665edo

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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 log23, 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 11/8: a sharp one from the optimal patent val, and a flat one from the 665e val. In the 11-limit, 41503/41472, 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 665e val.