  Given a filtration of a commutative monoid $A$ in a symmetric monoidal stable
model category $\mathcal{C}$, we construct a spectral sequence analogous to the
May spectral sequence whose input is the higher order topological Hochschild
homology of the associated graded commutative monoid of $A$, and whose output
is the higher order topological Hochschild homology of $A$. We then construct
examples of such filtrations and derive some consequences: for example, given a
connective commutative graded ring $R$, we get an upper bound on the size of
the $THH$-groups of $E_{\infty}$-ring spectra $A$ such that $\pi_*(A) \cong R$.
