This work develops a novel framework which can automatically detect, parameterize and interpolate between periodic motion patterns obtained from a motion capture sequence. Using our framework, periodic motions such as various stylised walking and running gaits or other motion sequences with periodic structure such as cleaning, dancing etc. can be detected automatically and without manual marking of the period start and end points. Our approach constructs an intrinsic parameterization of the motion and is computationally fast. Using this parameterization, we can generate prototypical motions. Additionally, we are able to easily interpolate between various periodic motions, yielding a rich class of `mixed' periodic actions. Our approach is based on ideas from applied algebraic topology, in particular, a novel method involving persistent cohomology to recover circular coordinates is applied for the first time in a graphics application. Additionally, we develop a suitable notion of homotopy which can be used to interpolate between periodic motion patterns. Our framework is directly applicable to the construction of walk cycles for animating character motions with motion graphs or state machine driven animation engines and processed our examples in approximately one minute or at a rate of about 10 frames per second.