3rd-octave temperaments
3edo is much less used as a scale, rather as a chord. In many diatonic-based interval region schemes, one step of 3edo is known as a major third, and the stacking of them is the augmented triad.
Usage of the 5/4 major third as one step of 3edo by tempering out 128/125, and therefore using 3edo as a augmented triad produced by stacking two major thirds is one of the features of standard Western music theory, and is supported by 12edo. See Augmented family for a collection of such temperaments.
24/19 is much closer to 1/3-octave than 5/4, and while it is not a microtemperament, a lot of equal divisions support it.
An interval closer to 1\3 is 63/50, with the associated comma being the landscape comma. See Landscape family for a collection of rank-3 temperaments tempering it out.
There are nonetheless other less common temperaments which divide the octave in three.
Temperaments discussed elsewhere are: