200edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
200edo contains a [[perfect fifth]] of exactly 702 cents and a [[perfect fourth]] of exactly 498 cents, which is accurate due to 200 being the denominator of a continued fraction convergent to log<sub>2</sub>(3/2). The error is only about 1/22 cents. In light of having its perfect fifth precise and the step divisible by 9, it is essentially a perfect edo for [[Carlos Alpha]], even up many octaves (the difference between 13 steps of 200edo and 1 step of Carlos Alpha is only 0.03501 cents). | 200edo contains a [[perfect fifth]] of exactly 702 cents and a [[perfect fourth]] of exactly 498 cents, which is accurate due to 200 being the denominator of a continued fraction convergent to log<sub>2</sub>(3/2). The error is only about 1/22 cents. In light of having its perfect fifth precise and the step divisible by 9, it is essentially a perfect edo for [[Carlos Alpha]], even up many octaves (the difference between 13 steps of 200edo and 1 step of Carlos Alpha is only 0.03501 cents). | ||
It [[tempering out|tempers out]] the [[schisma]] (32805/32768) and the quartemka, {{monzo| 2 -32 21 }} in the 5-limit, and the [[gamelisma]], 1029/1024, in the [[7-limit]], so that it [[support]]s the [[guiron]] temperament. | |||
One step of 200edo is close to [[289/288]]. Unfortunately, it is not compatible with any obvious 2.3.17 subgroup mappings of 200edo. | One step of 200edo is close to [[289/288]]. Unfortunately, it is not compatible with any obvious 2.3.17 subgroup mappings of 200edo. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
200 factorizes as | 200 factorizes as 2<sup>3</sup> × 5<sup>2</sup>, and has subset edos {{EDOs| 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 }}. | ||
[[400edo]], which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system. | [[400edo]], which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system. | ||
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== 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 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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| {{monzo| 317 -200 }} | | {{monzo| 317 -200 }} | ||
| {{mapping| 200 317 }} | | {{mapping| 200 317 }} | ||
| | | −0.0142 | ||
| 0.0142 | | 0.0142 | ||
| 0.24 | | 0.24 | ||
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=== 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 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| 498.00 | | 498.00 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
* 34 34 15 34 34 34 15 = [[ | * 34 34 15 34 34 34 15 = [[5L 2s|Pythagorean tuning]] | ||
* 32 32 20 32 32 32 20 = [[ | * 32 32 20 32 32 32 20 = [[5L 2s|Meantone tuning]] in the same way of [[50edo]] | ||
* 27 27 27 27 27 27 27 11 = [[ | * 27 27 27 27 27 27 27 11 = [[7L 1s|Porcupine tuning]] | ||
* 26 26 26 9 26 26 26 26 9 = [[ | * 26 26 26 9 26 26 26 26 9 = [[7L 2s|Superdiatonic tuning]] | ||
* 24 24 24 16 24 24 24 24 16 = [[ | * 24 24 24 16 24 24 24 24 16 = [[7L 2s|Superdiatonic tuning]] in the same way of [[25edo]] | ||
* 22 22 8 22 22 22 8 22 22 22 8 = [[ | * 22 22 8 22 22 22 8 22 22 22 8 = [[8L 3s|Sensi]] | ||
* 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = [[ | * 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = [[11L 3s|Ketradektriatoh tuning]] | ||
== Music == | == Music == | ||
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; [[Claudi Meneghin]] | ; [[Claudi Meneghin]] | ||
* ''Fugue on Elgar’s Enigma Theme'' | * ''Fugue on Elgar’s Enigma Theme'' – [https://www.youtube.com/watch?v=h4rjMFAzjow YouTube] | [http://soonlabel.com/xenharmonic/archives/1324 soonlabel archive]{{dead link}} | [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Claudi_Meneghin_Enigma_Fugue.mp3 play]{{dead link}} | ||
[[Category:3-limit record edos|###]] <!-- 3-digit number --> | |||
[[Category:Listen]] | [[Category:Listen]] | ||