Deresonating a Tate period

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.