# 1665edo

 ← 1664edo 1665edo 1666edo →
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 1665ed2), also called 1665-tone equal temperament (1665tet) or 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 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 (%) +0.0 +3.7 -1.0 -24.6 +4.6 -23.2 +37.4 +20.1 +26.9 +46.2 +26.3
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