400edo: Difference between revisions

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== Theory ==
== Theory ==
400edo is [[consistent]] in the [[21-odd-limit]]. It 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.  
400edo is a strong 17- and 19-limit system, [[consistency|distinctly consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1]] 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.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
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 contains a semiconvergent approximation to 3/2.
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 }}.  


== Interval table ==
== Interval table ==
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! Step
! Step
! Eliora's Naming System
! Eliora's Naming System
! Associated ratio
! Associated Ratio
|-
|-
| 0
| 0
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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
| -0.1080
| 0.1331
| 0.1331
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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
| -0.0965
| 0.1170
| 0.1170
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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
| -0.1166
| 0.1121
| 0.1121
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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
| -0.0734
| 0.1407
| 0.1407
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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
| -0.0645
| 0.1321
| 0.1321
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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
| -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 ===
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Octogintic]]
| [[Octogintic]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Scales ==
== Scales ==
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== Music ==
== Music ==
; [[Eliora]]
; [[Eliora]]
* [https://www.youtube.com/watch?v=av_RLK68ZUY ''Etude in Monzismic'' (Op. 2, No. 3)] (2023)
* [https://www.youtube.com/watch?v=av_RLK68ZUY ''Etude in Monzismic''] (2023)


; [[Francium]]
; [[Francium]]
* [https://www.youtube.com/watch?v=aTo2zfCWP9M ''thank you all''] (2023)
* [https://www.youtube.com/watch?v=aTo2zfCWP9M ''thank you all''] (2023)

Revision as of 16:19, 6 November 2023

← 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

Template:EDO intro

Theory

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

The equal temperament 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)

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.

Interval table

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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* 		Temperaments
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 it is distinct

Scales

Music

Eliora
Francium
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