This dissertation explores the consequences of cumulative interaction among markedness constraints in Harmonic Grammar (HG; Legendre, Miyata & Smolensky 1990, Smolensky & Legendre 2006), showing how HG inherently restricts cumulativity, allowing clear predictions about possible and impossible interactions to be made. Chapter 2 addresses the ability of HG to model certain patterns with a more limited constraint set than is feasible in Optimality Theory (OT; Prince & Smolensky 1993/2004). The particular focus here is on positional licensing constraints. The partially-overlapping contexts of these constraints lead to typologically-meaningful gang effects, allowing HG to capture patterns of disjunctive licensing that require the addition of positional faithfulness constraints in OT. With a smaller constraint set, the HG typology is restricted beyond what is possible in OT where more constraints are necessary.