Personally, I don't think this is intuitive at all. Every linear system has what is called normal modes$^\dagger$, solutions to the system where every component oscillates with the same frequency. A linear system is one where you can add two solutions together and still have a solution to the system.
So if we have such a system, there exist solutions where every part of the system oscillates with the same frequency. You can imagine a complicated setup of lots of springs and pendulums and somehow this has solutions where every part oscillates with the same frequency. This is quite amazing I think.
However, if you add two normal modes with different frequencies, this will break the pattern. There is not a single frequency anymore with which the system oscillates.
$\dagger$ I'm not entirely sure what the precise requirements are for a system having normal modes. I think it was that the system is linear. If someone can verify that it would be nice. This happens when every term in the equation is just $\theta(t)$ times a constant or a derivative of $\theta(t)$ times a constant.