  We develop a general algebraic framework involving "Poincar\'e--Novikov
structures" and "filtered matched pairs" to provide an abstract approach to the
barcodes associated to the homologies of interlevel sets of $\mathbb{R}$- or
$S^1$-valued Morse functions, which can then be applied to Floer-theoretic
situations where no readily apparent analogue of an interlevel set is
available. The resulting barcodes satisfy abstract versions of stability and
duality theorems, and in the case of Morse or Novikov theory they coincide with
the standard barcodes coming from interlevel persistence. In the case of
Hamiltonian Floer theory, the lengths of the bars yield multiple quantities
that are reminiscent of the spectral norm of a Hamiltonian diffeomorphism.
