Discussion of time in interactive computer music systems engineering has been largely limited to data acquisition rates and latency. Since music is an inherently time-based medium, we believe that time plays a more important role in both the usability and implementation of these systems. In this paper, we present a time design space, which we use to expose some of the challenges of developing computer music systems with time-based interaction. We describe and analyze the time-related issues we encountered whilst designing and building a series of interactive music exhibits that fall into this design space. These issues often occur because of the varying and sometimes conflicting conceptual models of time in the three domains of user, application (music), and engineering. We present some of our latest work in conducting gesture interpretation and frameworks for digital audio, which attempt to analyze and address these conflicts in temporal conceptual models. Keywords time design, con...