  We introduce a technique to compute monodromy periods in certain families of
algebraic varieties by perturbing (`deresonating') the fiberwise Betti to de
Rham comparison off the motivic setting. As an application, we find Apery
periods of Grassmannians G(2,N) and identify the Apery numbers for the
equations D3 of the Mukai threefolds with certain $L$--values. We show that the
argument of the $L$--function is 3 for the rational and 2 for the non--rational
Mukai threefolds.
