Canopic clan: Difference between revisions

(13 intermediate revisions by 2 users not shown)

Line 1:

The '''~~mirkwai~~ clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[~~mirkwai~~ comma]] ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.

The '''canopic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[canopic comma]] ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.

== Canopus ==

Line 9:

[[Comma list]]: 16875/16807

: mapping generators: ~3, ~7/5

: mapping generators: ~3, ~15/7

[[Optimal tuning]] ([[~~POTE~~]]): ~3 = 1901.9550{{c}}, ~7/5 = ~~583~~.~~9584~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}

: [[error map]]: {{val| +1.785 -0.771 -2.248 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}

: error map: {{val| 0.000 -0.381 +1.093 }}

[[Badness]] (Sintel): 0.0996

=== Overview to extensions ===

The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.

The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament clan and collection pages.

* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]

* ''[[Octoid]]'' (+4375/4374) → [[~~Ragismic microtemperaments~~ #Octoid|~~Ragismic microtemperaments~~]]

* [[Octoid]] (+4375/4374) → [[Stearnsmic clan #Octoid|Stearnsmic clan]]

The others are weak extensions. Mirkat tempers out [[19683/19600]], splitting the generator in two with a semitwelfth period. Sqrtphi tempers out [[15625/15552]], splitting the period in six. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Kwai tempers out [[5120/5103]], splitting the generator in ten. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Grendel tempers out [[6144/6125]], splitting the generator in eleven. Finally, eris tempers out [[65625/65536]], splitting the generator in sixteen.

Members of the clan discussed elsewhere are:

* ''[[Kwai]]'' (+5120/5103) → [[~~Hemifamity~~ temperaments #Kwai|~~Hemifamity~~ temperaments]]

* ''[[Kwai]]'' (+5120/5103) → [[Aberschismic temperaments #Kwai|Aberschismic temperaments]]

* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]

* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]

Line 31:

Line 37:

* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]

* ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]

* ''[[Subsedia]]'' (+65536/64827) → [[Buzzardsmic clan #Subsedia|Buzzardsmic clan]]

* ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]

* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]

Line 37:

Line 44:

* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]

* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma family]]

* ''[[Browser]]'' (+78732/78125) → [[Sensipent family #Browser|Sensipent family]]

* ''[[Quintiquart]]'' (+390625000/387420489) → [[Quartonic family #Quintiquart|Quartonic family]]

For ''no-~~twos~~'' extensions, see [[No-twos subgroup temperaments #Canopus]].

For ''no-2's'' extensions, see [[No-twos subgroup temperaments #Canopus]].

Considered below are mirkat, eris, subsemifourth, septendesemi, gaster~~, subsedia~~, hemiseptisix~~, browser~~, and grazer.

Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, and grazer, in the order of increasing [[badness]].

== Mirkat ==

Mirkat tempers out 19683/19600, the [[cataharry comma]], as well as 250047/250000, the [[landscape comma]], and may be described as the {{nowrap| 72 & 111 }} temperament with a [[ploidacot]] signature of triploid alpha-hexacot.

[[Subgroup]]: 2.3.5.7

Line 49:

Line 59:

: mapping generators: ~63/50, ~10/9

[[Optimal tuning]] ([[~~POTE~~]]): ~63/50 = 400.~~000~~{{c}}, ~10/9 = 183.~~539~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}

: [[error map]]: {{val| +0.083 -0.591 -0.117 +0.950 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5470{{c}}

: error map: {{val| 0.000 -0.673 -0.203 +0.831 }}

[[Badness]] (~~Smith~~): 0.~~059376~~

[[Badness]] (Sintel): 1.50

=== 11-limit ===

Line 61:

Line 76:

Comma list: 540/539, 1375/1372, 8019/8000

~~Mapping:~~ {{~~mapping~~| 3 2 1 2 9 | 0 6 13 14 3 }}

Optimal ~~tuning (POTE)~~: ~63/50 = 400.~~000~~{{c}}, ~10/9 = 183.~~528~~{{c}}

Optimal tunings:

* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5391{{c}}

Badness (~~Smith~~): 0.~~022126~~

Badness (Sintel): 0.731

=== 13-limit ===

Line 74:

Line 91:

Comma list: 351/350, 540/539, 676/675, 1375/1372

~~Mapping:~~ {{~~mapping~~| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}

Optimal ~~tuning (POTE)~~: ~63/50 = 400.~~000~~{{c}}, ~10/9 = 183.~~577~~{{c}}

Optimal tunings:

* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}

Badness (~~Smith~~): 0.~~018632~~

Badness (Sintel): 0.770

=== 17-limit ===

Line 87:

Line 106:

Comma list: 351/350, 442/441, 540/539, 561/560, 715/714

~~Mapping:~~ {{~~mapping~~| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}

Optimal ~~tuning (POTE)~~: ~34/27 = 400.~~000~~{{c}}, ~10/9 = 183.~~579~~{{c}}

Optimal tunings:

* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}

* CWE: ~34/27 = 400.0000{{c}}, ~10/9 = 183.5843{{c}}

Badness (~~Smith~~): 0.~~011775~~

Badness (Sintel): 0.600

== Eris ==

The [[2.5.7 subgroup]] restriction of this temperament is [[exodia]].

Eris tempers out 65625/65536, the [[horwell comma]], and may be described as the {{nowrap| 31 & 224 }} temperament. The [[2.5.7-subgroup|2.5.7 subgroup]] restriction of this temperament is [[exodia]].

