  We study the finite model property of subframe logics with expressible
transitive reflexive closure modality. For $m>0$, let $\mathrm{L}_m$ be the
logic given by axiom $\lozenge^{m+1} p\to \lozenge p\vee p$. We construct
filtrations for the logics $\mathrm{L}_m$. It follows that these logics and
their tense counterparts have the finite model property. Then we show that
every canonical subframe logic that contains $\mathrm{L}_m$ have the finite
model property.
