  Operadic tangent cohomology generalizes the existing theories of Harrison
cohomology, Chevalley--Eilenberg cohomology and Hochschild cohomology. These
are usually non-trivial to compute. We complement the existing computational
techniques by producing a spectral sequence that converges to the operadic
tangent cohomology of a fixed algebra. Our main technical tool is that of
filtrations arising from towers of cofibrations of algebras, which play the
same role cell attaching maps and skeletal filtrations do for topological
spaces.
  As an application, we consider the rational Adams--Hilton construction on
topological spaces, where our spectral sequence gives rise to a seemingly new
and completely algebraic description of the Serre spectral sequence, which we
also show is multiplicative and converges to the Chas--Sullivan loop product.
Finally, we consider relative Sullivan--de Rham models of a fibration $p$,
where our spectral sequence converges to the rational homotopy groups of the
identity component of the space of self-fiber homotopy equivalences of $p$.
