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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|251}}
{{ED intro}}
 
== Theory ==
== Theory ==
251et tempers out 1600000/1594323 (amity comma) and 562949953421312/556182861328125 (maquila comma) in the 5-limit; 4375/4374, 5120/5103, and 40500000/40353607 in the 7-limit.  
251et [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 49 -6 -17 }} ([[maquila comma]]) in the 5-limit; [[4375/4374]], [[5120/5103]], and 40500000/40353607 in the 7-limit, [[support]]ing [[amity]], [[supermajor (temperament)|supermajor]], and [[acrokleismic]].  


Using the patent val, it tempers out 1331/1323, 1375/1372, 16896/16807, and 24057/24010 in the 11-limit; 352/351, 676/675, 847/845, and 1573/1568 in the 13-limit.  
Using the [[patent val]] {{val| 251 398 583 705 '''868''' }}, it tempers out 1331/1323, 1375/1372, 16896/16807, and 24057/24010 in the 11-limit; [[352/351]], [[676/675]], [[847/845]], and [[1573/1568]] in the 13-limit.  


Using the 251e val, it tempers out 540/539, 5632/5625, 6250/6237, and 12005/11979 in the 11-limit; 364/363, 676/675, 1716/1715, and 3584/3575 in the 13-limit.
Using the 251e val {{val| 251 398 583 705 '''869''' }}, it tempers out [[540/539]], [[5632/5625]], [[6250/6237]], and 12005/11979 in the 11-limit; [[364/363]], [[676/675]], [[1716/1715]], and 3584/3575 in the 13-limit.


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
251edo is the 54th [[prime EDO]].
251edo is the 54th [[prime edo]].


==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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|398 -251}}
! rowspan="2" | [[Comma list]]
|{{val|251 398}}
! rowspan="2" | [[Mapping]]
| -0.2630
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 398 -251 }}
| {{Mapping| 251 398 }}
| −0.2630
| 0.2630
| 0.2630
| 5.50
| 5.50
|-
|-
|2.3.5
| 2.3.5
|{{monzo|9 -13 5}}, {{monzo|49 -6 -17}}
| {{Monzo| 9 -13 5 }}, {{monzo| 49 -6 -17 }}
|{{val|251 398 583}}
| {{Mapping| 251 398 583 }}
| -0.3099
| −0.3099
| 0.2247
| 0.2247
| 4.70
| 4.70
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 5120/5103, 6144/6125
| 4375/4374, 5120/5103, 40500000/40353607
|{{val|251 398 583 705}}
| {{Mapping| 251 398 583 705 }}
| -0.3830
| −0.3830
| 0.2322
| 0.2322
| 4.86
| 4.86
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|71\251
| 66\251
|339.44
| 315.54
|243/200
| 6/5
|[[Amity]]
| [[Acrokleismic]]
|-
|1
|75\251
|358.57
|315/256
|[[Restles]]
|-
|-
|1
| 1
|91\251
| 71\251
|435.06
| 339.44
|9/7
| 243/200
|[[Supermajor]]
| [[Amity]]
|-
|-
|1
| 1
|96\251
| 91\251
|458.96
| 435.06
|125/96
| 9/7
|[[Majvam]]
| [[Supermajor (temperament)|Supermajor]]
|-
|-
|1
| 1
|112\251
| 96\251
|535.46
| 458.96
|512/375
| 125/96
|[[Maquila]]
| [[Majvam]]
|-
|-
|1
| 1
|117\251
| 112\251
|559.26
| 535.46
|864/625
| 512/375
|[[Tritriple]]
| [[Maquila]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct


== Music ==
== Music ==
; [[Francium]]
; [[Francium]]
* "hope in dark times" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/1zSiYXz46ugWAmqySaEN2A Spotify] | [https://francium223.bandcamp.com/track/hope-in-dark-times Bandcamp] | [https://www.youtube.com/watch?v=yHl3oku_NmY YouTube]
* "hope in dark times" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/1zSiYXz46ugWAmqySaEN2A Spotify] | [https://francium223.bandcamp.com/track/hope-in-dark-times Bandcamp] | [https://www.youtube.com/watch?v=yHl3oku_NmY YouTube]
* "The Living Space" from ''Check Us Out'' (2026) – [https://open.spotify.com/track/66H2fQ5ogT1j33ludkD0ug Spotify] | [https://francium223.bandcamp.com/track/the-living-space Bandcamp] | [https://www.youtube.com/watch?v=Q4O5l0SnZMI YouTube]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/251edo"