It is ever possible to have an equal temperament without using the pure octave. In this case we can produce equal temperament where an universal half tone is used. It will permit to stay nearest possible to the just intonation and in the same time to modulate in all tonalities. This temperament exist and is named TEDJ. French name where TemperamentEquale with DifferencialeJust. Who know more about this temperament?
31 tone et gets very close to providing all of the just intervals that can be produced from voices and fretless instruments. But it is unwieldy to play a keyboard with 31 keys to the octave. I would suggest getting a piano tuned to well temperament. While some of the keys contain wildly sharp Pythagorean thirds they at least want to go outward and suggest movement to the fifth versus the sickly sounding thirds of et. Also, keys have different feeling in well temperament due to the different intonation they possess. Of course this can be grouped between closely related keys. Also c major and it's closely related keys contain practically perfect triads that sound sweet and glorious. I think overall the temperament that satisfied Bach's disappointment with mean tone is the best solution for keyboard instruments.