Canopic clan: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The ''' | 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= | {{Mapping|legend=3| 1 -2 -1 | 0 5 4 }} | ||
: mapping generators: ~3, ~15/7 | : mapping generators: ~3, ~15/7 | ||
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=== 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) → [[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) → [[ | * ''[[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]] | ||
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* [[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- | For ''no-2's'' extensions, see [[No-twos subgroup temperaments #Canopus]]. | ||
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster | 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|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|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|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|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|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|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|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 == | ||
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|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|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|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 }} | ||
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 307: | Line 311: | ||
Comma list: 540/539, 1375/1372, 14348907/14336000 | Comma list: 540/539, 1375/1372, 14348907/14336000 | ||
{{Mapping|legend=0| 1 -8 -34 -32 8 | 0 19 72 69 -9 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 322: | Line 326: | ||
Comma list: 540/539, 729/728, 1375/1372, 2200/2197 | Comma list: 540/539, 729/728, 1375/1372, 2200/2197 | ||
{{Mapping|legend=0| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 337: | 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|legend=0| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 352: | 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|legend=0| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 367: | 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|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 382: | 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|legend=0| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 397: | 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|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: | Optimal tunings: | ||
| Line 412: | 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|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: | Optimal tunings: | ||
| Line 427: | 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|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: | Optimal tunings: | ||
| Line 436: | Line 440: | ||
Badness (Sintel): 1.57 | Badness (Sintel): 1.57 | ||
== Hemiseptisix == | == 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 542: | Line 466: | ||
Comma list: 540/539, 1375/1372, 2734375/2725888 | Comma list: 540/539, 1375/1372, 2734375/2725888 | ||
{{Mapping|legend=0| 1 -19 -7 -17 -28 | 0 53 24 51 81 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 557: | Line 481: | ||
Comma list: 540/539, 625/624, 1375/1372, 2200/2197 | Comma list: 540/539, 625/624, 1375/1372, 2200/2197 | ||
{{Mapping|legend=0| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 572: | 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|legend=0| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 582: | Line 506: | ||
Badness (Sintel): 0.948 | 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 | [[Subgroup]]: 2.3.5.7 | ||
| Line 687: | Line 531: | ||
Comma list: 540/539, 1375/1372, 218750/216513 | Comma list: 540/539, 1375/1372, 218750/216513 | ||
{{Mapping|legend=0| 1 -3 -4 -5 -1 | 0 37 51 63 36 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 702: | Line 546: | ||
Comma list: 325/324, 364/363, 540/539, 2200/2197 | Comma list: 325/324, 364/363, 540/539, 2200/2197 | ||
{{Mapping|legend=0| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 717: | 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|legend=0| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 732: | 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|legend=0| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 742: | Line 586: | ||
Badness (Sintel): 1.37 | Badness (Sintel): 1.37 | ||
[[Category:Canopic clan| ]] <!-- main article --> | |||
[[Category:Canopic| ]] <!-- key article --> | |||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||