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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 LeapWeek[71] scale with 231\400 as the generator, which is close to 5/12 syntonic comma meantone. Likewise, 400edo contains LeapDay[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]], making it an approximation of [[18/17 equal-step tuning]]. Since it tempers out the 93347/93312, a stack of three 18/17's is equated with 19/16.


=== 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.


== Selected intervals ==
== Interval table ==
=== All intervals ===
See [[Table of 400edo 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
|-
|-
Line 35: Line 37:
| 33
| 33
| small septendecimal semitone
| small septendecimal semitone
| [[18/17]]
| [[18/17]], [[55/52]]
|-
|-
| 35
| 35
Line 59: Line 61:
| 231
| 231
| Gregorian leap week fifth
| Gregorian leap week fifth
| 118/79
| 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]]
! 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 91: 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 98: 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 105: 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 112: 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 119: 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
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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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| [[Monzismic]]
| [[Monzismic]]
|-
|-
|1
| 1
|33\400
| 33\400
|99.00
| 99.00
|18/17
| 18/17
|Gregorian leap day
| [[Gregorian leap day]]
|-
|-
| 1
| 1
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| 459.00
| 459.00
| 125/96
| 125/96
| [[Majvam]]
| [[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
| 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 ==
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* [[Huntington10]]
* [[Huntington10]]
* [[Huntington17]]
* [[Huntington17]]
* LeapWeek[71]
* Monzismic[29]
* LeapDay[97]
* 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]]

Latest revision as of 12:33, 14 August 2025

← 399edo 400edo 401edo →
Prime factorization 24 × 52
Step size 3 ¢ 
Fifth 234\400 (702 ¢) (→ 117\200)
Semitones (A1:m2) 38:30 (114 ¢ : 90 ¢)
Consistency limit 21
Distinct consistency limit 21

400 equal divisions of the octave (abbreviated 400edo or 400ed2), also called 400-tone equal temperament (400tet) or 400 equal temperament (400et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 400 equal parts of exactly 3 ¢ each. Each step represents a frequency ratio of 21/400, or the 400th root of 2.

Theory

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

As an equal temperament, it tempers out the unidecma, [-7 22 -12⟩, and the quintosec comma, [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 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

Approximation of prime harmonics in 400edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41
Error Absolute (¢) +0.00 +0.04 +0.69 +0.17 +0.68 -0.53 +0.04 -0.51 -1.27 -0.58 +0.96 +0.66 -0.06
Relative (%) +0.0 +1.5 +22.9 +5.8 +22.7 -17.6 +1.5 -17.1 -42.5 -19.2 +32.1 +21.9 -2.1
Steps
(reduced) 		400
(0) 		634
(234) 		929
(129) 		1123
(323) 		1384
(184) 		1480
(280) 		1635
(35) 		1699
(99) 		1809
(209) 		1943
(343) 		1982
(382) 		2084
(84) 		2143
(143)
Approximation of prime harmonics in 400edo (continued)
Harmonic 43 47 53 59 61 67 71 73 79 83 89 97 101
Error Absolute (¢) +1.48 +0.49 -0.50 -0.17 -0.88 -1.31 +0.30 +0.21 +1.46 -0.05 -0.88 +0.10 -0.85
Relative (%) +49.4 +16.4 -16.8 -5.7 -29.5 -43.6 +10.1 +7.0 +48.8 -1.6 -29.3 +3.5 -28.5
Steps
(reduced) 		2171
(171) 		2222
(222) 		2291
(291) 		2353
(353) 		2372
(372) 		2426
(26) 		2460
(60) 		2476
(76) 		2522
(122) 		2550
(150) 		2590
(190) 		2640
(240) 		2663
(263)

Subsets and supersets

Since 400 factors into 24 × 52, 400edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200.

Of edos that are a multiple of 400, 1600 and 2000 are notable for their high consistency limits, as interval size measures, and perhaps as ways of tuning various temperaments.

Interval table

All intervals

See Table of 400edo intervals.

Selected intervals

Step Eliora's naming system Associated ratio
0 unison 1/1
28 5/12-meantone semitone 6561/6250
33 small septendecimal semitone 18/17, 55/52
35 septendecimal semitone 17/16
37 diatonic semitone 16/15
99 undevicesimal minor third 19/16
100 symmetric minor third
200 symmetric tritone 99/70, 140/99
231 Gregorian leap week fifth 525/352, 3/2 / (81/80)^(5/12)
234 perfect fifth 3/2
323 harmonic seventh 7/4
372 5/12-meantone seventh 12500/6561
400 octave 2/1

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-7 22 -12⟩, [47 -15 -10⟩ [⟨400 634 929]] −0.1080 0.1331 4.44
2.3.5.7 2401/2400, 1959552/1953125, 14348907/14336000 [⟨400 634 929 1123]] −0.0965 0.1170 3.90
2.3.5.7.11 2401/2400, 5632/5625, 9801/9800, 46656/46585 [⟨400 634 929 1123 1384]] −0.1166 0.1121 3.74
2.3.5.7.11.13 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325 [⟨400 634 929 1123 1384 1480]] −0.0734 0.1407 4.69
2.3.5.7.11.13.17 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095 [⟨400 634 929 1123 1384 1480 1635]] −0.0645 0.1321 4.40
2.3.5.7.11.13.17.19 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715 [⟨400 634 929 1123 1384 1480 1635 1699]] −0.0413 0.1380 4.60
  • 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat 354 in the 17-limit, and 311 in the 19-limit; it is bettered by 422 in either subgroup.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 83\400 249.00 [-26 18 -1⟩ Monzismic
1 33\400 99.00 18/17 Gregorian leap day
1 101\400 303.00 25/21 Quinmite
1 153\400 459.00 125/96 Majvamic
1 169\400 507.00 525/352 Gregorian leap week
2 61\400 183.00 10/9 Unidecmic
5 123\400
(37\400) 		369.00
(111.00) 		1024/891
(16/15) 		Quintosec
10 83\400
(3\400) 		249.00
(9.00) 		15/13
(176/175) 		Decoid
80 166\400
(1\400) 		498.00
(3.00) 		4/3
(245/243) 		Octogintic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Music

Eliora
Francium
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