Canopic clan: Difference between revisions

{{Technical data page}}

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.

{{Main| Canopus }}

[[Subgroup]]: 3.5.7

{{Mapping|legend=3| 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}}

[[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 clan and collection pages.

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.

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

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

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

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

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

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

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

[[Optimal tuning]]s:

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

{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}

[[Badness]] (Sintel): 1.50

{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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 }}

[[Optimal tuning]]s:  

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

* [[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|legend=0| 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|legend=0| 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 }}

[[Optimal tuning]]s:

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

{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}

[[Badness]] (Sintel): 3.42

{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}

Optimal tunings:  

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

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

 

Badness (Sintel): 1.03

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

{{Main| Gaster temperament }}

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.  

 

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:  

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

* [[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]] (Sintel): 3.91

{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|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 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|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 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|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 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|legend=0| 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|legend=0| 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|legend=0| 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

== Grazer ==

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

[[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}}

{{Optimal ET sequence|legend=1| 113, 121, 234 }}

[[Badness]] (Sintel): 5.50

{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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

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

[[Category:Canopic| ]] <!-- key article -->

[[Category:Temperament clans]]

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