[[Subgroup]]: 2.3.5.7

Line 102:

Line 123:

[[Comma list]]: 16875/16807, 65625/65536

: mapping generators: ~2, ~49/30

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~60/49 = ~~348~~.~~216~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0256{{c}}, ~49/30 = 851.8023{{c}}

: [[error map]]: {{val| +0.026 -0.173 -0.528 +0.872 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/30 = 851.7845{{c}}

: error map: {{val| 0.000 -0.204 -0.590 +0.804 }}

{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224~~, 1823cd, 2271cd~~ }}

[[Badness]] (~~Smith~~): 0.~~074719~~

[[Badness]] (Sintel): 1.89

=== 11-limit ===

Line 115:

Line 141:

Comma list: 540/539, 1375/1372, 65625/65536

~~Mapping:~~ {{~~mapping~~| 1 ~~10 0 6 20~~ | 0 -29 8 -11 -57 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~11/9 = ~~348~~.~~219~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0218{{c}}, ~18/11 = 851.7963{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 851.7812{{c}}

{{Optimal ET sequence|legend=0| 31, 193, 224, 703, 927d~~, 1151cd~~ }}

Badness (~~Smith~~): 0.~~027621~~

Badness (Sintel): 0.913

=== 13-limit ===

Line 128:

Line 156:

Comma list: 540/539, 625/624, 1375/1372, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 ~~10 0 6 20~~ -14 | 0 -29 8 -11 -57 61 }}

Optimal tuning ~~(POTE)~~: ~2 = 1200.~~000~~{{c}}, ~11/9 = ~~348~~.~~213~~{{c}}

Optimal tuning:

* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}

* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}

Badness (~~Smith~~): 0.~~025137~~

Badness (Sintel): 1.04

== Subsemifourth ==

Line 141:

Line 171:

[[Comma list]]: 16875/16807, 26873856/26796875

: mapping generators: ~2, ~125~~/72~~

: mapping generators: ~2, ~144/125

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~144/125 = 244.~~719~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}

: [[error map]]: {{val| -0.082 -0.305 -0.223 +1.037 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/125 = 244.7172{{c}}

: error map: {{val| 0.000 -0.248 -0.082 +1.184 }}

[[Badness]] (~~Smith~~): 0.~~135173~~

[[Badness]] (Sintel): 3.42

=== 11-limit ===

Line 155:

Line 189:

Comma list: 540/539, 1375/1372, 234375/234256

~~Mapping:~~ {{~~mapping~~| 1 ~~39 27 45 56~~ | 0 -47 -31 -53 -66 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~121/105 = 244.~~719~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}

{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407~~, 966d~~ }}

Badness (~~Smith~~): 0.~~034276~~

Badness (Sintel): 1.13

=== 13-limit ===

Line 168:

Line 204:

Comma list: 540/539, 847/845, 1375/1372, 1575/1573

~~Mapping:~~ {{~~mapping~~| 1 ~~39 27 45 56 65~~ | 0 -47 -31 -53 -66 -77 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~15/13 = 244.~~714~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}

{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f~~, 662df~~ }}

Badness (~~Smith~~): 0.~~028387~~

Badness (Sintel): 1.17

== Septendesemi ==

~~The name ''septendesemi'' means a septendecimal semitone ([[17/16]]). The septendesemi temperament (80 & 103)~~ tempers out the mirkwai comma and 1959552/1953125 (''parkleiness'' comma~~, zotritrigu~~) in the 7-limit. [[183edo]] provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi.

Septendesemi tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma) in the 7-limit, and may be described as the {{nowrap| 80 & 103 }} temperament. [[183edo]] provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi. Septendesemi was named by [[Xenllium]] in 2021; the name ''septendesemi'' refers to a septendecimal semitone ([[17/16]]).

[[Subgroup]]: 2.3.5.7

Line 183:

Line 221:

[[Comma list]]: 16875/16807, 1959552/1953125

: mapping generators: ~2, ~~~648~~/~~343~~

: mapping generators: ~2, ~343/324

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~343/324 = 104.~~916~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}

: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 104.9134{{c}}

: error map: {{val| 0.000 -0.506 +0.395 +1.410 }}

[[Badness]] (~~Smith~~): 0.~~146795~~

[[Badness]] (Sintel): 3.71

=== 11-limit ===

Line 197:

Line 239:

Comma list: 540/539, 1375/1372, 43923/43750

~~Mapping:~~ {{~~mapping~~| 1 ~~39 37 53 50~~ | 0 -41 -38 -55 -51 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~35/33 = 104.~~916~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}

Badness (~~Smith~~): 0.~~041554~~

Badness (Sintel): 1.37

=== 13-limit ===

Line 210:

Line 254:

Comma list: 351/350, 540/539, 1375/1372, 4225/4224

~~Mapping:~~ {{~~mapping~~| 1 ~~39 37 53 50 11~~ | 0 -41 -38 -55 -51 -8 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~35/33 = 104.~~908~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}

{{Optimal ET sequence|legend=0| 80, 103, 183, 469f~~, 652def~~ }}

Badness (~~Smith~~): 0.~~027908~~

Badness (Sintel): 1.15

=== 17-limit ===

Line 223:

Line 269:

Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224

~~Mapping:~~ {{~~mapping~~| 1 ~~39 37 53 50 11 5~~ | 0 -41 -38 -55 -51 -8 -1 }}

Optimal tunings:

* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}

{{Optimal ET sequence|legend=0| 80, 103, 183, 469f~~, 652def~~ }}

Badness (~~Smith~~): 0.~~020128~~

Badness (Sintel): 1.03

== Gaster ==

Line 235:

Line 285:

~~The gaster temperament (111 & 113)~~ tempers out {{monzo| -70 72 -19 }} ~~(quadbila-negu)~~ in the 5-limit; mirkwai comma (16875/16807) and [[scheme comma]] (14348907/14336000) in the 7-limit. ~~The~~ word "[[Wiktionary: gaster|gaster]]" means [[Wiktionary: abdomen|abdomen]] or [[Wiktionary: stomach|stomach]], but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals [[111edo|111]], [[113edo|113]] and [[224edo|224]] support it even in the 41-limit.

Gaster tempers out {{monzo| -70 72 -19 }} in the 5-limit, mirkwai comma (16875/16807) and [[scheme comma]] (14348907/14336000) in the 7-limit, and may be described as the {{nowrap| 111 & 113 }} temperament.

It was named by [[Xenllium]] in 2022; the word "[[Wiktionary: gaster|gaster]]" means [[Wiktionary: abdomen|abdomen]] or [[Wiktionary: stomach|stomach]], but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals [[111edo|111]], [[113edo|113]] and [[224edo|224]] support it even in the 41-limit.

