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 (abbreviated 1665edo), or 1665-tone equal temperament (1665tet), 1665 equal temperament (1665et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1665 equal parts of about 0.721 ¢ each. Each step of 1665edo represents a frequency ratio of 21/1665, or the 1665th root of 2.

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)

Subsets and supersets

Since 1665 factors into 32 × 5 × 37, 1665edo has subset edos 3, 5, 9, 15, 37, 45, 111, 185, 333, and 555.

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* 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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct