Xenharmonic Wiki

400edo: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,351 edits
No edit summary
Tristanbay (talk | contribs)
457 edits
→Subsets and supersets: Added 1600edo and 2000edo as notable 400edo supersets
Tags: Mobile edit Mobile web edit
 
(57 intermediate revisions by 10 users not shown)
Line 1: Line 1:
'''400edo''' is the [[EDO|equal division of the octave]] into 400 parts of exact 3 cents each. It tempers out the unidecma, 31381059609/31250000000 and the qintosec comma, 140737488355328/140126044921875 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, 4096/4095, and 39366/39325 in the 13-limit, supporting the [[Breedsmic temperaments|decoid temperament]] and the [[Breedsmic temperaments|quinmite temperament]].
{{Infobox ET}}
{{ED intro}}


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.
== Theory ==
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.  


==Scales==
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.
*[[Huntington7]]
*[[Huntington10]]
*[[Huntington17]]


[[Category:Equal divisions of the octave]]
=== 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.
 
== Interval table ==
=== 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"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
| {{mapping| 400 634 929 }}
| &minus;0.1080
| 0.1331
| 4.44
|-
| 2.3.5.7
| 2401/2400, 1959552/1953125, 14348907/14336000
| {{mapping| 400 634 929 1123 }}
| &minus;0.0965
| 0.1170
| 3.90
|-
| 2.3.5.7.11
| 2401/2400, 5632/5625, 9801/9800, 46656/46585
| {{mapping| 400 634 929 1123 1384 }}
| &minus;0.1166
| 0.1121
| 3.74
|-
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
| {{mapping| 400 634 929 1123 1384 1480 }}
| &minus;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
| {{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 ==
* [[Huntington7]]
* [[Huntington10]]
* [[Huntington17]]
* Monzismic[29]
* 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:Listen]]
Retrieved from "https://en.xen.wiki/w/400edo"