Canopic clan: Difference between revisions

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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 '''canopus clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the '''canopus''' or '''[[mirkwai comma]]''' ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.

== Canopus ==

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* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]

* ''[[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]]

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

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For ''no-twos'' 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, browser, and grazer, in the order of increasing [[badness]].

== Mirkat ==

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== 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

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~~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

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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 }}~~

~~: mapping generators: ~2, ~15/14~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.2693{{c}}, ~15/14 = 118.8923{{c}}

~~: [[error map]]: {{val| -0.731 +0.322 -0.060 +1.543 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 118.9682{{c}}

~~: error map: {{val| 0.000 +1.536 +1.545 +3.556 }}~~

~~{{Optimal ET sequence|legend=1| 10, 91cd, 101, 111, 121, 232d }}~~

~~[[Badness]] (Sintel): 3.99~~

~~=== 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 tunings:~~

* WE: ~2 = 1199.2891{{c}}, ~15/14 = 118.8978{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9662{{c}}

~~{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}~~

~~Badness (Sintel): 2.21~~

~~=== 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 tunings:~~

* WE: ~2 = 1199.2920{{c}}, ~15/14 = 118.8980{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9666{{c}}

~~{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}~~

~~Badness (Sintel): 1.31~~

~~=== 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 tunings:~~

* WE: ~2 = 1199.2648{{c}}, ~15/14 = 118.8946{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9655{{c}}

~~{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232dg }}~~

~~Badness (Sintel): 1.00~~

~~=== 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 tunings:~~

* WE: ~2 = 1199.2847{{c}}, ~15/14 = 118.8929{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9644{{c}}

~~{{Optimal ET sequence|legend=0| 10, 111, 121, 232dg }}~~

~~Badness (Sintel): 1.09~~

== 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) 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

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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.

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

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== Grazer ==

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

[[Subgroup]]: 2.3.5.7

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 {{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 '''canopus clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the '''canopus''' or '''[[mirkwai comma]]''' ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.
 == Canopus ==
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 * ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
 * ''[[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]]
 * ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
@@ Line 47: / Line 48: @@
 For ''no-twos'' 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, browser, and grazer, in the order of increasing [[badness]].
 == Mirkat ==
@@ Line 213: / Line 214: @@
 == 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
@@ Line 283: / Line 284: @@
 {{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
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 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 }}
-: mapping generators: ~2, ~15/14
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.2693{{c}}, ~15/14 = 118.8923{{c}}
-: [[error map]]: {{val| -0.731 +0.322 -0.060 +1.543 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 118.9682{{c}}
-: error map: {{val| 0.000 +1.536 +1.545 +3.556 }}
-{{Optimal ET sequence|legend=1| 10, 91cd, 101, 111, 121, 232d }}
-[[Badness]] (Sintel): 3.99
-=== 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 tunings:
-* WE: ~2 = 1199.2891{{c}}, ~15/14 = 118.8978{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9662{{c}}
-{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
-Badness (Sintel): 2.21
-=== 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 tunings:
-* WE: ~2 = 1199.2920{{c}}, ~15/14 = 118.8980{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9666{{c}}
-{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
-Badness (Sintel): 1.31
-=== 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 tunings:
-* WE: ~2 = 1199.2648{{c}}, ~15/14 = 118.8946{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9655{{c}}
-{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232dg }}
-Badness (Sintel): 1.00
-=== 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 tunings:
-* WE: ~2 = 1199.2847{{c}}, ~15/14 = 118.8929{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9644{{c}}
-{{Optimal ET sequence|legend=0| 10, 111, 121, 232dg }}
-Badness (Sintel): 1.09
 == 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) 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
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 {{See also| Sensipent family }}
-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.
+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
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 == Grazer ==
+Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.
 [[Subgroup]]: 2.3.5.7