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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)
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