631edo

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← 630edo631edo632edo →
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