[[Subgroup]]: 2.3.5.7

Line 241:

Line 293:

[[Comma list]]: 16875/16807, 14348907/14336000

: mapping generators: ~2, ~567/400

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~~~800~~/~~567~~ = ~~594~~.~~641~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9920{{c}}, ~567/400 = 605.3546{{c}}

: [[error map]]: {{val| -0.008 -0.152 -0.506 +0.902 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~567/400 = 605.3586{{c}}

: error map: {{val| 0.000 -0.142 -0.497 +0.915 }}

[[Badness]] (~~Smith~~): 0.~~154521~~

[[Badness]] (Sintel): 3.91

=== 11-limit ===

Line 254:

Line 311:

Comma list: 540/539, 1375/1372, 14348907/14336000

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -1 | 0 -19 -72 -69 9 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~512~~/~~363~~ = ~~594~~.~~639~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}

{{Optimal ET sequence|legend=0| 111, 224, 783d~~, 1007d, 1231dd~~ }}

Badness (~~Smith~~): 0.~~054060~~

Badness (Sintel): 1.79

=== 13-limit ===

Line 267:

Line 326:

Comma list: 540/539, 729/728, 1375/1372, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26~~ | 0 -19 -72 -69 9 -45 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~55~~/39~~ = ~~594~~.~~639~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}

{{Optimal ET sequence|legend=0| 111, 224, 783df~~, 1007df, 1231ddf~~ }}

Badness (~~Smith~~): 0.~~024882~~

Badness (Sintel): 1.03

=== 17-limit ===

Line 280:

Line 341:

Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26 14~~ | 0 -19 -72 -69 9 -45 -20 }}

{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~24/17 = ~~594~~.~~636~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}

Badness (~~Smith~~): 0.~~021436~~

Badness (Sintel): 1.09

=== 19-limit ===

Line 293:

Line 356:

Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26 14~~ 32 | 0 -19 -72 -69 9 -45 -20 -56 }}

{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~24/17 = ~~594~~.~~636~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}

Badness (~~Smith~~): 0.~~018370~~

Badness (Sintel): 1.12

=== 23-limit ===

Line 306:

Line 371:

Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26 14~~ 32 7 | 0 -19 -72 -69 9 -45 -20 -56 -5 }}

{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~24/17 = ~~594~~.~~641~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.8733{{c}}, ~17/12 = 605.2946{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}

Badness (~~Smith~~): 0.~~017619~~

Badness (Sintel): 1.26

=== 29-limit ===

Line 319:

Line 386:

Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26 14~~ 32 7 -11 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 }}

{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~24/17 = ~~594~~.~~646~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9442{{c}}, ~17/12 = 605.3263{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3541{{c}}

Badness (~~Smith~~): 0.~~016815~~

Badness (Sintel): 1.41

=== 31-limit ===

Line 332:

Line 401:

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26 14~~ 32 7 -~~11 0~~ | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 }}

{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 | 0 19 72 69 -9 45 20 56 5 -32 -10 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~24/17 = ~~594~~.~~644~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9100{{c}}, ~17/12 = 605.3107{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3556{{c}}

Badness (~~Smith~~): 0.~~014790~~

Badness (Sintel): 1.42

=== 37-limit ===

Line 345:

Line 416:

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26 14~~ 32 7 -~~11 0~~ -27 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 }}

{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~24/17 = ~~594~~.~~644~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9087{{c}}, ~17/12 = 605.3101{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}

Badness (~~Smith~~): 0.~~014377~~

Badness (Sintel): 1.56

=== 41-limit ===

Line 358:

Line 431:

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666

~~Mapping:~~ {{~~mapping~~| 1 ~~11 38 37~~ -~~1 26 14~~ 32 7 -~~11 0~~ -~~27 45~~ | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80 }}

{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 -35 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 80 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~24/17 = ~~594~~.~~643~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}

Badness (~~Smith~~): ~~0.012858~~

Badness (Sintel): 1.57

~~== Subsedia ==~~

The generator for subsedia (10 & 111) is 0.5 cents flat of [[15/14]]-wide semitone and tempers out the mirkwai comma and [[buzzardsma]]. In this temperament, three generators makes ~[[16/13]], five of them equals ~[[24/17]], twelve of them equals ~[[16/7]], sixteen of them equals ~[[3/1]], and 45 of them equals ~22/1.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 16875/16807, 65536/64827~~

~~{{Mapping|legend=1| 1 0 5 4 | 0 16 -27 -12 }}~~

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 118.965{{c}}~~

~~{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}~~

~~[[Badness]] (Smith): 0.157658~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 540/539, 1375/1372, 65536/64827~~

~~Mapping: {{mapping| 1 0 5 4 -1 | 0 16 -27 -12 45 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}~~

~~{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}~~

~~Badness (Smith): 0.066838~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 352/351, 540/539, 676/675, 1375/1372~~

~~Mapping: {{mapping|~~ 1 ~~0 5 4 -1 4 | 0 16 -27 -12 45 -3 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}~~

~~{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}~~

~~Badness (Smith): 0.031635~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 256/255, 352/351, 442/441, 540/539, 715/714~~

~~Mapping: {{mapping| 1 0 5 4 -1 4 3 | 0 16 -27 -12 45 -3 11 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}~~

~~{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232dg }}~~

~~Badness (Smith): 0.019707~~

~~=== 19-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17.19~~

~~Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714~~

~~Mapping: {{mapping| 1 0 5 4 -1 4 3 10 | 0 16 -27 -12 45 -3 11 -58 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.964{{c}}~~

~~{{Optimal ET sequence|legend=0| 10, 101h, 111, 121, 232dg }}~~

~~Badness (Smith): 0~~.~~017935~~

== Hemiseptisix ==

~~The name ''hemiseptisix'' means a half of septimal major sixth ([[12/7]]). The hemiseptisix temperament (103 & 121)~~ tempers out the mirkwai comma and 95703125/95551488 (''pontiqak'' comma~~, lazozotritriyo~~) in the 7-limit. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix.

Hemiseptisix tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma) in the 7-limit, and may be described as the {{nowrap| 103 & 121 }} temperament. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix. Hemiseptisix was named by [[Xenllium]] in 2021; the name ''hemiseptisix'' refers to a half of septimal major sixth ([[12/7]]).

