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


== Theory ==
== Theory ==
981edo is a good 13- and 17-limit system, [[consistent]] to the [[17-odd-limit]]. The equal temperament [[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 = {{factorization|981}}, 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 }}
| &minus;0.0586
| −0.0586
| 0.0586
| 0.0586
| 4.79
| 4.79
|-
|-
| 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 }}
| &minus;0.0722
| −0.0722
| 0.0515
| 0.0515
| 4.21
| 4.21
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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 }}
| &minus;0.0385
| −0.0385
| 0.0562
| 0.0562
| 4.59
| 4.59
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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 }}
| &minus;0.0630
| −0.0630
| 0.0545
| 0.0545
| 4.46
| 4.46
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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 }}
| &minus;0.0453
| −0.0453
| 0.0636
| 0.0636
| 5.20
| 5.20
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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 }}
| &minus;0.0473
| −0.0473
| 0.0591
| 0.0591
| 4.83
| 4.83
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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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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]]
Retrieved from "https://en.xen.wiki/w/981edo"