981edo: Difference between revisions

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== Theory ==
== Theory ==
981edo is a good 13- and 17-limit system, [[consistent]] to the [[17-odd-limit]]. As an equal temperament, it [[tempering out|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.  
981edo is a good [[13-limit|13-]] and [[17-limit]] system, [[consistent]] to the [[17-odd-limit]]. As an equal temperament, it [[tempering out|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 [[ennealympic]], which additionally tempers out 4096/4095.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 981 factors into primes as 3<sup>2</sup> × 109, 981edo has subset edos 3, 9, 109, and 327.
Since 981 factors into primes as {{nowrap| 3<sup>2</sup> × 109 }}, 981edo has subset edos {{EDOs| 3, 9, 109, and 327 }}.


== Regular temperament properties ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 1555 -981 }}
| {{Monzo| 1555 -981 }}
| {{mapping| 981 1555 }}
| {{Mapping| 981 1555 }}
| −0.0586
| −0.0586
| 0.0586
| 0.0586
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| {{Monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| {{mapping| 981 1555 2278 }}
| {{Mapping| 981 1555 2278 }}
| −0.0722
| −0.0722
| 0.0515
| 0.0515
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}
| 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}
| {{mapping| 981 1555 2278 2754 }}
| {{Mapping| 981 1555 2278 2754 }}
| −0.0385
| −0.0385
| 0.0562
| 0.0562
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 4375/4374, 131072/130977, 1771561/1771470
| 2401/2400, 4375/4374, 131072/130977, 1771561/1771470
| {{mapping| 981 1555 2278 2754 3394 }}
| {{Mapping| 981 1555 2278 2754 3394 }}
| −0.0630
| −0.0630
| 0.0545
| 0.0545
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470
| 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470
| {{mapping| 981 1555 2278 2754 3394 3630 }}
| {{Mapping| 981 1555 2278 2754 3394 3630 }}
| −0.0453
| −0.0453
| 0.0636
| 0.0636
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913
| 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913
| {{mapping| 981 1555 2278 2754 3394 3630 4010 }}
| {{Mapping| 981 1555 2278 2754 3394 3630 4010 }}
| −0.0473
| −0.0473
| 0.0591
| 0.0591
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 9
| 9
| 258\981<br />(40\981)
| 258\981<br>(40\981)
| 315.60<br />(48.93)
| 315.60<br>(48.93)
| 6/5<br />(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimmia
| [[Ennealimmal]] / ennealympic
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:Ennealimmic]]
[[Category:Ennealimmic]]
[[Category:Ennealimmia]]
[[Category:Ennealimmal]]

Latest revision as of 09:32, 19 May 2026

← 980edo 981edo 982edo →
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 981ed2), also called 981-tone equal temperament (981tet) or 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 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. As an equal temperament, it 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 ennealympic, which additionally 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 (%) +0.0 +15.2 +18.9 -1.5 +29.8 -13.1 +19.9 -21.7 +38.6 +32.1 -6.7
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 factors into primes as 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

Table of rank-2 temperaments by generator
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 / ennealympic

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

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