[[Subgroup]]: 2.3.5.7

Line 440:

Line 448:

[[Comma list]]: 16875/16807, 95703125/95551488

: mapping generators: ~2, ~75~~/49~~

: mapping generators: ~2, ~98/75

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~98/75 = 466.~~071~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.2693{{c}}, ~98/75 = 466.0801{{c}}

: [[error map]]: {{val| +0.023 -0.149 -0.553 +0.866 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 466.0715{{c}}

: error map: {{val| 0.000 -0.167 -0.598 +0.819 }}

[[Badness]] (~~Smith~~): 0.~~162826~~

[[Badness]] (Sintel): 4.12

=== 11-limit ===

Line 454:

Line 466:

Comma list: 540/539, 1375/1372, 2734375/2725888

~~Mapping:~~ {{~~mapping~~| 1 34 17 ~~34 53~~ | 0 -53 -24 -51 -81 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~55/42 = 466.~~070~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0183{{c}}, ~55/42 = 466.0767{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0699{{c}}

Badness (~~Smith~~): 0.~~043381~~

Badness (Sintel): 1.43

=== 13-limit ===

Line 467:

Line 481:

Comma list: 540/539, 625/624, 1375/1372, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 34 17 ~~34 53 30~~ | 0 -53 -24 -51 -81 -43 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~55/42 = 466.~~071~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9784{{c}}, ~55/42 = 466.0622{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0703{{c}}

Badness (~~Smith~~): 0.~~021127~~

Badness (Sintel): 0.873

=== 17-limit ===

Line 480:

Line 496:

Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 34 17 ~~34 53 30 31~~ | 0 -53 -24 -51 -81 -43 -44 }}

{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~17/13 = 466.~~074~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.8544{{c}}, ~17/13 = 466.0174{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 466.0718{{c}}

Badness (~~Smith~~): 0.~~018611~~

Badness (Sintel): 0.948

== ~~Browser~~ ==

== Grazer ==

~~{{See also| Sensipent family }}~~

Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.

~~This can also be considered a~~ [[~~non-over-1 temperament~~]]~~, with considerable scope for harmony~~ in ~~the 2.5/3.7/3.11/3.13/3.17/3 subgroup with MOSes of 8~~, ~~15, 23, and 31 notes despite no harmonics from~~ the ~~root. It can be considered a detemperament of 8et, with a generator very slightly sharp of 1\8~~.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 16875/16807, 78732/78125~~

~~{{Mapping|legend=1| 1 6 8 10 | 0 -35 -45 -57 }}~~

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 151.399{{c}}~~

~~{{Optimal ET sequence|legend=1| 103, 111, 214 }}~~

~~[[Badness]] (Smith): 0.180645~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 540/539, 1375/1372, 78732/78125~~

~~Mapping: {{mapping| 1 6 8 10 8 | 0 -35 -45 -57 -36 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.405{{c}}~~

~~{{Optimal ET sequence|legend=0| 103, 214 }}~~

~~Badness (Smith): 0.057634~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 351/350, 540/539, 847/845, 1375/1372~~

~~Mapping: {{mapping| 1 6 8 10 8 9 | 0 -35 -45 -57 -36 -42 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.403{{c}}~~

~~{{Optimal ET sequence|legend=0| 103, 111, 214 }}~~

~~Badness (Smith): 0.028822~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 351/350, 540/539, 561/560, 715/714, 847/845~~

~~Mapping: {{mapping| 1 6 8 10 8 9 8 | 0 -35 -45 -57 -36 -42 -31 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.397{{c}}~~

~~{{Optimal ET sequence|legend=0| 103, 111, 214 }}~~

~~Badness (Smith): 0.020384~~

~~=== 19-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17.19~~

~~Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714~~

~~Mapping: {{mapping| 1 6 8 10 8 9 8 18 | 0 -35 -45 -57 -36 -42 -31 -109 }}~~

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.396{{c}}~~

~~{{Optimal ET sequence|legend=0| 103h, 111, 214 }}~~

~~Badness (Smith): 0.017570~~

~~== Grazer ==~~

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807, 1071875/1062882

: mapping generators: ~2, ~90/49

: mapping generators: ~2, ~49/45

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~49/45 = 148.~~719~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0310{{c}}, ~49/45 = 148.7229{{c}}

: [[error map]]: {{val| +0.031 +0.700 -1.561 +0.563 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/45 = 148.7198{{c}}

: error map: {{val| 0.000 +0.676 -1.606 +0.519 }}

[[Badness]] (~~Smith~~): 0.~~217166~~

[[Badness]] (Sintel): 5.50

=== 11-limit ===

Line 576:

Line 531:

Comma list: 540/539, 1375/1372, 218750/216513

~~Mapping:~~ {{~~mapping~~| 1 ~~34 47 58 35~~ | 0 -37 -51 -63 -36 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~12/11 = 148.~~729~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.7242{{c}}, ~12/11 = 148.6946{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}

{{Optimal ET sequence|legend=0| 113, 121, 234~~, 355e, 589cee~~ }}

Badness (~~Smith~~): 0.~~076062~~

Badness (Sintel): 2.51

=== 13-limit ===

Line 589:

Line 546:

Comma list: 325/324, 364/363, 540/539, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 ~~34 47 58 35 44~~ | 0 -37 -51 -63 -36 -46 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~12/11 = 148.~~729~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.7257{{c}}, ~12/11 = 148.6947{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}

{{Optimal ET sequence|legend=0| 113, 121, 234~~, 355e, 589cee~~ }}

Badness (~~Smith~~): 0.~~036248~~

Badness (Sintel): 1.50

=== 17-limit ===

Line 602:

Line 561:

Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989

~~Mapping:~~ {{~~mapping~~| 1 ~~34 47 58 35 44 33~~ | 0 -37 -51 -63 -36 -46 -33 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~12/11 = 148.~~735~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.5690{{c}}, ~12/11 = 148.6815{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7267{{c}}

Badness (~~Smith~~): 0.~~025410~~

Badness (Sintel): 1.29

=== 19-limit ===

Line 615:

Line 576:

Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663

~~Mapping:~~ {{~~mapping~~| 1 ~~34 47 58 35 44 33 6~~ | 0 -37 -51 -63 -36 -46 -33 ~~-2 }}~~

{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }}

~~Optimal tuning (POTE): ~~~2 ~~= 1200.000{{c}}, ~12/11 = 148.727{{c~~}}

{{~~Optimal ET sequence|legend~~=~~0| 113~~, ~~121, 234g, 355eg, 589ceegg~~ }}

Optimal tunings:

* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}

~~Badness (Smith):~~ 0~~.022574~~

~~== References ==~~

Badness (Sintel): 1.37

[[Category:Canopic clan| ]]

[[Category:Canopic| ]]

[[Category:Temperament clans]]

