  The provability logic of a theory $T$ captures the structural behavior of
formalized provability in $T$ as provable in $T$ itself. Like provability, one
can formalize the notion of relative interpretability giving rise to
interpretability logics. Where provability logics are the same for all
moderately sound theories of some minimal strength, interpretability logics do
show variations.
  The logic IL(All) is defined as the collection of modal principles that are
provable in any moderately sound theory of some minimal strength. In this paper
we raise the previously known lower bound of IL(All) by exhibiting two series
of principles which are shown to be provable in any such theory. Moreover, we
compute the collection of frame conditions for both series.
