An individual can interact with the same set of people over many different scales simultaneously. Four people might interact as a group of four and, at the same time, in pairs and triads. What is the relationship between different parallel interaction scales, and how might those scales themselves interact? We devised a four-player experimental game, the Modular Stag Hunt, in which participants chose not just whether to coordinate, but with whom, and at what scale. Our results reveal coordination behavior with such a strong preference for dyads that undermining pairwise coordination actually improves group-scale outcomes. We present these findings as experimental evidence for competition, as opposed to complementarity, between different possible scales of multi-player coordination. This result undermines a basic premise of approaches, like those of network science, that fail to model the interacting effects of dyadic, triadic, and group-scale structure on group outcomes.