[[Category:~~Mirkwai clan| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The '''mirkwai clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[mirkwai comma]] ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.
+The '''canopic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[canopic comma]] ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.
 == Canopus ==
@@ Line 9: / Line 9: @@
 [[Comma list]]: 16875/16807
-{{Mapping|legend=2| 1 3 3 | 0 -5 -4 }}
+{{Mapping|legend=3| 1 -2 -1 | 0 5 4 }}
-: mapping generators: ~3, ~7/5
+: mapping generators: ~3, ~15/7
-[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550{{c}}, ~7/5 = 583.9584{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
+: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
+* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
+: error map: {{val| 0.000 -0.381 +1.093 }}
 [[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
+[[Badness]] (Sintel): 0.0996
 === Overview to extensions ===
-The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.
+The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament clan and collection pages.
 * ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
-* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
+* [[Octoid]] (+4375/4374) → [[Stearnsmic clan #Octoid|Stearnsmic clan]]
 The others are weak extensions. Mirkat tempers out [[19683/19600]], splitting the generator in two with a semitwelfth period. Sqrtphi tempers out [[15625/15552]], splitting the period in six. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Kwai tempers out [[5120/5103]], splitting the generator in ten. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Grendel tempers out [[6144/6125]], splitting the generator in eleven. Finally, eris tempers out [[65625/65536]], splitting the generator in sixteen.
 Members of the clan discussed elsewhere are:
-* ''[[Kwai]]'' (+5120/5103) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]]
+* ''[[Kwai]]'' (+5120/5103) → [[Aberschismic temperaments #Kwai|Aberschismic temperaments]]
 * ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
 * ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
@@ Line 31: / Line 37: @@
 * ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
 * ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]
+* ''[[Subsedia]]'' (+65536/64827) → [[Buzzardsmic clan #Subsedia|Buzzardsmic clan]]
 * ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]
 * ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
@@ Line 37: / Line 44: @@
 * [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
 * ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma family]]
+* ''[[Browser]]'' (+78732/78125) → [[Sensipent family #Browser|Sensipent family]]
 * ''[[Quintiquart]]'' (+390625000/387420489) → [[Quartonic family #Quintiquart|Quartonic family]]
-For ''no-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]].
+For ''no-2's'' extensions, see [[No-twos subgroup temperaments #Canopus]].
-Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer.
+Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, and grazer, in the order of increasing [[badness]].
 == Mirkat ==
+Mirkat tempers out 19683/19600, the [[cataharry comma]], as well as 250047/250000, the [[landscape comma]], and may be described as the {{nowrap| 72 & 111 }} temperament with a [[ploidacot]] signature of triploid alpha-hexacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 49: / Line 59: @@
 {{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}
+: mapping generators: ~63/50, ~10/9
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~10/9 = 183.539{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}
+: [[error map]]: {{val| +0.083 -0.591 -0.117 +0.950 }}
+* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5470{{c}}
+: error map: {{val| 0.000 -0.673 -0.203 +0.831 }}
 {{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}
-[[Badness]] (Smith): 0.059376
+[[Badness]] (Sintel): 1.50
 === 11-limit ===
@@ Line 61: / Line 76: @@
 Comma list: 540/539, 1375/1372, 8019/8000
-Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
+{{Mapping|legend=0| 3 2 1 2 9 | 0 6 13 14 3 }}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~10/9 = 183.528{{c}}
+Optimal tunings:
+* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5391{{c}}
 {{Optimal ET sequence|legend=0| 39d, 72, 111, 183, 255 }}
-Badness (Smith): 0.022126
+Badness (Sintel): 0.731
 === 13-limit ===
@@ Line 74: / Line 91: @@
 Comma list: 351/350, 540/539, 676/675, 1375/1372
-Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}
+{{Mapping|legend=0| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~10/9 = 183.577{{c}}
+Optimal tunings:
+* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}
 {{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}
-Badness (Smith): 0.018632
+Badness (Sintel): 0.770
 === 17-limit ===
@@ Line 87: / Line 106: @@
 Comma list: 351/350, 442/441, 540/539, 561/560, 715/714
-Mapping: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}
+{{Mapping|legend=0| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}
-Optimal tuning (POTE): ~34/27 = 400.000{{c}}, ~10/9 = 183.579{{c}}
+Optimal tunings:
+* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}
+* CWE: ~34/27 = 400.0000{{c}}, ~10/9 = 183.5843{{c}}
 {{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}
-Badness (Smith): 0.011775
+Badness (Sintel): 0.600
 == Eris ==
-The [[2.5.7 subgroup]] restriction of this temperament is [[exodia]].
+Eris tempers out 65625/65536, the [[horwell comma]], and may be described as the {{nowrap| 31 & 224 }} temperament. The [[2.5.7-subgroup|2.5.7 subgroup]] restriction of this temperament is [[exodia]].
 [[Subgroup]]: 2.3.5.7
@@ Line 102: / Line 123: @@
 [[Comma list]]: 16875/16807, 65625/65536
-{{Mapping|legend=1| 1 10 0 6 | 0 -29 8 -11 }}
+{{Mapping|legend=1| 1 -19 8 -5 | 0 29 -8 11 }}
+: mapping generators: ~2, ~49/30
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~60/49 = 348.216{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0256{{c}}, ~49/30 = 851.8023{{c}}
+: [[error map]]: {{val| +0.026 -0.173 -0.528 +0.872 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/30 = 851.7845{{c}}
+: error map: {{val| 0.000 -0.204 -0.590 +0.804 }}
-{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224, 1823cd, 2271cd }}
+{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}
-[[Badness]] (Smith): 0.074719
+[[Badness]] (Sintel): 1.89
 === 11-limit ===
@@ Line 115: / Line 141: @@
 Comma list: 540/539, 1375/1372, 65625/65536
-Mapping: {{mapping| 1 10 0 6 20 | 0 -29 8 -11 -57 }}
+{{Mapping|legend=0| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 348.219{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0218{{c}}, ~18/11 = 851.7963{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 851.7812{{c}}
