Research on neural models for cognition suggests that thought is far from a simply serial process. Nonetheless, there has been relatively little work on which parameters govern just what aspects of thought are parallel and what are serial. Clearly, speech as such is serial. In consequence, interior monologue (understood as subvocalised speech) is serial. Moreover, stream of consciousness – mental experience not confined to subvocalised speech – must be articulated in serial form in a novel. Due to this constraint on representation, it seems that novelists commonly imagine that stream of consciousness itself really is serial. Joyce, however, developed a sense of parallel cognitive processing in the course of Ulysses. Specifically, in the “Wandering Rocks” episode, he explored spatiotemporally parallel events in complex social systems. In the following chapter, “Sirens,” he in effect transferred this treatment of external parallelism to the human mind, systematically developing cognitive parallelism in his representation of Leopold Bloom. This development was perhaps reinforced by ideas of harmony and counterpoint associated with the episode's musical model. Understanding Joyce's exploration of parallel and serial processes in thought is important not only for what it tells us about Ulysses. It is also important for what it contributes to our understanding of cognitive parallelism.