Silver proved that GCH can fail at a measurable cardinal; using Prikry forcing we get the failure of SCH at a singular cardinal of cofinality $\omega$. It seems that to bring that down to $\aleph_\omega$, one either uses Prikry forcing interleaved with collapse or some extender based forcing. My question is why the most naive thing--Prikry forcing followed by collapse-- not work.
Namely, suppose $\kappa=\lim_n\kappa_n$ is strong limit, $\kappa_0=\omega$, $2^\kappa=\kappa^{++}$. Let $\mathbb{P}=\prod_n\text{Col}(\kappa_n^+,\kappa_{n+1})$. By choosing $\kappa_n$s appropriately, this forcing should preserve both $\kappa$ and $\kappa^+$, using similar argument as in Easton forcing. It has size $\kappa^{++}$ so cardinals starting from $\kappa^{+++}$ are also preserved. Must it collapse $\kappa^{++}$? All I can see is that if there is a scale on $\prod_n\kappa_n^+$ of length $\kappa^+$ then the forcing seems to collapse $\prod_n\kappa_{n+1}=\kappa^{++}$.