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{{Technical data page}}
{{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 ==
== Canopus ==
Line 9: Line 9:
[[Comma list]]: 16875/16807
[[Comma list]]: 16875/16807


{{Mapping|legend=2| 1 -2 -1 | 0 5 4 }}
{{Mapping|legend=3| 1 -2 -1 | 0 5 4 }}
: mapping generators: ~3, ~15/7
: mapping generators: ~3, ~15/7


Line 23: Line 23:


=== Overview to extensions ===
=== 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]]
* ''[[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.  
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:  
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]]
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
Line 37: Line 37:
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]
* ''[[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]]
* ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
Line 43: Line 44:
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma 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]]
* ''[[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 ==
Line 74: Line 76:
Comma list: 540/539, 1375/1372, 8019/8000
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 tunings:  
Optimal tunings:  
Line 89: Line 91:
Comma list: 351/350, 540/539, 676/675, 1375/1372
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 tunings:  
Optimal tunings:  
Line 104: Line 106:
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714
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 tunings:  
Optimal tunings:  
Line 139: Line 141:
Comma list: 540/539, 1375/1372, 65625/65536
Comma list: 540/539, 1375/1372, 65625/65536


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


Optimal tunings:  
Optimal tunings:  
Line 154: Line 156:
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
Comma list: 540/539, 625/624, 1375/1372, 4096/4095


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


Optimal tuning:  
Optimal tuning:  
Line 187: Line 189:
Comma list: 540/539, 1375/1372, 234375/234256
Comma list: 540/539, 1375/1372, 234375/234256


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


Optimal tunings:  
Optimal tunings:  
Line 202: Line 204:
Comma list: 540/539, 847/845, 1375/1372, 1575/1573
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 }}
{{Mapping|legend=0| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


Optimal tunings:  
Optimal tunings:  
Line 213: Line 215:


== Septendesemi ==
== 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) 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
[[Subgroup]]: 2.3.5.7
Line 237: Line 239:
Comma list: 540/539, 1375/1372, 43923/43750
Comma list: 540/539, 1375/1372, 43923/43750


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


Optimal tunings:  
Optimal tunings:  
Line 252: Line 254:
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
Comma list: 351/350, 540/539, 1375/1372, 4225/4224


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


Optimal tunings:  
Optimal tunings:  
Line 267: Line 269:
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224


Mapping: {{mapping| 1 -2 -1 -2 -1 3 4 | 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:  
Optimal tunings:  
Line 283: Line 285:
{{Main| Gaster temperament }}
{{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
[[Subgroup]]: 2.3.5.7
Line 289: Line 293:
[[Comma list]]: 16875/16807, 14348907/14336000
[[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 }}
{{Optimal ET sequence|legend=1| 111, 224 }}


[[Badness]] (Smith): 0.154521
[[Badness]] (Sintel): 3.91


=== 11-limit ===
=== 11-limit ===
Line 302: Line 311:
Comma list: 540/539, 1375/1372, 14348907/14336000
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 ===
=== 13-limit ===
Line 315: Line 326:
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
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 ===
=== 17-limit ===
Line 328: Line 341:
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197
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 }}
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}


Badness (Smith): 0.021436
Badness (Sintel): 1.09


=== 19-limit ===
=== 19-limit ===
Line 341: Line 356:
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
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 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness (Smith): 0.018370
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 354: Line 371:
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
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 }}
{{Optimal ET sequence|legend=0| 111, 224 }}


Badness (Smith): 0.017619
Badness (Sintel): 1.26


=== 29-limit ===
=== 29-limit ===
Line 367: Line 386:
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
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 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness (Smith): 0.016815
Badness (Sintel): 1.41


=== 31-limit ===
=== 31-limit ===
Line 380: Line 401:
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714
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 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness (Smith): 0.014790
Badness (Sintel): 1.42


=== 37-limit ===
=== 37-limit ===
Line 393: Line 416:
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714
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 }}
{{Optimal ET sequence|legend=0| 111, 113, 224 }}


Badness (Smith): 0.014377
Badness (Sintel): 1.56


=== 41-limit ===
=== 41-limit ===
Line 406: 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
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 }}
{{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 ==
== 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
[[Subgroup]]: 2.3.5.7
Line 488: Line 448:
[[Comma list]]: 16875/16807, 95703125/95551488
[[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 }}
{{Optimal ET sequence|legend=1| 103, 121, 224 }}


[[Badness]] (Smith): 0.162826
[[Badness]] (Sintel): 4.12


=== 11-limit ===
=== 11-limit ===
Line 502: Line 466:
Comma list: 540/539, 1375/1372, 2734375/2725888
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 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness (Smith): 0.043381
Badness (Sintel): 1.43


=== 13-limit ===
=== 13-limit ===
Line 515: Line 481:
Comma list: 540/539, 625/624, 1375/1372, 2200/2197
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 }}
{{Optimal ET sequence|legend=0| 103, 121, 224 }}


Badness (Smith): 0.021127
Badness (Sintel): 0.873


=== 17-limit ===
=== 17-limit ===
Line 528: Line 496:
Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197
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 }}
{{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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807, 1071875/1062882
[[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 }}
{{Optimal ET sequence|legend=1| 113, 121, 234 }}


[[Badness]] (Smith): 0.217166
[[Badness]] (Sintel): 5.50


=== 11-limit ===
=== 11-limit ===
Line 624: Line 531:
Comma list: 540/539, 1375/1372, 218750/216513
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 ===
=== 13-limit ===
Line 637: Line 546:
Comma list: 325/324, 364/363, 540/539, 2200/2197
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 ===
=== 17-limit ===
Line 650: Line 561:
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989
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 ===
=== 19-limit ===
Line 663: Line 576:
Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663
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 tunings:
* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}


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


Badness (Smith): 0.022574
Badness (Sintel): 1.37
 
== References ==


[[Category:Canopic clan| ]] <!-- main article -->
[[Category:Canopic| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Canopic_clan"