  The family of Thom spectra $y(n)$ interpolate between the sphere spectrum and
the mod two Eilenberg-MacLane spectrum. Computations of Mahowald, Ravenel, and
Shick and the authors show that the $E_1$ ring spectrum $y(n)$ has chromatic
complexity $n$. We show that topological periodic cyclic homology of $y(n)$ has
chromatic complexity $n+1$. This gives evidence that topological periodic
cyclic homology shifts chromatic height at all chromatic heights, supporting a
variant of the Ausoni--Rognes red-shift conjecture. We also show that relative
algebraic K-theory, topological cyclic homology, and topological negative
cyclic homology of $y(n)$ at least preserve chromatic complexity.
