  In this paper, we investigate the relationship between the Tukey order and
PCF theory, as applied to sets of regular cardinals. We show that it is
consistent that for all sets $A$ of regular cardinals that the Tukey spectrum
of $A$, denoted $\operatorname{spec}(A)$, is equal to the set of possible
cofinalities of $A$, denoted $\operatorname{pcf}(A)$; this is to be read in
light of the $\mathsf{ZFC}$ fact that
$\operatorname{pcf}(A)\subseteq\operatorname{spec}(A)$ holds for all $A$. We
also prove results about when regular limit cardinals must be in the Tukey
spectrum or must be out of the Tukey spectrum of some $A$, and we show the
relevance of these for forcings which might separate $\operatorname{spec}(A)$
from $\operatorname{pcf}(A)$. Finally, we show that the strong part of the
Tukey spectrum can be used in place of PCF-theoretic scales to lift the
existence of Jonsson algebras from below a singular to hold at its successor.
We close with a list of questions.
