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 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.

 

 

[[Comma list]]: 16875/16807, 19683/19600

 

 

 

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

[[Badness]] (Sintel): 1.50

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

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

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 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

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 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

== Eris ==

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

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

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

: mapping generators: ~2, ~49/30

[[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 }}

[[Badness]] (Sintel): 1.89

Subgroup: 2.3.5.7.11

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

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

* 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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

== Subsemifourth ==

[[Subgroup]]: 2.3.5.7

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

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

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

* [[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]] (Sintel): 3.42

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

* 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 }}

Badness (Sintel): 1.13

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

* 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 }}

Badness (Sintel): 1.17

== 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

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

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

[[Optimal tuning]]s:

* [[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

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

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

=== 13-limit ===

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

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

=== 17-limit ===

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

 

 

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

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

 

 

 

 

 

 

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.

 

 

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

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

[[Optimal tuning]]s:  

* [[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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 111, 224, 783d }}

Badness (Sintel): 1.79

=== 13-limit ===

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

 

 

 

 

 

 

 

 

{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}

Badness (Sintel): 1.09

Subgroup: 2.3.5.7.11.13.17.19

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

 

* 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

Subgroup: 2.3.5.7.11.13.17.19.23

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

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

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

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

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

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

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:  

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

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

Badness (Sintel): 1.42

=== 37-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

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 -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:  

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

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

Badness (Sintel): 1.56

=== 41-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

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 -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 ==

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

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

{{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

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

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

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

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

 

 

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

Badness (Sintel): 0.948

{{See also| Sensipent family }}

Named by [[Xenllium]] in 2022, browser may be described as the {{nowrap| 103 & 111 }} 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 mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]]s:

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

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

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

[[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

 

 

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

 

 

 

 

 

 

 

 

 

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]]