# 641edo

 ← 640edo 641edo 642edo →
Prime factorization 641 (prime)
Step size 1.87207¢
Fifth 375\641 (702.028¢)
Semitones (A1:m2) 61:48 (114.2¢ : 89.86¢)
Consistency limit 5
Distinct consistency limit 5

641 equal divisions of the octave (abbreviated 641edo or 641ed2), also called 641-tone equal temperament (641tet) or 641 equal temperament (641et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 641 equal parts of about 1.87 ¢ each. Each step represents a frequency ratio of 21/641, or the 641st root of 2.

## Theory

641edo is only consistent to the 5-odd-limit. Since both harmonics 7 and 11 are about halfway between its steps, and since harmonic 5 is also off by more than a third step, it can be used as a dual-5 dual-7 dual-11 temperament. Alternatively, it can be used as a 2.3.5.13.17.19 subgroup temperament, as it is consistent in the no-7 no-11 19-odd-limit.

To start with, consider the 641d val 641 1016 1488 1799 2217 2372] in the 13-limit, which tempers out 625/624, 2200/2197, 4459/4455, 14641/14625, and 19712/19683. The alternative 641df val, 641 1016 1488 1799 2217 2371], tempers out 676/675, 1001/1000, 19712/19683, 31213/31104, and 983125/979776. The 641ce val, 641 1016 1489 1800 2218 2372], tempers out 676/675, 1001/1000, 6144/6125, 10985/10976, and 85294/85184.

### Odd harmonics

Approximation of odd harmonics in 641edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.073 -0.666 +0.909 +0.146 -0.928 +0.034 -0.593 -0.119 +0.147 -0.890 +0.743
Relative (%) +3.9 -35.6 +48.5 +7.8 -49.6 +1.8 -31.7 -6.4 +7.8 -47.5 +39.7
Steps
(reduced)
1016
(375)
1488
(206)
1800
(518)
2032
(109)
2217
(294)
2372
(449)
2504
(581)
2620
(56)
2723
(159)
2815
(251)
2900
(336)

### Subsets and supersets

641edo is the 116th prime edo. 1282edo, which doubles it, gives a good correction to the harmonics 7 and 11.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [1016 -641 [641 1016]] -0.0231 0.0231 1.23
2.3.5 [24 -21 4, [-56 -13 33 [641 1016 1488]] +0.0803 0.1474 7.87

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 254\641 475.507 320/243 Vulture

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