Scales at אω

We show that it is consistent relative to a supercompact cardinal that $${N_\omega }$$ is a strong limit, $${2^{{N_\omega }}} = {N_{\omega + 2}}$$ and $${\square _{{N_\omega },{N_n}}}$$ fails for all n < ω. This gives a partial answer to an old question of Woodin about the consistency of failure of SCH and failure of weak square.