Frequency of Pendulum Outside the Small Angle Approximation

Question

I understand that if one makes the small angle approximation, the angular frequency of, say, a pendulum, is easy to find as $\sqrt{k/m}$. Without making any approximations and sticking with the original equation $m\ddot \theta=-k\sin \theta$, we cannot analytically solve the system as we have an elliptic integral. However, I was wondering if it was possible to at least determine properties of the exact frequency of the oscillation of this system (e.g. is it rational, transcendental, etc.), or more generally: are there any significant statements we can make at all about the system that aren't based on numerical approximations?