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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|400}}
{{ED intro}}


== Theory ==
== Theory ==
400edo is a strong 17- and 19-limit system, [[consistency|distinctly consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.  
400edo is a strong 17- and 19-limit system, [[consistency|distinctly and purely consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.  


The equal temperament [[tempering out|tempers out]] the unidecma, {{monzo| -7 22 -12 }}, and the quintosec comma, {{monzo| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.  
As an equal temperament, it [[tempering out|tempers out]] the unidecma, {{monzo| -7 22 -12 }}, and the quintosec comma, {{monzo| 47 -15 -10 }}, in the [[5-limit]]; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the [[7-limit]]; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the [[13-limit]], [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|400|columns=13}}
{{Harmonics in equal|400|columns=13}}
{{Harmonics in equal|400|start=14|columns=13|title=Approximation of prime harmonics in 400edo (continued)}}
{{Harmonics in equal|400|columns=13|start=14|collapsed=true|title=Approximation of prime harmonics in 400edo (continued)}}
{{Harmonics in equal|400|start=27|columns=11|collapsed=1|title=Approximation of prime harmonics in 400edo (103 to 157)}}
{{Harmonics in equal|400|start=38|columns=14|collapsed=1|title=Approximation of prime harmonics in 400edo (163 to 229)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 400 factors into {{factorization|400}}, 400edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.
Since 400 factors into 2<sup>4</sup> × 5<sup>2</sup>, 400edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.
 
Of edos that are a multiple of 400, {{EDOs| 1600 and 2000}} are notable for their high consistency limits, as [[Interval size measure|interval size measures]], and perhaps as ways of tuning various temperaments.


== Interval table ==
== Interval table ==
=== All intervals ===
=== All intervals ===
''see [[Table of 400edo intervals]]''
See [[Table of 400edo intervals]].


=== Selected intervals ===
=== Selected intervals ===
Line 24: Line 24:
|-
|-
! Step
! Step
! Eliora's Naming System
! Eliora's naming system
! Associated Ratio
! Associated ratio
|-
|-
| 0
| 0
Line 81: Line 81:


== Regular temperament properties ==
== 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
| 2.3.5
| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
| {{Monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
| {{mapping| 400 634 929 }}
| {{Mapping| 400 634 929 }}
| &minus;0.1080
| −0.1080
| 0.1331
| 0.1331
| 4.44
| 4.44
Line 92: Line 101:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 1959552/1953125, 14348907/14336000
| 2401/2400, 1959552/1953125, 14348907/14336000
| {{mapping| 400 634 929 1123 }}
| {{Mapping| 400 634 929 1123 }}
| &minus;0.0965
| −0.0965
| 0.1170
| 0.1170
| 3.90
| 3.90
Line 99: Line 108:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 5632/5625, 9801/9800, 46656/46585
| 2401/2400, 5632/5625, 9801/9800, 46656/46585
| {{mapping| 400 634 929 1123 1384 }}
| {{Mapping| 400 634 929 1123 1384 }}
| &minus;0.1166
| −0.1166
| 0.1121
| 0.1121
| 3.74
| 3.74
Line 106: Line 115:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
| {{mapping| 400 634 929 1123 1384 1480 }}
| {{Mapping| 400 634 929 1123 1384 1480 }}
| &minus;0.0734
| −0.0734
| 0.1407
| 0.1407
| 4.69
| 4.69
Line 113: Line 122:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
| {{mapping| 400 634 929 1123 1384 1480 1635 }}
| {{Mapping| 400 634 929 1123 1384 1480 1635 }}
| &minus;0.0645
| −0.0645
| 0.1321
| 0.1321
| 4.40
| 4.40
Line 120: Line 129:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715
| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715
| {{mapping| 400 634 929 1123 1384 1480 1635 1699 }}
| {{Mapping| 400 634 929 1123 1384 1480 1635 1699 }}
| &minus;0.0413
| −0.0413
| 0.1380
| 0.1380
| 4.60
| 4.60
{{comma basis end}}
|}
* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo|354]] in the 17-limit, and [[311edo|311]] in the 19-limit; it is bettered by [[422edo|422]] in either subgroup.  
* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo|354]] in the 17-limit, and [[311edo|311]] in the 19-limit; it is bettered by [[422edo|422]] in either subgroup.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{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*
! Temperament
|-
|-
| 1
| 1
| 83\400
| 83\400
| 249.00
| 249.00
| {{monzo| -26 18 -1 }}
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
| [[Monzismic]]
|-
|-
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| 25/21
| 25/21
| [[Quinmite]]
| [[Quinmite]]
|-
| 1
| 131\400
| 393.00
| 2744/2187
| [[Emmthird]] (7-limit)
|-
|-
| 1
| 1
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|-
|-
| 5
| 5
| 123\400<br />(37\400)
| 123\400<br>(37\400)
| 369.00<br />(111.00)
| 369.00<br>(111.00)
| 1024/891<br />(16/15)
| 1024/891<br>(16/15)
| [[Quintosec]]
| [[Quintosec]]
|-
|-
| 10
| 10
| 83\400<br />(3\400)
| 83\400<br>(3\400)
| 249.00<br />(9.00)
| 249.00<br>(9.00)
| 15/13<br />(176/175)
| 15/13<br>(176/175)
| [[Decoid]]
| [[Decoid]]
|-
|-
| 80
| 80
| 166\400<br />(1\400)
| 166\400<br>(1\400)
| 498.00<br />(3.00)
| 498.00<br>(3.00)
| 4/3<br />(245/243)
| 4/3<br>(245/243)
| [[Octogintic]]
| [[Octogintic]]
{{rank-2 end}}
|}
{{orf}}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Retrieved from "https://en.xen.wiki/w/400edo"