We study the interpolation of Hodge-Tate and de Rham periods in families of
Galois representations. Given a Galois representation on a coherent locally free sheaf
over a reduced rigid space and a bounded range of weights, we obtain a stratification of
this space by locally closed subvarieties where the Hodge-Tate and bounded de Rham
periods (within this range) form locally free sheaves. At every thickened geometric point
within one of the strata, we obtain a corresponding number of unbounded de Rham periods.
If the number of interpolated de Rham periods is the number of fixed Hodge-Tate-Sen
weights, we prove similar statements for non-reduced affinoid spaces. We also prove
strong vanishing results for higher cohomology. These results encapsulate a robust theory
of interpolation for Hodge-Tate and de Rham periods that simultaneously generalizes
results of Berger-Colmez and Kisin.