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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|214}}
{{ED intro}}
==Theory==
 
214et is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which tempers out the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; [[6144/6125]] (porwell), 16875/16807 (mirkwai), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]] and 1375/1372 in the 11-limit; 1188/1183, [[351/350]] and [[847/845]] in the 13-limit. It can be viewed as a 2.13/5 subgroup temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
== Theory ==
===Prime harmonics===
214edo is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which [[tempering out|tempers out]] the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 6144/6125 ([[porwell comma]]), 16875/16807 ([[mirkwai comma]]), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]], 1375/1372, [[5632/5625]], in the 11-limit; [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]], and [[4096/4095]] in the 13-limit. It can be viewed as a 2.3.5.13.19.23 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
 
=== Prime harmonics ===
{{Harmonics in equal|214}}
{{Harmonics in equal|214}}
===Subsets and supersets===
 
214 factors into 2 × 107, with [[2edo]] and [[107edo]] as its subset edos.
=== Subsets and supersets ===
==Regular temperament properties==
Since 214 factors into {{factorisation|214}}, 214edo contains [[2edo]] and [[107edo]] as its subsets.
 
== 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|-339 214}}
! rowspan="2" | [[Comma list]]
|{{val|214 339}}
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -339 214 }}
| {{mapping| 214 339 }}
| +0.3219
| +0.3219
| 0.3220
| 0.3220
| 5.74
| 5.74
|-
|-
|2.3.5
| 2.3.5
|78732/78125, {{monzo|-49 28 2}}
| 78732/78125, {{monzo| -49 28 2 }}
|{{val|214 339 497}}
| {{mapping| 214 339 497 }}
| +0.1281
| +0.1281
| 0.3797
| 0.3797
| 6.77
| 6.77
|-
|-
|2.3.5.7
| 2.3.5.7
|6144/6125, 16875/16807, 78732/78125
| 6144/6125, 16875/16807, 78732/78125
|{{val|214 339 497 601}}
| {{mapping| 214 339 497 601 }}
| -0.0169
| −0.0169
| 0.4137
| 0.4137
| 7.38
| 7.38
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 1375/1372, 3025/3024, 5632/5625
| 540/539, 1375/1372, 5632/5625, 72171/71680
|{{val|214 339 497 601 740}}
| {{mapping| 214 339 497 601 740 }}
| +0.0897
| +0.0897
| 0.4270
| 0.4270
| 7.61
| 7.61
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|540/539, 847/845, 1001/1000, 1375/1372, 5632/5625
| 351/350, 540/539, 847/845, 1375/1372, 4096/4095
|{{val|214 339 497 601 740 792}}
| {{mapping| 214 339 497 601 740 792 }}
| +0.0480
| +0.0480
| 0.4008
| 0.4008
| 7.15
| 7.15
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|351/350, 715/714, 936/935, 1275/1274, 5544/5525, 5850/5831
| 351/350, 540/539, 715/714, 847/845, 936/935, 4096/4095
|{{val|214 339 497 601 740 792 875}}
| {{mapping| 214 339 497 601 740 792 875 }}
| -0.0144
| −0.0144
| 0.4012
| 0.4012
| 7.15
| 7.15
|}
|}
=== 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
|69\214
| 27\214
|386.92
| 151.40
|5/4
| 12/11
|[[Grendel]]
| [[Browser]]
|-
| 1
| 69\214
| 386.92
| 5/4
| [[Grendel]]
|-
|-
|1
| 1
|79\214
| 79\214
|442.99
| 442.99
|9/7
| 162/125
|[[Sensi]]
| [[Sensipent]]
|-
|-
|1
| 1
|105\214
| 105\214
|588.79
| 588.79
|7/5
| 7/5
|[[Untriton]] / [[aufo]]
| [[Aufo]]
|-
|-
|2
| 2
|28\214
| 28\214
|157.01
| 157.01
|35/32
| 35/32
|[[Bison]]
| [[Bison]] (214e)
|-
|-
|2
| 2
|29\214
| 29\214
|162.62
| 162.62
|1125/1024
| 1125/1024
|[[Kwazy]]
| [[Kwazy]]
|}
|}
<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:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Browser]]
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