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=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}}

[[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 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]]

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

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

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 }}

[[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: {{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 }}

[[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: {{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 }}

[[Optimal tuning]]s:

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

{{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 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

{{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 }}

[[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: {{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 }}

 

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

 

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

 

 

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]]).

 

 

 

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

[[Optimal tuning]]s:  

* [[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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

=== 13-limit ===

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Badness]] (Sintel): 4.57

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

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

Optimal tunings:  

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

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

Badness (Sintel): 1.91

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

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

Optimal tunings:  

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

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

Badness (Sintel): 1.19

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

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

Optimal tunings:  

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

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

Badness (Sintel): 1.04

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

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

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

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]]s:

* [[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]] (Sintel): 5.50

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

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

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

 

 

 

 

 

 

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

 

 

 

 

 

 

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

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

[[Category:Temperament clans]]