665edo

The 665 equal temperament divides the octave into 665 equal parts of 1.80451 cents each. It is best known for its extremely accurate fifth, only 0.00011 cents flat. 665edo is the denominator of a convergent to log2(3), after 41edo, 53edo and 306edo, and before 15601edo. However, it also provides the optimal patent val for the rank four temperament tempering out 4000/3993. In the 3-limit it tempers out the 'satanic' comma, |-1054 665>; in the 5-limit the enneadeca, |-14 -19 19> and the monzisma, |54 -37 2>; in the 7-limit, 4375/4374 and the meter, | -11 2 7 -3 >; in the 11-limit, 4000/3993 and | 17 -5 0 -2 -1 > and | -1 2 -4 5 -2 >, providing the optimal patent val for 11-limit brahmagupta temperament. In the 13-limit it tempers out 1575/1573, 2080/2079, 4096/4095 and 4225/4224; since it tempers out 1575/1573, the nicola, it supports nicolic tempering and hence the nicolic tetrad, for which it provides an excellent tuning. In the 17-limit it tempers out 1156/1155, 1275/1274, 2058/2057, 2500/2499 and 5832/5831; in the 19-limit it tempers out 969/968, 1445/1444, 2432/2431, 3136/3135, 3250/3249 and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184 and 2737/2736.