200edo: Difference between revisions

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== 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 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). 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 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.
The equal temperament [[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 2.3.17 subgroup mapping 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.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
200's divisors are: {{EDOs| 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 }}. It factorizes as 5<sup>2</sup> × 2<sup>3</sup>.  
200 factorizes as 5<sup>2</sup> × 2<sup>3</sup>. 200edo's subset edos are: {{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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| 2.3
| 2.3
| {{monzo| 317 -200 }}
| {{monzo| 317 -200 }}
| {{val| 200 317 }}
| {{mapping| 200 317 }}
| -0.0142
| -0.0142
| 0.0142
| 0.0142
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| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 2 -32 21 }}
| 32805/32768, {{monzo| 2 -32 21 }}
| {{val| 200 317 464 }}
| {{mapping| 200 317 464 }}
| +0.3226
| +0.3226
| 0.4767
| 0.4767
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| 2.3.5.7
| 2.3.5.7
| 1029/1024, 10976/10935, 390625/387072
| 1029/1024, 10976/10935, 390625/387072
| {{val| 200 317 464 561 }}
| {{mapping| 200 317 464 561 }}
| +0.4937
| +0.4937
| 0.5082
| 0.5082
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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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| [[Helmholtz]]
| [[Helmholtz]]
|}
|}
<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 ==
* 34 34 15 34 34 34 15 = [[5L_2s|Pythagorean tuning]]
* 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]]
* 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]]
* 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]]
* 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]]
* 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]]
* 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]]
* 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = [[11L_3s|Ketradektriatoh tuning]]


== Music ==
== Music ==
; [[Francium]]
; [[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]
* "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]


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [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]
* [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]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Listen]]
[[Category:Listen]]

Revision as of 12:48, 15 April 2024

← 199edo 200edo 201edo →
Prime factorization 23 × 52
Step size 6 ¢ 
Fifth 117\200 (702 ¢)
(semiconvergent)
Semitones (A1:m2) 19:15 (114 ¢ : 90 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

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 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).

The equal temperament tempers out the schisma, 32805/32768 and the quartemka, [2 -32 21⟩ in the 5-limit, and the gamelisma, 1029/1024, in the 7-limit, so that it supports 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.

Prime harmonics

Approximation of prime harmonics in 200edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.04 -2.31 -2.83 +0.68 -0.53 -2.96 +2.49 +1.73 +2.42 +0.96
Relative (%) +0.0 +0.7 -38.6 -47.1 +11.4 -8.8 -49.3 +41.4 +28.8 +40.4 +16.1
Steps
(reduced) 		200
(0) 		317
(117) 		464
(64) 		561
(161) 		692
(92) 		740
(140) 		817
(17) 		850
(50) 		905
(105) 		972
(172) 		991
(191)

Subsets and supersets

200 factorizes as 52 × 23. 200edo's subset edos are: 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.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [317 -200⟩ [⟨200 317]] -0.0142 0.0142 0.24
2.3.5 32805/32768, [2 -32 21⟩ [⟨200 317 464]] +0.3226 0.4767 7.95
2.3.5.7 1029/1024, 10976/10935, 390625/387072 [⟨200 317 464 561]] +0.4937 0.5082 8.47

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

Music

Francium
Claudi Meneghin
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