Singular cardinals and square properties

We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then □_κ,ω holds in the bigger model. We extend this to the situation where every regular cardinal in an interval [κ, ν] is singularized, for some regular cardinal ν. More precisely, we show that if V ⊂ W, κ < ν are cardinals, where ν is regular in V, κ is a singular cardinal in W of countable cofinality, cf^W (τ) = ω for all V -regular κ ≤ τ ≤ ν, and (ν⁺)^V = (κ⁺)^W, then W |= □_κ,ω.