  The second law of thermodynamics implies that no macroscopic system may
oscillate indefinitely without consuming energy. The question of the number of
possible oscillations and the coherent quality of these oscillations remain
unanswered. This paper proves the upper-bounds on the number and quality of
such oscillations when the system in question is homogeneously driven and has a
discrete network of states. In a closed system, the maximum number of
oscillations is bounded by the number of states in the network. In open
systems, the size of the network bounds the quality factor of oscillation. This
work also explores how the quality factor of macrostate oscillations, such as
would be observed in chemical reactions, are bounded by the smallest equivalent
loop of the network, not the size of the entire system. The consequences of
this limit are explored in the context of chemical clocks and limit cycles.
