Gauss-Manin connection and noncommutative calculus, revisited

We review and revise the constructions of noncommutative calculus and of the Gauss-Manin superconnection in noncommutative geometry. In particular, we provide explicit formulas that are intriguing in various respects. Namely, they seem to have good convergence properties, both $p$-adically and in Archimedean metrics; and they look very much like something from microlocal analysis and mathematical physics (where the role of the Planck constant is played by the formal parameter $u$ from cyclic homology theory).