1665edo

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← 1664edo1665edo1666edo →
Prime factorization 32 × 5 × 37
Step size 0.720721¢
Fifth 974\1665 (701.982¢)
Semitones (A1:m2) 158:125 (113.9¢ : 90.09¢)
Consistency limit 15
Distinct consistency limit 15

1665 equal divisions of the octave (1665edo), or 1665-tone equal temperament (1665tet), 1665 equal temperament (1665et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 1665 equal parts of about 0.721 ¢ each.

Theory

1665edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the gross temperament.

1665edo provides the optimal patent val for the rhodium temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for dzelic temperament in the 13-limit.

1665cc val is a tuning for the roentgenium temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.

Prime harmonics

Approximation of prime harmonics in 1665edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.000 +0.027 -0.007 -0.177 +0.033 -0.167 +0.270 +0.145 +0.194 +0.333 +0.190
relative (%) +0 +4 -1 -25 +5 -23 +37 +20 +27 +46 +26
Steps
(reduced)
1665
(0)
2639
(974)
3866
(536)
4674
(1344)
5760
(765)
6161
(1166)
6806
(146)
7073
(413)
7532
(872)
8089
(1429)
8249
(1589)

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator
(Reduced)
Cents
(Reduced)
Associated
Ratio
Temperaments
1 127\1665 91.531 [9 -32 18 Gross
37 377\1665
(17\1665)
271.711
(12.252)
117/100
(?)
Dzelic
45 1301\1665
(6\1665)
937.657
(4.324)
55/32
(?)
Rhodium