Previous work on frequent item set mining has focused on finding all itemsets that are frequent in a specified part of a database. We motivate the dual question of finding under what circumstances a given item set satisfies a pattern of interest (e.g., frequency) in a database. Circumstances form a lattice that generalizes the instance lattice associated with datacube. Exploiting this, we adapt known cube algorithms and propose our own, minCirc, for mining the strongest (e.g., minimal) circumstances under which an itemset satisfies a pattern. Our experiments show that minCirc is competitive with the adapted algorithms. We motivate mining queries involving migration between item set and circumstance lattices and propose the notion of Armstrong Basis as a structure that provides efficient support for such migration queries, as well as a simple algorithm for computing it.