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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. In the 17-limit, it tempers out 4914/4913.  
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.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|400}}
{{Harmonics in equal|400|columns=13}}
{{Harmonics in equal|400|columns=13|start=14|collapsed=true|title=Approximation of prime harmonics in 400edo (continued)}}


=== Table of intervals ===
=== Subsets and supersets ===
{| class="wikitable"
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 }}.
|+
 
!Step
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.
!Name
 
!Associated ratio
== Interval table ==
!Notes
=== All intervals ===
See [[Table of 400edo intervals]].
 
=== Selected intervals ===
{| class="wikitable center-1"
|-
! 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 ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
|0
! rowspan="2" | [[Subgroup]]
|unison
! rowspan="2" | [[Comma list]]
|1/1 exact
! rowspan="2" | [[Mapping]]
|
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|28
! [[TE error|Absolute]] (¢)
|5/12-meantone semitone
! [[TE simple badness|Relative]] (%)
|6561/6250
|
|-
|-
|35
| 2.3.5
|septendecimal semitone
| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
|[[17/16]]
| {{mapping| 400 634 929 }}
|
| &minus;0.1080
| 0.1331
| 4.44
|-
|-
|231
| 2.3.5.7
|Gregorian leap week fifth
| 2401/2400, 1959552/1953125, 14348907/14336000
|118/79, twelfth root of 800000/6561
| {{mapping| 400 634 929 1123 }}
|
| &minus;0.0965
| 0.1170
| 3.90
|-
|-
|234
| 2.3.5.7.11
|perfect fifth
| 2401/2400, 5632/5625, 9801/9800, 46656/46585
|[[3/2]]
| {{mapping| 400 634 929 1123 1384 }}
|
| &minus;0.1166
| 0.1121
| 3.74
|-
|-
|372
| 2.3.5.7.11.13
|5/12-meantone seventh
| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
|12500/6561
| {{mapping| 400 634 929 1123 1384 1480 }}
|
| &minus;0.0734
| 0.1407
| 4.69
|-
|-
|400
| 2.3.5.7.11.13.17
|octave
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
|1/1 exact
| {{mapping| 400 634 929 1123 1384 1480 1635 }}
|
| &minus;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
| {{mapping| 400 634 929 1123 1384 1480 1635 1699 }}
| &minus;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 [[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 ===
{| 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
| 83\400
| 249.00
| {{monzo| -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<br />(37\400)
| 369.00<br />(111.00)
| 1024/891<br />(16/15)
| [[Quintosec]]
|-
| 10
| 83\400<br />(3\400)
| 249.00<br />(9.00)
| 15/13<br />(176/175)
| [[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 57: Line 206:
* [[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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