  A Poisson geometry arising from maximal commutative subalgebras is studied. A
spectral sequence convergent to Hochschild homology with coefficients in a
bimodule is presented. It depends on the choice of a maximal commutative
subalgebra inducing appropriate filtrations. Its E^{2}_{p,q}-groups are
computed in terms of canonical homology with values in a Poisson module defined
by a given bimodule and a maximal commutative subalgebra.