-{{Optimal ET sequence|legend=0| 31, 193, 224, 703, 927d, 1151cd }}
+{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}
-Badness (Smith): 0.027621
+Badness (Sintel): 0.913
 === 13-limit ===
@@ Line 128: / Line 156: @@
 Comma list: 540/539, 625/624, 1375/1372, 4096/4095
-Mapping: {{mapping| 1 10 0 6 20 -14 | 0 -29 8 -11 -57 61 }}
+{{Mapping|legend=0| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 348.213{{c}}
+Optimal tuning:
+* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}
+* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}
 {{Optimal ET sequence|legend=0| 31, 193, 224 }}
-Badness (Smith): 0.025137
+Badness (Sintel): 1.04
 == Subsemifourth ==
@@ Line 141: / Line 171: @@
 [[Comma list]]: 16875/16807, 26873856/26796875
-{{Mapping|legend=1| 1 39 27 45 | 0 -47 -31 -53 }}
+{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
-: mapping generators: ~2, ~125/72
+: mapping generators: ~2, ~144/125
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~144/125 = 244.719{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}
+: [[error map]]: {{val| -0.082 -0.305 -0.223 +1.037 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/125 = 244.7172{{c}}
+: error map: {{val| 0.000 -0.248 -0.082 +1.184 }}
 {{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}
-[[Badness]] (Smith): 0.135173
+[[Badness]] (Sintel): 3.42
 === 11-limit ===
@@ Line 155: / Line 189: @@
 Comma list: 540/539, 1375/1372, 234375/234256
-Mapping: {{mapping| 1 39 27 45 56 | 0 -47 -31 -53 -66 }}
+{{Mapping|legend=0| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~121/105 = 244.719{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}
-{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407, 966d }}
+{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407 }}
-Badness (Smith): 0.034276
+Badness (Sintel): 1.13
 === 13-limit ===
@@ Line 168: / Line 204: @@
 Comma list: 540/539, 847/845, 1375/1372, 1575/1573
-Mapping: {{mapping| 1 39 27 45 56 65 | 0 -47 -31 -53 -66 -77 }}
+{{Mapping|legend=0| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/13 = 244.714{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}
-{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f, 662df }}
+{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}
-Badness (Smith): 0.028387
+Badness (Sintel): 1.17
 == Septendesemi ==
-The name ''septendesemi'' means a septendecimal semitone ([[17/16]]). The septendesemi temperament (80 & 103) tempers out the mirkwai comma and 1959552/1953125 (''parkleiness'' comma, zotritrigu) in the 7-limit. [[183edo]] provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi.
+Septendesemi tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma) in the 7-limit, and may be described as the {{nowrap| 80 & 103 }} temperament. [[183edo]] provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi. Septendesemi was named by [[Xenllium]] in 2021; the name ''septendesemi'' refers to a septendecimal semitone ([[17/16]]).
 [[Subgroup]]: 2.3.5.7
@@ Line 183: / Line 221: @@
 [[Comma list]]: 16875/16807, 1959552/1953125
-{{Mapping|legend=1| 1 39 37 53 | 0 -41 -38 -55 }}
+{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
-: mapping generators: ~2, ~648/343
+: mapping generators: ~2, ~343/324
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~343/324 = 104.916{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}
+: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 104.9134{{c}}
+: error map: {{val| 0.000 -0.506 +0.395 +1.410 }}
 {{Optimal ET sequence|legend=0| 80, 103, 183 }}
-[[Badness]] (Smith): 0.146795
+[[Badness]] (Sintel): 3.71
 === 11-limit ===
@@ Line 197: / Line 239: @@
 Comma list: 540/539, 1375/1372, 43923/43750
-Mapping: {{mapping| 1 39 37 53 50 | 0 -41 -38 -55 -51 }}
+{{Mapping|legend=0| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 104.916{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}
 {{Optimal ET sequence|legend=0| 80, 103, 183 }}
-Badness (Smith): 0.041554
+Badness (Sintel): 1.37
 === 13-limit ===
@@ Line 210: / Line 254: @@
 Comma list: 351/350, 540/539, 1375/1372, 4225/4224
-Mapping: {{mapping| 1 39 37 53 50 11 | 0 -41 -38 -55 -51 -8 }}
+{{Mapping|legend=0| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 104.908{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}
-{{Optimal ET sequence|legend=0| 80, 103, 183, 469f, 652def }}
+{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
-Badness (Smith): 0.027908
+Badness (Sintel): 1.15
 === 17-limit ===
@@ Line 223: / Line 269: @@
 Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224
-Mapping: {{mapping| 1 39 37 53 50 11 5 | 0 -41 -38 -55 -51 -8 -1 }}
+{{Mapping|legend=0| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}
+Optimal tunings:
+* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}
 Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}
-{{Optimal ET sequence|legend=0| 80, 103, 183, 469f, 652def }}
+{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
-Badness (Smith): 0.020128
+Badness (Sintel): 1.03
 == Gaster ==
@@ Line 235: / Line 285: @@
 {{Main| Gaster temperament }}
-The gaster temperament (111 & 113) tempers out {{monzo| -70 72 -19 }} (quadbila-negu) in the 5-limit; mirkwai comma (16875/16807) and [[scheme comma]] (14348907/14336000) in the 7-limit. The word "[[Wiktionary: gaster|gaster]]" means [[Wiktionary: abdomen|abdomen]] or [[Wiktionary: stomach|stomach]], but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals [[111edo|111]], [[113edo|113]] and [[224edo|224]] support it even in the 41-limit.
+Gaster tempers out {{monzo| -70 72 -19 }} in the 5-limit, mirkwai comma (16875/16807) and [[scheme comma]] (14348907/14336000) in the 7-limit, and may be described as the {{nowrap| 111 & 113 }} temperament.
+It was named by [[Xenllium]] in 2022; the word "[[Wiktionary: gaster|gaster]]" means [[Wiktionary: abdomen|abdomen]] or [[Wiktionary: stomach|stomach]], but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals [[111edo|111]], [[113edo|113]] and [[224edo|224]] support it even in the 41-limit.
 [[Subgroup]]: 2.3.5.7
@@ Line 241: / Line 293: @@
 [[Comma list]]: 16875/16807, 14348907/14336000
-{{Mapping|legend=1| 1 11 38 37 | 0 -19 -72 -69 }}
+{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
+: mapping generators: ~2, ~567/400
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~800/567 = 594.641{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9920{{c}}, ~567/400 = 605.3546{{c}}
+: [[error map]]: {{val| -0.008 -0.152 -0.506 +0.902 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~567/400 = 605.3586{{c}}
+: error map: {{val| 0.000 -0.142 -0.497 +0.915 }}
 {{Optimal ET sequence|legend=1| 111, 224 }}
