Xenharmonic Wiki

200edo: Difference between revisions

← Older edit
VisualWikitext
KingHyperio (talk | contribs)
autopatrolled, emailconfirmed
216 edits
Eufalesio (talk | contribs)
autopatrolled
641 edits
m Next best fifth after 53edo
Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit
 
(25 intermediate revisions by 8 users not shown)
Line 1: Line 1:
{{EDO intro|200}}
{{Infobox ET}}
 
{{ED intro}}
One step of 200edo is close to [[289/288]].


== 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 log2(3/2). The error is only about 1/22 cent. In light of having its perfect fifth precise and the step divisibly 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). Only about 0.045 cents sharp, it is the next best fifth in absolute error after [[53edo]]'s. 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 tempers out the schisma, 32805/32768 and the quartemka, |2 -32 21&gt; in the 5-limit and the gamelisma, 1029/1024, in the [[7-limit]], so that it [[support]]s [[guiron]] temperament.
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.


200's divisors are: {{EDOs|2, 4, 5, 8, 10, 20, 25, 40, 50, 100}}. It factorizes as 5^2 * 2^3.
One step of 200edo is close to [[289/288]]. Unfortunately, it is not compatible with any obvious 2.3.17 subgroup mappings of 200edo.  


=== Odd harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|200}}
{{Harmonics in equal|200}}


== Scales ==
=== Subsets and supersets ===
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 }}.


* 34 34 15 34 34 34 15 = [[5L_2s|Pythagorean tuning]]
[[400edo]], which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.


* 32 32 20 32 32 32 20 = [[5L_2s|Meantone tuning]] in the same way of [[50edo]]
== 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
| {{monzo| 317 -200 }}
| {{mapping| 200 317 }}
| −0.0142
| 0.0142
| 0.24
|-
| 2.3.5
| 32805/32768, {{monzo| 2 -32 21 }}
| {{mapping| 200 317 464 }}
| +0.3226
| 0.4767
| 7.95
|-
| 2.3.5.7
| 1029/1024, 10976/10935, 390625/387072
| {{mapping| 200 317 464 561 }}
| +0.4937
| 0.5082
| 8.47
|}


* 27 27 27 27 27 27 27 11 = [[7L_1s|Porcupine tuning]]
=== 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*
! Temperaments
|-
| 1
| 23\200
| 138.00
| 27/25
| [[Quartemka]]
|-
| 1
| 39\200
| 234.00
| 8/7
| [[Guiron]]
|-
| 1
| 83\200
| 498.00
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


* 26 26 26 9 26 26 26 26 9 = [[7L_2s|Superdiatonic tuning]]
== Scales ==
* 34 34 15 34 34 34 15 = [[5L 2s|Pythagorean tuning]]
* 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 = [[7L 1s|Porcupine tuning]]
* 26 26 26 9 26 26 26 26 9 = [[7L 2s|Superdiatonic tuning]]
* 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 = [[8L 3s|Sensi]]
* 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = [[11L 3s|Ketradektriatoh tuning]]


* 24 24 24 16 24 24 24 24 16 = [[7L_2s|Superdiatonic tuning]] in the same way of [[25edo]]
== Music ==
; [[Francium]]
* "On Fire" from ''Mysteries'' (2023) – [https://open.spotify.com/track/6janPwh3S8FLgIzWf9S0oQ Spotify] | [https://francium223.bandcamp.com/track/on-fire Bandcamp] | [https://www.youtube.com/watch?v=S1NKb_EoYrw YouTube]


* 22 22 8 22 22 22 8 22 22 22 8 = [[8L_3s|Sensi]]
; [[Claudi Meneghin]]
 
* ''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}}
* 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = [[11L_3s|Ketradektriatoh tuning]]
 
== Music ==
* [http://soonlabel.com/xenharmonic/archives/1324 Fugue on Elgar’s Enigma Theme] [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Claudi_Meneghin_Enigma_Fugue.mp3 play] by [[Claudi Meneghin]]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:3-limit record edos|###]] <!-- 3-digit number -->
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/200edo"