User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20190803082138/@comment-11227630-20190803160750

Levels of "Stability" usually jump from a-stable to (a+1)-stable. But there are finer structure between them. The concept of "β-Πn-reflecting" split every stable level into ω levels, which improve little, but "β-Πn-reflecting on A" is much much finer. Finer structures enable smooth travel between the ordinals.

Also, the concept of "β-Πn[η] reflecting" is much finer than "(β,η)-stable" (η<ω).

But I don't know finer structure of "gap" beyond "α is gap length α". I don't know such concept X that "X vs. gap is similar to reflecting vs. stable".