4th-octave temperaments
4edo is much less used as a scale, rather as a chord. In many diatonic-based interval region schemes, one step of 4edo is known as a minor third, and the stacking of them is the diminished seventh chord.
Usage of the 6/5 minor third as one step of 4edo by tempering out 648/625, and therefore using 4edo as a diminished seventh chord produced by stacking three minor thirds is one of the features of standard Western music theory, and is supported by 12edo. See Diminished family for a collection of such temperaments.
19/16, the 19th harmonic octave-reduced, is much closer to quarter-octave than 6/5, and while it is not a microtemperament, a lot of equal divisions support it.
An interval closer to 1\4 is 25/21, with the associated comma being the dimcomp comma. See Dimcomp family for a collection of rank-3 temperaments tempering it out.
There are nonetheless other less common temperaments which divide the octave in four.
Temperaments discussed elsewhere are: