What is comparison scale?

Near a point $a$ ($a \in \R$ or $\pm\infty$), for functions (or sequences, with $n \to \infty$): $f = o(g)$, $f = O(g)$, $f \sim g$ as in the Year 1 volume. A comparison scale at $a$ is a family of positive functions, pairwise comparable, totally ordered by $o(\cdot)$ — the standard scale at $+\infty$ being

ordered lexicographically in $(\alpha, \beta)$, refined when needed by exponentials $\eu^{\gamma x}$.