Canopic clan: Difference between revisions

{{Technical data page}}

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

{{Main| Canopus }}

[[Subgroup]]: 3.5.7

{{Mapping|legend=2| 1 -2 -1 | 0 5 4 }}

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

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

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

[[Badness]] (Sintel): 0.0996

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

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

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.

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

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

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

Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, 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

{{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}

* [[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}}

[[Badness]] (Sintel): 1.50

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 39d, 72, 111, 183, 255 }}

Badness (Sintel): 0.731

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}

Badness (Sintel): 0.770

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}

Badness (Sintel): 0.600

 

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 -19 8 -5 | 0 29 -8 11 }}

: [[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 }}

[[Badness]] (Sintel): 1.89

Mapping: {{mapping| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}

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 }}

Badness (Sintel): 0.913

Mapping: {{mapping| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}

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 (Sintel): 1.04

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}

* [[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}}

[[Badness]] (Sintel): 3.42

Mapping: {{mapping| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}

Optimal tunings:  

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

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

Badness (Sintel): 1.13

Mapping: {{mapping| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}

Optimal tunings:  

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

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

Badness (Sintel): 1.17

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

{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}

: [[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]] (Sintel): 3.71

Mapping: {{mapping| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 80, 103, 183 }}

Badness (Sintel): 1.37

Mapping: {{mapping| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}

Optimal tunings:  

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

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

Badness (Sintel): 1.15

Mapping: {{mapping| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}

 

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

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

Badness (Sintel): 1.03

: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''

{{Main| Gaster 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

{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}

* [[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]] (Sintel): 3.91

Mapping: {{mapping| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}

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 }}

Badness (Sintel): 1.79

Mapping: {{mapping| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}

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 }}

Badness (Sintel): 1.03

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

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 (Sintel): 1.09

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

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 (Sintel): 1.12

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

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 (Sintel): 1.26

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

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 (Sintel): 1.41

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

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 (Sintel): 1.42

Mapping: {{mapping| 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 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 (Sintel): 1.56

Mapping: {{mapping| 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 tunings:  

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

{{Optimal ET sequence|legend=0| 111, 113, 224 }}

Badness (Sintel): 1.57

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

{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}

: [[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]] (Sintel): 4.12

Mapping: {{mapping| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 103, 121, 224 }}

Badness (Sintel): 1.43

Mapping: {{mapping| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 103, 121, 224 }}

Badness (Sintel): 0.873

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 103, 121, 224 }}

Badness (Sintel): 0.948

{{See also| Sensipent family }}

 

 

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 -29 -37 -47 | 0 35 45 57 }}

* [[WE]]: ~2 = 1199.9313{{c}}, ~90/49 = 1048.5414{{c}}

: [[error map]]: {{val| -0.069 -1.013 +0.592 +1.264 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~90/49 = 1048.5998{{c}}

: error map: {{val| 0.000 -0.962 +0.677 +1.362 }}

[[Badness]] (Sintel): 4.57

Mapping: {{mapping| 1 -29 -37 -47 -28 | 0 35 45 57 36 }}

Optimal tunings:  

* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5981{{c}}

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

Badness (Sintel): 1.91

Mapping: {{mapping| 1 -29 -37 -47 -28 -33 | 0 35 45 57 36 42 }}

Optimal tunings:  

* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5984{{c}}

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

Badness (Sintel): 1.19

Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}

Optimal tunings:  

* WE: ~2 = 1199.9191{{c}}, ~11/6 = 1048.5324{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6014{{c}}

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

Badness (Sintel): 1.04

Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 -91 | 0 35 45 57 36 42 31 109 }}

Optimal tunings:  

* WE: ~2 = 1199.9145{{c}}, ~11/6 = 1048.5290{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6021{{c}}

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

Badness (Sintel): 1.07

 

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 -3 -4 -5 | 0 37 51 63 }}

[[Optimal tuning]]s:

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

[[Badness]] (Sintel): 5.50

Mapping: {{mapping| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 113, 121, 234 }}

Badness (Sintel): 2.51

Mapping: {{mapping| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 113, 121, 234 }}

Badness (Sintel): 1.50

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 113, 121, 234g }}

Badness (Sintel): 1.29

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 113, 121, 234g }}

Badness (Sintel): 1.37