256edo
The 256 equal division divides the octave into 256 equal parts of exactly 4.6875 cents each. It is contorted in the 5-limit, and the error of harmonic 7 leads to inconsistency, which is likely one of the reasons this EDO attracts little interest.
Theory
Script error: No such module "primes_in_edo". 256edo is good at the 2.23.43.47 subgroup. If the error below 40% is considered "good", 256edo can be used to play no-fives 17 limit. If the "prime number obsession" approach found in math circles is applied - then 256edo can be played using the coprime harmonics, no matter how bad the approximation. Under such rule, 256edo supports the 2.7.13.19 subgroup.
In the 2.7.13.19 subgroup in the patent val, 256edo tempers out 32851/32768, and supports the corresponding 20 & 73 & 256 rank 3 temperament.