981edo

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← 980edo981edo982edo →
Prime factorization 32 × 109
Step size 1.22324¢
Fifth 574\981 (702.141¢)
Semitones (A1:m2) 94:73 (115¢ : 89.3¢)
Consistency limit 17
Distinct consistency limit 17

981 equal divisions of the octave (abbreviated 981edo), or 981-tone equal temperament (981tet), 981 equal temperament (981et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 981 equal parts of about 1.22 ¢ each. Each step of 981edo represents a frequency ratio of 21/981, or the 981st root of 2.

Theory

981edo is a good 13- and 17-limit system, consistent to the 17-odd-limit. The equal temperament tempers out 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4225/4224, 4375/4374, 4459/4455 and 4914/4913 in the 17-limit. It provides the optimal patent val for 13-limit ennealimmic, the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit ennealimmia, which also tempers out 4096/4095.

Prime harmonics

Approximation of prime harmonics in 981edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.000 +0.186 +0.231 -0.019 +0.364 -0.161 +0.243 -0.265 +0.472 +0.392 -0.081
relative (%) +0 +15 +19 -2 +30 -13 +20 -22 +39 +32 -7
Steps
(reduced)
981
(0)
1555
(574)
2278
(316)
2754
(792)
3394
(451)
3630
(687)
4010
(86)
4167
(243)
4438
(514)
4766
(842)
4860
(936)

Subsets and supersets

Since 981 = 32 × 109, 981edo has subset edos 3, 9, 109, and 327.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [1555 -981 [981 1555]] -0.0586 0.0586 4.79
2.3.5 [1 -27 18, [85 -17 -25 [981 1555 2278]] -0.0722 0.0515 4.21
2.3.5.7 2401/2400, 4375/4374, [79 -25 -23 5 [981 1555 2278 2754]] -0.0385 0.0562 4.59
2.3.5.7.11 2401/2400, 4375/4374, 131072/130977, 1771561/1771470 [981 1555 2278 2754 3394]] -0.0630 0.0545 4.46
2.3.5.7.11.13 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470 [981 1555 2278 2754 3394 3630]] -0.0453 0.0636 5.20
2.3.5.7.11.13.17 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913 [981 1555 2278 2754 3394 3630 4010]] -0.0473 0.0591 4.83

Rank-2 temperaments

Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 409\981 500.306 8192/6137 Protolangwidge
9 258\981
(40\981)
315.60
(48.93)
6/5
(36/35)
Ennealimmal / ennealimmia

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