814edo: Difference between revisions

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== Theory ==

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 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 ===

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== Regular temperament properties ==

{~~{comma basis begin}}~~

{| 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]] (%)

|-

| 2.3.5.7

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| 0.0595

| 4.04

~~{{comma basis end}~~}

|}

* 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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Note: 5-limit temperaments supported by 407edo are not included.

{{rank-2 ~~begin}}~~

{| 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

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| 448/405

| [[Sesquiquartififths]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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]]

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 {{Infobox ET}}
-{{EDO intro|814}}
+{{ED intro}}
 == Theory ==
-edo 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 171 &amp; 643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].
+edo 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 &amp; 643}} temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].
 === Prime harmonics ===
@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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]] (%)
 |-
 | 2.3.5.7
@@ Line 55: / Line 64: @@
 | 0.0595
 | 4.04
-{{comma basis end}}
+|}
 * 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.
@@ Line 61: / Line 70: @@
 Note: 5-limit temperaments supported by 407edo are not included.
-{{rank-2 begin}}
+{| 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
@@ Line 68: / Line 84: @@
 | 448/405
 | [[Sesquiquartififths]]
-{{rank-2 end}}
+|}
-{{orf}}
+<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]]