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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|814}}
{{ED intro}}


== Theory ==
== Theory ==
814edo is distinctly [[consistent]] to the [[17-odd-limit]] and is a strong 17-limit system. The equal temperament is [[enfactoring|enfactored]] in the 5-limit, tempering out the [[schisma]] as does 407et. In the 7-limit it tempers out [[2401/2400]] so that it [[support]]s and gives a good tuning for [[sesquiquartififths]]. In the 11-limit it tempers out [[9801/9800]], in the 13-limit [[4225/4224]] and [[6656/6655]], and in the 17-limit [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[5832/5831]]. The 171 & 643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].
814edo is [[consistency|distinctly consistent]] to the [[17-odd-limit]] and is a strong 17-limit system. The equal temperament is [[enfactoring|enfactored]] in the 5-limit, tempering out the [[schisma]] as does 407et. In the 7-limit it tempers out [[2401/2400]] so that it [[support]]s and gives a good tuning for [[sesquiquartififths]]. In the 11-limit it tempers out [[9801/9800]], in the 13-limit [[4225/4224]] and [[6656/6655]], and in the 17-limit [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[5832/5831]]. The {{nowrap|171 & 643}} temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|814|columns=11}}
{{Harmonics in equal|814}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 814 factors into 2 × 11 × 37, 814edo has subset edos {{EDOs| 2, 11, 22, 37, 74, and 407 }}.  
Since 814 factors into {{factorization|814}}, 814edo has subset edos {{EDOs| 2, 11, 22, 37, 74, and 407 }}.  


== 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" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 32805/32768, {{monzo| 25 20 -22 -2 }}
| 2401/2400, 32805/32768, {{monzo| 25 20 -22 -2 }}
| [{{val| 814 1290 1890 2285 }}]
| {{mapping| 814 1290 1890 2285 }}
| +0.0695
| +0.0695
| 0.0577
| 0.0577
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 9801/9800, 32805/32768, 20155392/20131375
| 2401/2400, 9801/9800, 32805/32768, 20155392/20131375
| [{{val| 814 1290 1890 2285 2816 }}]
| {{mapping| 814 1290 1890 2285 2816 }}
| +0.0536
| +0.0536
| 0.0605
| 0.0605
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2401/2400, 4225/4224, 6656/6655, 9801/9800, 32805/32768
| 2401/2400, 4225/4224, 6656/6655, 9801/9800, 32805/32768
| [{{val| 814 1290 1890 2285 2816 3012 }}]
| {{mapping| 814 1290 1890 2285 2816 3012 }}
| +0.0552
| +0.0552
| 0.0554
| 0.0554
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4225/4224, 6656/6655
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4225/4224, 6656/6655
| [{{val| 814 1290 1890 2285 2816 3012 3327 }}]
| {{mapping| 814 1290 1890 2285 2816 3012 3327 }}
| +0.0573
| +0.0573
| 0.0528
| 0.0528
| 3.50
| 3.50
|-
| 2.3.5.7.11.13.17.19
| 1445/1444, 1521/1520, 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2601/2600
| {{mapping| 814 1290 1890 2285 2816 3012 3327 3458 }}
| +0.0421
| 0.0629
| 4.27
|-
| 2.3.5.7.11.13.17.19.23
| 1445/1444, 1521/1520, 1701/1700, 1863/1862, 2058/2057, 2376/2375, 2401/2400, 2601/2600
| {{mapping| 814 1290 1890 2285 2816 3012 3327 3682 }}
| +0.0439
| 0.0595
| 4.04
|}
|}
* 814et is notable in the 17- and 23-limit with lower absolute errors than any previous equal temperaments, beating [[764edo|764]] in the 17-limit and [[742edo|742i]] in the 23-limit, and is only bettered by [[935edo|935]] in either subgroup.  
* 814et is notable in the 17- and 23-limit with lower absolute errors than any previous equal temperaments, beating [[764edo|764]] in the 17-limit and [[742edo|742i]] in the 23-limit, and is only bettered by [[935edo|935]] in either subgroup.  
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{| 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 Octave
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Sesquiquartififths]]
| [[Sesquiquartififths]]
|}
|}
<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:Sesquiquartififths]]
[[Category:Sesquiquartififths]]
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