200edo: Difference between revisions

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== 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 ~~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 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).

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 ===

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=== Subsets and supersets ===

200's ~~divisors~~ are: {{EDOs| 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 }}~~. It factorizes as 52 × 23~~.

200 factorizes as 52 × 23. 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.

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| 2.3

| {{monzo| 317 -200 }}

| {{~~val~~| 200 317 }}

| {{mapping| 200 317 }}

| -0.0142

| 0.0142

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| 2.3.5

| 32805/32768, {{monzo| 2 -32 21 }}

| {{~~val~~| 200 317 464 }}

| {{mapping| 200 317 464 }}

| +0.3226

| 0.4767

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| 2.3.5.7

| 1029/1024, 10976/10935, 390625/387072

| {{~~val~~| 200 317 464 561 }}

| {{mapping| 200 317 464 561 }}

| +0.4937

| 0.5082

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[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 ==

* 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]]

== 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]

* "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]]

* [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|###]] ~~

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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).
+edo 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 ===
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-'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>.
+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.
@@ Line 30: / Line 30: @@
 | 2.3
 | {{monzo| 317 -200 }}
-| {{val| 200 317 }}
+| {{mapping| 200 317 }}
 | -0.0142
 | 0.0142
@@ Line 37: / Line 37: @@
 | 2.3.5
 | 32805/32768, {{monzo| 2 -32 21 }}
-| {{val| 200 317 464 }}
+| {{mapping| 200 317 464 }}
 | +0.3226
 | 0.4767
@@ Line 44: / Line 44: @@
 | 2.3.5.7
 | 1029/1024, 10976/10935, 390625/387072
-| {{val| 200 317 464 561 }}
+| {{mapping| 200 317 464 561 }}
 | +0.4937
 | 0.5082
@@ Line 54: / Line 54: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 77: / Line 77: @@
 | [[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 ==
 * 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]]
 == 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]
+* "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]]
-* [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]]