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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|331}}
{{ED intro}}
 
== Theory ==
== Theory ==
331et tempers out [[78125000/78121827]], [[5120/5103]] and 1959552/1953125 in the 7-limit; 806736/805255, 1835008/1830125, 1019215872/1019046875, 12005/11979, [[16384/16335]], 2359296/2358125, 42875/42768, 180224/180075, 1684375/1679616, 968000/964467, [[3025/3024]], 78408/78125, 1362944/1361367, 4108797/4096000 and 43923/43750 in the 11-limit.
331edo is only [[consistent]] to the [[5-odd-limit]] and the errors of both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are quite large, commending itself as a temperament of the 2.9.15.7.11.13.17.19 [[subgroup]].
===Odd harmonics===
 
Using the [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] [[5120/5103]], 1959552/1953125 and [[78125000/78121827]] in the 7-limit; [[3025/3024]], 12005/11979, [[16384/16335]], 42875/42768, 43923/43750, 78408/78125, and 180224/180075 in the 11-limit.
 
=== Odd harmonics ===
{{Harmonics in equal|331}}
{{Harmonics in equal|331}}
===Subsets and supersets===
 
=== Subsets and supersets ===
331edo is the 67th [[prime edo]]. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.  
331edo is the 67th [[prime edo]]. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.  
==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]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|-1049 331}}
| {{monzo| -1049 331 }}
|{{val|331 1049}}
| {{mapping| 331 1049 }}
| 0.1402
| +0.1402
| 0.1402
| 0.1402
| 3.87
| 3.87
|-
|-
|2.9.15
| 2.9.15
|{{monzo|-7 17 -12}}, {{monzo|-74 -5 23}}
| {{monzo| -7 17 -12 }}, {{monzo| -81 12 11 }}
|{{val|331 1049 1293}}
| {{mapping| 331 1049 1293 }}
| 0.1494
| +0.1238
| 0.1152
| 0.1168
| 3.18
| 3.22
|-
|-
|2.9.15.7
| 2.9.15.7
|65625/65536, 420175/419904, 80387359983/80000000000
| 65625/65536, 420175/419904, 80387359983/80000000000
|{{val|331 1049 1293 929}}
| {{mapping| 331 1049 1293 929 }}
| 0.1878
| +0.1685
| 0.1199
| 0.1275
| 3.52
|-
| 2.9.15.7.11
| 9801/9800, 41503/41472, 137781/137500, 759375/758912
| {{mapping| 331 1049 1293 929 1145 }}
| +0.1499
| 0.1200
| 3.31
| 3.31
|-
|-
|2.9.15.7.11
| 2.9.15.7.11.13
|9801/9800, 41503/41472, 137781/137500, 759375/758912
| 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875
|{{val|331 1049 1293 929 1145}}
| {{mapping| 331 1049 1293 929 1145 1225 }}
| 0.1653
| +0.0997
| 0.1163
| 3.21
|-
|2.9.15.7.11.13
|729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875, 53361/53248, 20336647/2028000
|{{val|331 1049 1293 929 1145 1225}}
| 0.1125
| 0.1587
| 4.38
|-
|2.9.15.7.11.13.17
|729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 14161/14157, 14175/14144, 43904/43875, 18816/18785, 92823/95744
|{{val|331 1049 1293 929 1145 1225 1353}}
| 0.0901
| 0.1568
| 0.1568
| 4.33
| 4.33
|-
| 2.9.15.7.11.13.17
| 729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 18816/18785
| {{mapping| 331 1049 1293 929 1145 1225 1353 }}
| +0.0791
| 0.1537
| 4.24
|}
|}


=== 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
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|89\331
| 107\331
|322.66
| 387.92
|6/5
| 5/4
|[[Magicaltet]]
| [[Würschmidt]] (331, 5-limit)
|-
|1
|107\331
|387.92
|5/4
|[[Würschmidt]]
|-
|1
|137\331
|496.68
|5457/4096
|[[Edson]]
|}
|}
<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
== Scales ==
* [[Magicaltet7]]
* [[Magicaltet11]]
* [[Magicaltet15]]


==Scales==
== Music ==
*[[Magicaltet7]]
; [[User:Francium|Francium]]
*[[Magicaltet11]]
* "Silent Silence" from ''Edson EP'' (2023) – [https://open.spotify.com/track/6q3xr4E4QIL9BMaZyf6LXd Spotify] | [https://francium223.bandcamp.com/track/silent-silence Bandcamp] | [https://www.youtube.com/watch?v=g3FF6oqnilk YouTube] – in Edson, 331edo tuning
*[[Magicaltet15]]
* "Moth Mustard" from ''Unsuspecting Tyrant Double-Decker Beef Fort'' (2026) – [https://open.spotify.com/track/6M1I3YWmWvHM5bnMUz4bYg Spotify] | [https://francium223.bandcamp.com/track/moth-mustard Bandcamp] | [https://www.youtube.com/watch?v=arg4hjvpeuY YouTube]


==Music==
[[Category:Listen]]
*[https://www.youtube.com/watch?v=g3FF6oqnilk Silent Silence] by [[User:Francium|Francium]]
Retrieved from "https://en.xen.wiki/w/331edo"