An Optimality Theoretic (OT) grammar arises from the comparison of candidates over a set of constraints, oriented toward obtaining certain of those candidates as optimal. The typology of a specified system collects its grammars, encompassing all total domination orders among the posited constraints. Considerable progress has been made in understanding the internal structure of OT grammars. Here we move up a level, from grammar to typology, probing the structure that emerges from the most basic commitments of the theory. Comparison is once again central: a constraint viewed at the typological level turns out to rate entire grammars against each other. From this perspective, the constraint goes beyond its familiar role as an engine of comparison based on quantitative penalties and instead takes the form of a more abstract order and equivalence structure. This we call an EPO, an 'Equivalence-augmented Privileged Order', presentable as a kind of enriched Hasse diagram. The collection of the EPOs, one for each constraint, forms the MOAT, the Mother of All Tableaux. The EPOs of a typology's unique MOAT are respected in every violation tableau associated with it. With the MOAT concept in place, it becomes possible to understand exactly which sets of disjoint grammars constitute valid typologies. This finding gives us the conditions under which grammars of a given typology can merge to produce another simpler typology and thereby abstract away informatively from various differences between them. Geometrically, the MOAT concept enables us to show, following the insights of Riggle (2010, 2012), that the grammars of a typology neatly partition its representation on the permutohedron into connected, spherically convex regions. Discussion proceeds along both concrete and abstract lines, aiming to facilitate access for readers across a wide range of interests.Nazarre Merchant is an associate professor of mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University. Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.