# 631edo

 ← 630edo 631edo 632edo →
Prime factorization 631 (prime)
Step size 1.90174¢
Fifth 369\631 (701.743¢)
Semitones (A1:m2) 59:48 (112.2¢ : 91.28¢)
Consistency limit 9
Distinct consistency limit 9

631 equal divisions of the octave (abbreviated 631edo or 631ed2), also called 631-tone equal temperament (631tet) or 631 equal temperament (631et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 631 equal parts of about 1.9 ¢ each. Each step represents a frequency ratio of 21/631, or the 631st root of 2.

## Theory

631edo is consistent to the 9-odd-limit, with all of the odd harmonics having a flat tendency. Using the patent val, the equal temperament tempers out 4375/4374, 41503/41472, 32805/32768 and 12005/11979 in the 11-limit; 1575/1573, 4375/4374, 4459/4455, 4225/4224 and 83349/83200 in the 13-limit.

### Odd harmonics

Approximation of odd harmonics in 631edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.212 -0.260 -0.839 -0.423 +0.188 +0.043 -0.472 -0.360 -0.841 +0.851 -0.699
Relative (%) -11.1 -13.7 -44.1 -22.3 +9.9 +2.3 -24.8 -18.9 -44.2 +44.8 -36.8
Steps
(reduced)
1000
(369)
1465
(203)
1771
(509)
2000
(107)
2183
(290)
2335
(442)
2465
(572)
2579
(55)
2680
(156)
2772
(248)
2854
(330)

### Subsets and supersets

631edo is the 115th prime EDO.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-1000 631 [631 1000]] 0.0668 0.0668 3.51
2.3.5 32805/32768, [-50 -71 70 [631 1000 1465]] 0.0818 0.0585 3.08
2.3.5.7 4375/4374, 32805/32768, 678223072849/675000000000 [631 1000 1465 1771]] 0.1361 0.1067 5.61
2.3.5.7.11 4375/4374, 41503/41472, 32805/32768, 12005/11979 [631 1000 1465 1771 2183]] 0.0980 0.1221 6.42
2.3.5.7.11.13 1575/1573, 4375/4374, 4459/4455, 4225/4224, 83349/83200 [631 1000 1465 1771 2183 2335]] 0.0797 0.1187 6.24
2.3.5.7.11.13.17 1225/1224, 1701/1700, 833/832, 1575/1573, 4459/4455, 4225/4224 [631 1000 1465 1771 2183 2335 2579]] 0.0809 0.1099 5.78

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 262\631 498.257 4/3 Helmholtz / Pontiac

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