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.

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

==Regular temperament properties==

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

== Regular temperament properties ==

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

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

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

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

~~! rowspan="2" |Optimal 8ve Stretch (¢)~~

~~! colspan="2" |Tuning Error~~

|-

~~![[TE error|Absolute]] (¢)~~

~~![[TE simple badness|Relative]] (%)~~

|-

|2.3

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

|{{monzo|317 -200}}

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

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

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

| -0.0142

! 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

| 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

| 0.4767

| 7.95

|-

|2.3.5.7

| 2.3.5.7

|1029/1024, 10976/10935, ~~32805~~/~~32768~~

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

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

| {{mapping| 200 317 464 561 }}

| +0.4937

| 0.5082

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=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

|1

| 1

|23\200

| 23\200

|138.00

| 138.00

|~~625~~/~~576~~

| 27/25

|[[Quartemka]]

| [[Quartemka]]

|-

|1

| 1

|39\200

| 39\200

|234.00

| 234.00

|~~9375~~/~~8192~~

| 8/7

|[[Guiron]]

| [[Guiron]]

|-

|1

| 1

|83\200

| 83\200

|498.00

| 498.00

|4/3

| 4/3

|[[Helmholtz]]

| [[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

== 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]]

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

== Music ==

; [[Francium]]

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

* "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]

* 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~~]]

; [[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}}

=~~= 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: @@
 {{Infobox ET}}
-{{EDO intro|200}}
+{{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.
 === 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 2<sup>3</sup> × 5<sup>2</sup>, and has subset edos {{EDOs| 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 }}.
-==Regular temperament properties==
+[[400edo]], which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
-! colspan="2" |Tuning Error
-|-
-![[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
 |-
-|2.3
+! rowspan="2" | [[Subgroup]]
-|{{monzo|317 -200}}
+! rowspan="2" | [[Comma list]]
-|{{val|200 317}}
+! rowspan="2" | [[Mapping]]
-| -0.0142
+! 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
+| 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
 | 0.4767
 | 7.95
 |-
-|2.3.5.7
+| 2.3.5.7
-|1029/1024, 10976/10935, 32805/32768
+| 1029/1024, 10976/10935, 390625/387072
-|{{val|200 317 464 561}}
+| {{mapping| 200 317 464 561 }}
 | +0.4937
 | 0.5082
@@ Line 50: / Line 53: @@
 === Rank-2 temperaments ===
 {| 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
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
-|1
+| 1
-|23\200
+| 23\200
-|138.00
+| 138.00
-|625/576
+| 27/25
-|[[Quartemka]]
+| [[Quartemka]]
 |-
-|1
+| 1
-|39\200
+| 39\200
-|234.00
+| 234.00
-|9375/8192
+| 8/7
-|[[Guiron]]
+| [[Guiron]]
 |-
-|1
+| 1
-|83\200
+| 83\200
-|498.00
+| 498.00
-|4/3
+| 4/3
-|[[Helmholtz]]
+| [[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
 == 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]]
-* 34 34 15 34 34 34 15 = [[5L_2s|Pythagorean tuning]]
+== Music ==
+; [[Francium]]
-* 32 32 20 32 32 32 20 = [[5L_2s|Meantone tuning]] in the same way of [[50edo]]
+* "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]
-* 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]]
+; [[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}}
-== 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]]