra. We also discuss links with Fedosov quantisation.
