Eno's "Music for Airports" famously uses a system of multiple tape loops that produce sequences of different periods. As you listen, you can hear phrases that occur nearly together and then later, well-separated in time, as the periods of these loops go in and out of phase:
"One of the notes repeats every 23 1/2 seconds. It is in fact a long loop running around a series of tubular aluminum chairs in Conny Plank's studio. The next lowest loop repeats every 25 7/8 seconds or something like that. The third one every 29 15/16 seconds or something. What I mean is they all repeat in cycles that are called incommensurable — they are not likely to come back into sync again."
This interaction of periods will have long memory. (If the tape lengths are L and L+d, for small d, then the repeat time could easily be as long as L*(L/d), and even longer if d does not divide L evenly.) Thus, it is very different from what you get with a Markov chain.