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). 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, {{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.

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 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.

=== ~~Odd~~ harmonics ===

=== Prime harmonics ===

=== ~~Miscellaneous properties~~ ===

=== 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 23 × 52, and has subset edos {{EDOs| 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 }}.

~~== Scales ==~~

[[400edo]], which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.

* 34 34 15 34 34 34 15 = [[~~5L_2s~~|~~Pythagorean tuning~~]]

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 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

|}

* ~~32 32 20 32 32 32 20 =~~ [[~~5L_2s~~|~~Meantone tuning~~]] in the ~~same way of~~ [[~~50edo~~]]

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | 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 (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

* 27 27 27 27 27 27 27 11 = [[~~7L_1s~~|Porcupine 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]]

* 26 26 26 9 26 26 26 26 9 = [[~~7L_2s~~|~~Superdiatonic 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]

* 24 24 24 16 24 24 24 24 16 = [[~~7L_2s|Superdiatonic tuning]] in the same way of [[25edo~~]]

; [[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}}

* ~~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 ==~~

* [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~~|###]]

[[Category:3-limit record edos|###]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-{{EDO intro|200}}
+{{Infobox ET}}
+{{ED intro}}
 == 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). 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, {{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.
+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 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.
-=== Odd harmonics ===
+=== Prime harmonics ===
 {{Harmonics in equal|200}}
-=== Miscellaneous properties ===
+=== 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 2<sup>3</sup> × 5<sup>2</sup>, and has subset edos {{EDOs| 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 }}.
-== Scales ==
+[[400edo]], which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.
-* 34 34 15 34 34 34 15 = [[5L_2s|Pythagorean tuning]]
+== 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
+|}
-* 32 32 20 32 32 32 20 = [[5L_2s|Meantone tuning]] in the same way of [[50edo]]
+=== 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
-* 27 27 27 27 27 27 27 11 = [[7L_1s|Porcupine 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]]
-* 26 26 26 9 26 26 26 26 9 = [[7L_2s|Superdiatonic 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]
-* 24 24 24 16 24 24 24 24 16 = [[7L_2s|Superdiatonic tuning]] in the same way of [[25edo]]
+; [[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}}
-* 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 ==
-* [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]]