-[[Badness]] (Smith): 0.154521
+[[Badness]] (Sintel): 3.91
 === 11-limit ===
@@ Line 254: / Line 311: @@
 Comma list: 540/539, 1375/1372, 14348907/14336000
-Mapping: {{mapping| 1 11 38 37 -1 | 0 -19 -72 -69 9 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~512/363 = 594.639{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}
-{{Optimal ET sequence|legend=0| 111, 224, 783d, 1007d, 1231dd }}
+{{Optimal ET sequence|legend=0| 111, 224, 783d }}
-Badness (Smith): 0.054060
+Badness (Sintel): 1.79
 === 13-limit ===
@@ Line 267: / Line 326: @@
 Comma list: 540/539, 729/728, 1375/1372, 2200/2197
-Mapping: {{mapping| 1 11 38 37 -1 26 | 0 -19 -72 -69 9 -45 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/39 = 594.639{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}
-{{Optimal ET sequence|legend=0| 111, 224, 783df, 1007df, 1231ddf }}
+{{Optimal ET sequence|legend=0| 111, 224, 783df }}
-Badness (Smith): 0.024882
+Badness (Sintel): 1.03
 === 17-limit ===
@@ Line 280: / Line 341: @@
 Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197
-Mapping: {{mapping| 1 11 38 37 -1 26 14 | 0 -19 -72 -69 9 -45 -20 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.636{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}
 {{Optimal ET sequence|legend=0| 111, 224, 559dgg }}
-Badness (Smith): 0.021436
+Badness (Sintel): 1.09
 === 19-limit ===
@@ Line 293: / Line 356: @@
 Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
-Mapping: {{mapping| 1 11 38 37 -1 26 14 32 | 0 -19 -72 -69 9 -45 -20 -56 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.636{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}
 {{Optimal ET sequence|legend=0| 111, 224 }}
-Badness (Smith): 0.018370
+Badness (Sintel): 1.12
 === 23-limit ===
@@ Line 306: / Line 371: @@
 Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
-Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 | 0 -19 -72 -69 9 -45 -20 -56 -5 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.641{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8733{{c}}, ~17/12 = 605.2946{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}
 {{Optimal ET sequence|legend=0| 111, 224 }}
-Badness (Smith): 0.017619
+Badness (Sintel): 1.26
 === 29-limit ===
@@ Line 319: / Line 386: @@
 Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
-Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.646{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9442{{c}}, ~17/12 = 605.3263{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3541{{c}}
 {{Optimal ET sequence|legend=0| 111, 113, 224 }}
-Badness (Smith): 0.016815
+Badness (Sintel): 1.41
 === 31-limit ===
@@ Line 332: / Line 401: @@
 Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714
-Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 | 0 19 72 69 -9 45 20 56 5 -32 -10 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.644{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9100{{c}}, ~17/12 = 605.3107{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3556{{c}}
 {{Optimal ET sequence|legend=0| 111, 113, 224 }}
-Badness (Smith): 0.014790
+Badness (Sintel): 1.42
 === 37-limit ===
@@ Line 345: / Line 416: @@
 Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714
-Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 -27 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.644{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9087{{c}}, ~17/12 = 605.3101{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}
 {{Optimal ET sequence|legend=0| 111, 113, 224 }}
-Badness (Smith): 0.014377
+Badness (Sintel): 1.56
 === 41-limit ===
@@ Line 358: / Line 431: @@
 Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666
-Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 -27 45 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80 }}
+{{Mapping|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 -35 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 80 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 594.643{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}
 {{Optimal ET sequence|legend=0| 111, 113, 224 }}
-Badness (Smith): 0.012858
+Badness (Sintel): 1.57
-== Subsedia ==
-The generator for subsedia (10 & 111) is 0.5 cents flat of [[15/14]]-wide semitone and tempers out the mirkwai comma and [[buzzardsma]]. In this temperament, three generators makes ~[[16/13]], five of them equals ~[[24/17]], twelve of them equals ~[[16/7]], sixteen of them equals ~[[3/1]], and 45 of them equals ~22/1.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 16875/16807, 65536/64827
-{{Mapping|legend=1| 1 0 5 4 | 0 16 -27 -12 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 118.965{{c}}
-{{Optimal ET sequence|legend=1| 10, 101, 111, 121, 232d }}
-[[Badness]] (Smith): 0.157658
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 65536/64827
-Mapping: {{mapping| 1 0 5 4 -1 | 0 16 -27 -12 45 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}
-{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
-Badness (Smith): 0.066838
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 540/539, 676/675, 1375/1372
-Mapping: {{mapping| 1 0 5 4 -1 4 | 0 16 -27 -12 45 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}
-{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
-Badness (Smith): 0.031635
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 256/255, 352/351, 442/441, 540/539, 715/714
-Mapping: {{mapping| 1 0 5 4 -1 4 3 | 0 16 -27 -12 45 -3 11 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.968{{c}}
-{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232dg }}
-Badness (Smith): 0.019707
-=== 19-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714
-Mapping: {{mapping| 1 0 5 4 -1 4 3 10 | 0 16 -27 -12 45 -3 11 -58 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 118.964{{c}}
-{{Optimal ET sequence|legend=0| 10, 101h, 111, 121, 232dg }}
-Badness (Smith): 0.017935
 == Hemiseptisix ==
-The name ''hemiseptisix'' means a half of septimal major sixth ([[12/7]]). The hemiseptisix temperament (103 & 121) tempers out the mirkwai comma and 95703125/95551488 (''pontiqak'' comma, lazozotritriyo) in the 7-limit. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix.
+Hemiseptisix tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma) in the 7-limit, and may be described as the {{nowrap| 103 & 121 }} temperament. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix. Hemiseptisix was named by [[Xenllium]] in 2021; the name ''hemiseptisix'' refers to a half of septimal major sixth ([[12/7]]).
 [[Subgroup]]: 2.3.5.7
@@ Line 440: / Line 448: @@
 [[Comma list]]: 16875/16807, 95703125/95551488
-{{Mapping|legend=1| 1 34 17 34 | 0 -53 -24 -51 }}
+{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}
-: mapping generators: ~2, ~75/49
