For concreteness, let $A = \{\aleph_n : n < \omega\}$. We know $\max \mathrm{pcf}(A) \in [\aleph_{\omega+1},\Pi A]$. My question is, if $\Pi A$ is big (say, $\aleph_{\omega_1+1}$), then which cardinals in that interval can $\max \mathrm{pcf}(A)$ really be?

My second question, which motivates the first, is: Let $E = \{\aleph_{2n} : n < \omega\}$ and $O = \{\aleph_{2n+1} : n < \omega \}$; then is it possible for $\max \mathrm{pcf}(A) = \max \mathrm{pcf}(E) \gg \max \mathrm{pcf}(O)$ (where $\gg$ means "much larger than" in any way you care to make precise)?

If the answer to the first question is that $\max \mathrm{pcf}(A)$ can never really be that big, then the second question doesn't matter. But if it can be big, and the answer to the second question is that the even alephs and the odd alephs can have very different $\max\mathrm{pcf}$'s, I would find that somewhat disturbing.