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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>4</sup> × 5<sup>2</sup>
{{ED intro}}
| Step size = 3.00000¢
| Fifth = 234\400 (702.00¢) (→ [[200edo|117\200]])
| Semitones = 38:30 (114.00¢ : 90.00¢)
| Consistency = 21
}}
The '''400 equal divisions of the octave''' ('''400edo'''), or the '''400(-tone) equal temperament''' ('''400tet''', '''400et''') when viewed from a [[regular temperament]] perspective,  is the [[EDO|equal division of the octave]] into 400 parts of exact 3 [[cent]]s each.


== Theory ==
== Theory ==
400edo is [[consistent]] in the [[21-odd-limit]]. It tempers out the unidecma, {{monzo| -7 22 -12 }}, and the qintosec 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|supporting]] 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.  
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.  


400 factors into  2<sup>4</sup> × 5<sup>2</sup>, with subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}. Notably, 200edo holds a record for the best 3/2 fifth approximation.
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.  
 
400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the GregorianLeapWeek[71] scale with 231\400 as the generator, which is close to 5/12 syntonic comma meantone. An interesting variation upon the scale would be the ISOWeek[71], which contains all the years which [[Wikipedia:ISO week date|have 53 weeks]] in the current calendar system. Likewise, 400edo contains GregorianLeapDay[97] scale, which is a [[maximal evenness]] version of the leap rule currently in use in the world today. The scale has a 33\400 generator which is associated to [[18/17]] and [[55/52]], and the corresponding temperament is 97 & 400, with comma list 2432/2431, 2601/2600, 2926/2925, 6175/6174, 17689/17680, and 22477/22440.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|400|columns=13}}
{{Harmonics in equal|400|columns=13}}
{{Harmonics in equal|400|columns=13|start=14|collapsed=true|title=Approximation of prime harmonics in 400edo (continued)}}
=== Subsets and supersets ===
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 ==
=== All intervals ===
See [[Table of 400edo intervals]].


== Selected intervals ==
=== Selected intervals ===
{| class="wikitable center-1"
{| class="wikitable center-1"
|+
|-
! Step
! Step
! Eliora's Naming System
! Eliora's naming system
! Associated ratio
! Associated ratio
|-
|-
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| 33
| 33
| small septendecimal semitone
| small septendecimal semitone
| [[18/17]]
| [[18/17]], [[55/52]]
|-
|-
| 35
| 35
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| 231
| 231
| Gregorian leap week fifth
| Gregorian leap week fifth
| 85/57, 94/63
| 525/352, 3/2 / (81/80)^(5/12)
|-
|-
| 234
| 234
Line 80: Line 82:
== 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
|-
|-
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| 2.3.5
| 2.3.5
| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
| [{{val| 400 634 929 }}]
| {{mapping| 400 634 929 }}
| -0.1080
| &minus;0.1080
| 0.1331
| 0.1331
| 4.44
| 4.44
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 1959552/1953125, 14348907/14336000
| 2401/2400, 1959552/1953125, 14348907/14336000
| [{{val| 400 634 929 1123 }}]
| {{mapping| 400 634 929 1123 }}
| -0.0965
| &minus;0.0965
| 0.1170
| 0.1170
| 3.90
| 3.90
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| 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
| [{{val| 400 634 929 1123 1384 }}]
| {{mapping| 400 634 929 1123 1384 }}
| -0.1166
| &minus;0.1166
| 0.1121
| 0.1121
| 3.74
| 3.74
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| 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
| [{{val| 400 634 929 1123 1384 1480 }}]
| {{mapping| 400 634 929 1123 1384 1480 }}
| -0.0734
| &minus;0.0734
| 0.1407
| 0.1407
| 4.69
| 4.69
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| 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
| [{{val| 400 634 929 1123 1384 1480 1635 }}]
| {{mapping| 400 634 929 1123 1384 1480 1635 }}
| -0.0645
| &minus;0.0645
| 0.1321
| 0.1321
| 4.40
| 4.40
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| 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
| [{{val| 400 634 929 1123 1384 1480 1635 1699 }}]
| {{mapping| 400 634 929 1123 1384 1480 1635 1699 }}
| -0.0413
| &minus;0.0413
| 0.1380
| 0.1380
| 4.60
| 4.60
|}
|}
* 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 ===
{| 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*
! Temperaments
! Associated<br />ratio*
! Temperament
|-
|-
| 1
| 1
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| 125/96
| 125/96
| [[Majvamic]]
| [[Majvamic]]
|-
| 1
| 169\400
| 507.00
| 525/352
| [[Gregorian leap week]]
|-
|-
| 2
| 2
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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)
| 10125/8192<br>(16/15)
| 1024/891<br />(16/15)
| [[Qintosec]] (5-limit)
| [[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
| 234\400<br>(1\400)
| 166\400<br />(1\400)
| 702.00<br>(3.00)
| 498.00<br />(3.00)
| 3/2<br>(245/243)
| 4/3<br />(245/243)
| [[Octogintic]]
| [[Octogintic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
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* [[Huntington10]]
* [[Huntington10]]
* [[Huntington17]]
* [[Huntington17]]
* Monzismic[29]
* GregorianLeapWeek[71]
* GregorianLeapWeek[71]
* ISOWeek[71]
* ISOWeek[71]
* GregorianLeapDay[97]
* GregorianLeapDay[97]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=av_RLK68ZUY ''Etude in Monzismic''] (2023)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=aTo2zfCWP9M ''thank you all''] (2023)
 
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/400edo"