+: mapping generators: ~2, ~98/75
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~98/75 = 466.071{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.2693{{c}}, ~98/75 = 466.0801{{c}}
+: [[error map]]: {{val| +0.023 -0.149 -0.553 +0.866 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 466.0715{{c}}
+: error map: {{val| 0.000 -0.167 -0.598 +0.819 }}
 {{Optimal ET sequence|legend=1| 103, 121, 224 }}
-[[Badness]] (Smith): 0.162826
+[[Badness]] (Sintel): 4.12
 === 11-limit ===
@@ Line 454: / Line 466: @@
 Comma list: 540/539, 1375/1372, 2734375/2725888
-Mapping: {{mapping| 1 34 17 34 53 | 0 -53 -24 -51 -81 }}
+{{Mapping|legend=0| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/42 = 466.070{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0183{{c}}, ~55/42 = 466.0767{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0699{{c}}
 {{Optimal ET sequence|legend=0| 103, 121, 224 }}
-Badness (Smith): 0.043381
+Badness (Sintel): 1.43
 === 13-limit ===
@@ Line 467: / Line 481: @@
 Comma list: 540/539, 625/624, 1375/1372, 2200/2197
-Mapping: {{mapping| 1 34 17 34 53 30 | 0 -53 -24 -51 -81 -43 }}
+{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/42 = 466.071{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9784{{c}}, ~55/42 = 466.0622{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0703{{c}}
 {{Optimal ET sequence|legend=0| 103, 121, 224 }}
-Badness (Smith): 0.021127
+Badness (Sintel): 0.873
 === 17-limit ===
@@ Line 480: / Line 496: @@
 Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197
-Mapping: {{mapping| 1 34 17 34 53 30 31 | 0 -53 -24 -51 -81 -43 -44 }}
+{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/13 = 466.074{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8544{{c}}, ~17/13 = 466.0174{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 466.0718{{c}}
 {{Optimal ET sequence|legend=0| 103, 121, 224 }}
-Badness (Smith): 0.018611
+Badness (Sintel): 0.948
-== Browser ==
+== Grazer ==
-{{See also| Sensipent family }}
+Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.
-This can also be considered a [[non-over-1 temperament]], with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with MOSes of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8et, with a generator very slightly sharp of 1\8.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 16875/16807, 78732/78125
-{{Mapping|legend=1| 1 6 8 10 | 0 -35 -45 -57 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 151.399{{c}}
-{{Optimal ET sequence|legend=1| 103, 111, 214 }}
-[[Badness]] (Smith): 0.180645
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 78732/78125
-Mapping: {{mapping| 1 6 8 10 8 | 0 -35 -45 -57 -36 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.405{{c}}
-{{Optimal ET sequence|legend=0| 103, 214 }}
-Badness (Smith): 0.057634
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 351/350, 540/539, 847/845, 1375/1372
-Mapping: {{mapping| 1 6 8 10 8 9 | 0 -35 -45 -57 -36 -42 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.403{{c}}
-{{Optimal ET sequence|legend=0| 103, 111, 214 }}
-Badness (Smith): 0.028822
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
-Mapping: {{mapping| 1 6 8 10 8 9 8 | 0 -35 -45 -57 -36 -42 -31 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.397{{c}}
-{{Optimal ET sequence|legend=0| 103, 111, 214 }}
-Badness (Smith): 0.020384
-=== 19-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714
-Mapping: {{mapping| 1 6 8 10 8 9 8 18 | 0 -35 -45 -57 -36 -42 -31 -109 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 151.396{{c}}
-{{Optimal ET sequence|legend=0| 103h, 111, 214 }}
-Badness (Smith): 0.017570
-== Grazer ==
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 16875/16807, 1071875/1062882
-{{Mapping|legend=1| 1 34 47 58 | 0 -37 -51 -63 }}
+{{Mapping|legend=1| 1 -3 -4 -5 | 0 37 51 63 }}
-: mapping generators: ~2, ~90/49
+: mapping generators: ~2, ~49/45
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 148.719{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0310{{c}}, ~49/45 = 148.7229{{c}}
+: [[error map]]: {{val| +0.031 +0.700 -1.561 +0.563 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/45 = 148.7198{{c}}
+: error map: {{val| 0.000 +0.676 -1.606 +0.519 }}
 {{Optimal ET sequence|legend=1| 113, 121, 234 }}
-[[Badness]] (Smith): 0.217166
+[[Badness]] (Sintel): 5.50
 === 11-limit ===
@@ Line 576: / Line 531: @@
 Comma list: 540/539, 1375/1372, 218750/216513
-Mapping: {{mapping| 1 34 47 58 35 | 0 -37 -51 -63 -36 }}
+{{Mapping|legend=0| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.729{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7242{{c}}, ~12/11 = 148.6946{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}
-{{Optimal ET sequence|legend=0| 113, 121, 234, 355e, 589cee }}
+{{Optimal ET sequence|legend=0| 113, 121, 234 }}
-Badness (Smith): 0.076062
+Badness (Sintel): 2.51
 === 13-limit ===
@@ Line 589: / Line 546: @@
 Comma list: 325/324, 364/363, 540/539, 2200/2197
-Mapping: {{mapping| 1 34 47 58 35 44 | 0 -37 -51 -63 -36 -46 }}
+{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.729{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7257{{c}}, ~12/11 = 148.6947{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}
-{{Optimal ET sequence|legend=0| 113, 121, 234, 355e, 589cee }}
+{{Optimal ET sequence|legend=0| 113, 121, 234 }}
-Badness (Smith): 0.036248
+Badness (Sintel): 1.50
 === 17-limit ===
@@ Line 602: / Line 561: @@
 Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989
-Mapping: {{mapping| 1 34 47 58 35 44 33 | 0 -37 -51 -63 -36 -46 -33 }}
+{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.735{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.5690{{c}}, ~12/11 = 148.6815{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7267{{c}}
-{{Optimal ET sequence|legend=0| 113, 121, 234g, 355eg }}
+{{Optimal ET sequence|legend=0| 113, 121, 234g }}
-Badness (Smith): 0.025410
+Badness (Sintel): 1.29
 === 19-limit ===
@@ Line 615: / Line 576: @@
 Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663
-Mapping: {{mapping| 1 34 47 58 35 44 33 6 | 0 -37 -51 -63 -36 -46 -33 -2 }}
+{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 148.727{{c}}
-{{Optimal ET sequence|legend=0| 113, 121, 234g, 355eg, 589ceegg }}
+Optimal tunings:
+* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}
-Badness (Smith): 0.022574
+{{Optimal ET sequence|legend=0| 113, 121, 234g }}
-== References ==
+Badness (Sintel): 1.37
+[[Category:Canopic clan| ]] <!-- main article -->
+[[Category:Canopic| ]] <!-- key article -->
 [[Category:Temperament clans]]
-[[Category:Mirkwai clan| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]

Canopic clan: Difference between revisions

FloraC (talk | contribs)