Tight stationarity and tree-like scales

Let κ   be a singular cardinal of countable cofinality, 〈κn:n<ω〉〈κn:n<ω〉 be a sequence of regular cardinals which is increasing and cofinal in κ. Using a scale, we define a mapping μ   from ∏nP(κn)∏nP(κn) to P(κ+)P(κ+) which relates tight stationarity on κ   to the usual notion of stationarity on κ+κ+. We produce a model where all subsets of κ+κ+ are in the range of μ for some κ a singular. Using a version of the diagonal supercompact Prikry forcing, we obtain such a model where κ is strong limit. Then we construct a sequence of stationary sets that is not tightly stationary in a strong way, namely, its image under μ is empty. All of these results start from a model with a continuous tree-like scale on κ.