Here, we present a new perspective on an old question: how does the neurobiology of human language relate to brain systems in nonhuman primates? We argue that higher-order language combinatorics, including sentence and discourse processing, can be situated in a unified, cross-species dorsal-ventral streams architecture for higher auditory processing, and that the functions of the dorsal and ventral streams in higher-order language processing can be grounded in their respective computational properties in primate audition. This view challenges an assumption, common in the cognitive sciences, that a nonhuman primate model forms an inherently inadequate basis for modeling higher-level language functions.