981edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|981}}
{{ED intro}}


== Theory ==
== Theory ==
981edo is a good 13- and 17-limit system. 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 [[ennealimmia]], which also tempers out 4096/4095.  
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.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|981|columns=11}}
{{Harmonics in equal|981}}
 
=== Subsets and supersets ===
Since 981 factors into primes as 3<sup>2</sup> × 109, 981edo has subset edos 3, 9, 109, and 327.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 21: Line 25:
| 2.3
| 2.3
| {{monzo| 1555 -981 }}
| {{monzo| 1555 -981 }}
| [{{val| 981 1555 }}]
| {{mapping| 981 1555 }}
| -0.0586
| −0.0586
| 0.0586
| 0.0586
| 4.79
| 4.79
Line 28: Line 32:
| 2.3.5
| 2.3.5
| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| [{{val| 981 1555 2278 }}]
| {{mapping| 981 1555 2278 }}
| -0.0722
| −0.0722
| 0.0515
| 0.0515
| 4.21
| 4.21
Line 35: Line 39:
| 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 }}
| [{{val| 981 1555 2278 2754 }}]
| {{mapping| 981 1555 2278 2754 }}
| -0.0385
| −0.0385
| 0.0562
| 0.0562
| 4.59
| 4.59
Line 42: Line 46:
| 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
| [{{val| 981 1555 2278 2754 3394 }}]
| {{mapping| 981 1555 2278 2754 3394 }}
| -0.0630
| −0.0630
| 0.0545
| 0.0545
| 4.46
| 4.46
Line 49: Line 53:
| 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
| [{{val| 981 1555 2278 2754 3394 3630 }}]
| {{mapping| 981 1555 2278 2754 3394 3630 }}
| -0.0453
| −0.0453
| 0.0636
| 0.0636
| 5.20
| 5.20
Line 56: Line 60:
| 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
| [{{val| 981 1555 2278 2754 3394 3630 4010 }}]
| {{mapping| 981 1555 2278 2754 3394 3630 4010 }}
| -0.0473
| −0.0473
| 0.0591
| 0.0591
| 4.83
| 4.83
Line 64: Line 68:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Generator<br>(Reduced)
|-
! Cents<br>(Reduced)
! Periods<br />per 8ve
! Associated<br>Ratio
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
| 1
| 409\981
| 500.306
| 8192/6137
| [[Protolangwidge]]
|-
|-
| 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]] / ennealimmia
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Ennealimmic]]
[[Category:Ennealimmic]]
[[Category:Ennealimmia]]
[[Category:Ennealimmia]]

Latest revision as of 23:09, 20 February 2025

← 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 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 (%) +0.0 +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 / ennealimmia

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

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