111edo is the equal division of the octave into 111 parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent optimal patent val tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&111 temperament, and buzzard, the 58&111 temperament. The Trio piece below is in guanyin temperament, the planar temperament tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.
The prime factorization is
[math]111 = 3 \cdot 37[/math]