# 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 compressed. 665edo is the denominator of a convergent to log_{2}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. It tempers out the 'satanic' comma, |-1054 665> in the 3-limit; the enneadeca, |-14 -19 19> and the monzisma, |54 -37 2> in the 5-limit; the ragisma, 4375/4374, the meter, 703125/702464, and 68719476736/68641485507 in the 7-limit; 4000/3993, 46656/46585, 131072/130977 and 151263/151250 in the 11-limit, 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.

665edo provides excellent approximations for the 7-limit intervals and harmonics 13, 17, 19 and 23. It is considered as the excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the 27-odd-limit (with no elevens). Despite its division number of the octave, 665edo provides poor approximations for the 11-limit intervals, with two mappings possible for the 11/8 fourth: a sharp one from the patent val, and a flat one from the 665e val. Using the 665e val, 41503/41472, 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.