The formal study of polyrhythms (2:3, 3:4, 3:4:5 etc.) often begins in the typical four-semester undergraduate theory sequence. Polyphonic instrumentalists are, no doubt, introduced to these rhythms earlier. Commonly, the rhythms are taught with the aid of subdivisions, the least common denominators shared by the separate parts of a polyrhythm. For example, the polyrhythm 3:2 has six subdivisions (the product of its two separate parts). Typically, this information is presented as a linear index (Ex. 1a), or as a series of counts (Ex. 1b) indicating the exact moments each part is to be articulated.This paper suggests teaching and understanding polyrhythms in a somewhat different manner: looping the subdivisions around themselves so that they form repetitive cycles and, in the process, mimic the motions of rotating gears rather than a linear index or series of counts. One advantage is that oral counting of the subdivisions is unnecessary and thus, the polyrhythm is more easily integrated into the composition at hand. (It would be difficult, for instance, to simultaneously count the polyrhythms' subdivisions and the actual metrical units in Exx. 1a and 1b). A second advantage is that one of the "gears" always coincides with the metrical unit; this allows the polyrhythm to be prepared a beat or two prior to its occurrence, something that is of particular value to the ensemble performer. The process is easily demonstrated, learned, and becomes a valuable reference tool while motivating students to explore ever more complex polyrhythms.