Persistent homology is well-defined and well studied for tame functions, for example various ones arising from finite point sets. However, periodic functions -- for example used to study periodic point sets, like the atoms of a crystal -- are not tame. We therefore extend the definition of persistent homology to periodic functions, which is a non-trivial endeavor. In contrast to related work, we quantify how the multiplicities of persistence pairs tend to infinity with increasing window size, in a way that is stable under perturbations and invariant under different finite representations of the infinite periodic function. This project is still ongoing research, but I'll explain what we already know and what we don't know yet.