Most of this book concerns large deviations, in which we fix a triangle density t, typically different from $e^3$, and take a limit as the number n of nodes goes to infinity. In this section, however, we consider deviation of t from $e^3$ of order $n^{-\alpha}$ for some fixed number $\alpha>0$. In particular, we consider lower tails, estimating the probability that the triangle density of a graph with edge density e has triangle density at most $e^3-\Delta t$, where $\Delta t$ is a positive quantity. Estimates similar to the CLT were proven for $\alpha>1$ and more recently for $\alpha>3/4$. We prove a different set of estimates when $\alpha<3/4$, showing that the probability of a deviation of size $\Delta t$ is governed by the behavior of the Boltzmann entropy as t approaches $e^3$ from below. The estimates are sharpest for e > ½, where we also have the best estimates of B(e,t). Similar but slightly different estimates apply to moderate deviations of other odd cycles.