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'''251edo''' is the [[EDO|equal division of the octave]] into 251 parts of 4.7809 [[cent]]s each. It 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. 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 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.
{{Infobox ET}}
{{ED intro}}


251edo is the 54th [[prime EDO]].
== Theory ==
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]], and [[acrokleismic]].  


[[Category:Edo]]
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.
[[Category:Prime EDO]]
 
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 ===
{{Harmonics in equal|251}}
 
=== Subsets and supersets ===
251edo is the 54th [[prime edo]].
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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
| 5.50
|-
| 2.3.5
| {{monzo| 9 -13 5 }}, {{monzo| 49 -6 -17 }}
| {{mapping| 251 398 583 }}
| −0.3099
| 0.2247
| 4.70
|-
| 2.3.5.7
| 4375/4374, 5120/5103, 40500000/40353607
| {{mapping| 251 398 583 705 }}
| −0.3830
| 0.2322
| 4.86
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 66\251
| 315.54
| 6/5
| [[Acrokleismic]]
|-
| 1
| 71\251
| 339.44
| 243/200
| [[Amity]]
|-
| 1
| 91\251
| 435.06
| 9/7
| [[Supermajor]]
|-
| 1
| 96\251
| 458.96
| 125/96
| [[Majvam]]
|-
| 1
| 112\251
| 535.46
| 512/375
| [[Maquila]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[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]
 
[[Category:Listen]]
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