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The '''400 equal divisions of the octave''' ('''400edo''') is the [[EDO|equal division of the octave]] into 400 parts of exact 3 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== 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, supporting the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out 4914/4913 and [[24576/24565]] in the 17-limit, and [[1729/1728]] with 93347/93312 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.  


400edo doubles [[200edo]], which holds a record for the best 3/2 fifth approximation. 400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the Sym454 calendar scale with 231\400 as the generator, which can be treated as 5/12 syntonic comma meantone, which is the first meantone in the continued fraction that offers good precision. Other items like 1/3 and 2/5 eventually become inconsistent with the edo.  
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.  


The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to [[18/17]], making it an interpretation of [[18/17s equal temperament]]. Since it tempers out the 93347/93312, a stack of three 18/17s is equated with 19/16.  
=== Prime harmonics ===
{{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.


=== Prime harmonics ===
== Interval table ==
{{Primes in edo|400|columns=15}}
=== All intervals ===
See [[Table of 400edo intervals]].


=== Table of intervals ===
=== Selected intervals ===
{| class="wikitable"
{| class="wikitable center-1"
|+
!Step
!Name
!Associated ratio
!Notes
|-
|-
|0
! Step
|unison
! Eliora's naming system
|1/1 exact
! Associated ratio
|
|-
|-
|28
| 0
|5/12-meantone semitone
| unison
|6561/6250
| 1/1
|
|-
|-
|33
| 28
|small septendecimal semitone
| 5/12-meantone semitone
|[[18/17]]
| 6561/6250
|
|-
|-
|35
| 33
|septendecimal semitone
| small septendecimal semitone
|[[17/16]]
| [[18/17]], [[55/52]]
|
|-
|-
|37
| 35
|diatonic semitone
| septendecimal semitone
|[[16/15]]
| [[17/16]]
|
|-
|-
|99
| 37
|undevicesimal minor third
| diatonic semitone
|[[19/16]]
| [[16/15]]
|
|-
|-
|100
| 99
|symmetric minor third
| undevicesimal minor third
|
| [[19/16]]
|
|-
|-
|200
| 100
|symmetric tritone
| symmetric minor third
|[[99/70]], [[140/99]]
|  
|
|-
|-
|231
| 200
|Gregorian leap week fifth
| symmetric tritone
|118/79, twelfth root of 800000/6561
| [[99/70]], [[140/99]]
|
|-
|-
|234
| 231
|perfect fifth
| Gregorian leap week fifth
|[[3/2]]
| 525/352, 3/2 / (81/80)^(5/12)
|
|-
|-
|323
| 234
|harmonic seventh
| perfect fifth
|[[7/4]]
| [[3/2]]
|
|-
|-
|372
| 323
|5/12-meantone seventh
| harmonic seventh
|12500/6561
| [[7/4]]
|
|-
|-
|400
| 372
|octave
| 5/12-meantone seventh
|2/1 exact
| 12500/6561
|
|-
| 400
| octave
| 2/1
|}
|}


== 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]]
! 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
|-
|-
Line 98: Line 94:
| 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
Line 105: Line 101:
| 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
Line 112: 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
| [{{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
Line 119: 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
| [{{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
Line 126: 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
| [{{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
Line 133: 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
| [{{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
| 83\270
| 83\400
| 249.00
| 249.00
| {{monzo| -26 18 -1 }}
| {{monzo| -26 18 -1 }}
| [[Monzismic]]
| [[Monzismic]]
|-
| 1
| 33\400
| 99.00
| 18/17
| [[Gregorian leap day]]
|-
|-
| 1
| 1
Line 164: Line 168:
| 459.00
| 459.00
| 125/96
| 125/96
| [[Majvam]]
| [[Majvamic]]
|-
| 1
| 169\400
| 507.00
| 525/352
| [[Gregorian leap week]]
|-
|-
| 2
| 2
Line 173: Line 183:
|-
|-
| 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
| 166\400<br />(1\400)
| 498.00<br />(3.00)
| 4/3<br />(245/243)
| [[Octogintic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Line 189: Line 206:
* [[Huntington10]]
* [[Huntington10]]
* [[Huntington17]]
* [[Huntington17]]
* LeapWeek[71]{{clarify}}
* Monzismic[29]
* LeapDay[97]{{clarify}}
* GregorianLeapWeek[71]
* ISOWeek[71]
* GregorianLeapDay[97]
 
== 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:Equal divisions of the octave]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/400edo"