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Canopic clan: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-26 14:23:57 UTC</tt>.<br>
: The original revision id was <tt>189995434</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">This tempers out the //mirkwai comma//, 16875/16807, a no-twos comma. Members of the clan include grendel and kwai, considered below, as well as miracle, octokaidecal, nusecond, semisept and bohpier, considered elsewhere.


==Grendel==
== Canopus ==
Commas: 6144/6125, 16875/16807
{{Main| Canopus }}


[[POTE tuning|POTE generator]]: 386.863
[[Subgroup]]: 3.5.7


Map: [&lt;1 9 2 7|, &lt;0 -23 1 -13|]
[[Comma list]]: 16875/16807
Wedgie: &lt;&lt;23 -1 13 -55 -44 33||
EDOs: 31, 90, 121, 152, 335


==Kwai==
{{Mapping|legend=2| 1 -2 -1 | 0 5 4 }}
Commas: 5120/5103, 16875/16807
: mapping generators: ~3, ~15/7


[[POTE tuning|POTE generator]]: 702.616
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
: error map: {{val| 0.000 -0.381 +1.093 }}


Map: [&lt;1 0 -50 -40|, &lt;0 1 33 27|]
[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
Wedgie: &lt;&lt;1 33 27 50 40 -30||
 
EDOs: 41, 111, 152, 345, 497</pre></div>
[[Badness]] (Sintel): 0.0996
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Mirkwai clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This tempers out the &lt;em&gt;mirkwai comma&lt;/em&gt;, 16875/16807, a no-twos comma. Members of the clan include grendel and kwai, considered below, as well as miracle, octokaidecal, nusecond, semisept and bohpier, considered elsewhere.&lt;br /&gt;
=== Overview to extensions ===
&lt;br /&gt;
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.
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Grendel"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Grendel&lt;/h2&gt;
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
Commas: 6144/6125, 16875/16807&lt;br /&gt;
* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 386.863&lt;br /&gt;
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.
&lt;br /&gt;
 
Map: [&amp;lt;1 9 2 7|, &amp;lt;0 -23 1 -13|]&lt;br /&gt;
Members of the clan discussed elsewhere are:  
Wedgie: &amp;lt;&amp;lt;23 -1 13 -55 -44 33||&lt;br /&gt;
* ''[[Kwai]]'' (+5120/5103) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]]
EDOs: 31, 90, 121, 152, 335&lt;br /&gt;
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
&lt;br /&gt;
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Kwai"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Kwai&lt;/h2&gt;
* ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]]
Commas: 5120/5103, 16875/16807&lt;br /&gt;
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
&lt;br /&gt;
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.616&lt;br /&gt;
* ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]
&lt;br /&gt;
* ''[[Subsedia]]'' (+65536/64827) → [[Buzzardsmic clan #Subsedia|Buzzardsmic clan]]
Map: [&amp;lt;1 0 -50 -40|, &amp;lt;0 1 33 27|]&lt;br /&gt;
* ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]
Wedgie: &amp;lt;&amp;lt;1 33 27 50 40 -30||&lt;br /&gt;
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
EDOs: 41, 111, 152, 345, 497&lt;/body&gt;&lt;/html&gt;</pre></div>
* ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]]
* ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[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
 
[[Comma list]]: 16875/16807, 19683/19600
 
{{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}
: mapping generators: ~63/50, ~10/9
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}
: [[error map]]: {{val| +0.083 -0.591 -0.117 +0.950 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5470{{c}}
: error map: {{val| 0.000 -0.673 -0.203 +0.831 }}
 
{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}
 
[[Badness]] (Sintel): 1.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 8019/8000
 
Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
 
Optimal tunings:
* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}
* 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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}
 
{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}
 
Badness (Sintel): 0.770
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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:
* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}
* 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 }}
 
{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}
 
[[Badness]] (Sintel): 1.89
 
=== 11-limit ===
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 }}
 
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
 
=== 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 }}
: mapping generators: ~2, ~144/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}
: [[error map]]: {{val| -0.082 -0.305 -0.223 +1.037 }}
* [[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 }}
 
Optimal tunings:
* 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 }}
 
Optimal tunings:
* 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 }}
: mapping generators: ~2, ~343/324
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}
: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}
* [[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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 43923/43750
 
Mapping: {{mapping| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}
 
Optimal tunings:
* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}
 
{{Optimal ET sequence|legend=0| 80, 103, 183 }}
 
Badness (Sintel): 1.37
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
* 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 ===
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}
 
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}
 
Badness (Sintel): 1.03
 
== Gaster ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
{{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
 
[[Comma list]]: 16875/16807, 14348907/14336000
 
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
: mapping generators: ~2, ~567/400
 
[[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 }}
 
[[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:
* 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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
 
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
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197
 
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
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
 
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
 
=== 23-limit ===
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
 
=== 29-limit ===
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
 
=== 31-limit ===
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:
* 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
 
=== 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:
* 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
 
=== 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:
* 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 }}
 
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 }}
: mapping generators: ~2, ~98/75
 
[[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 }}
 
[[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:
* 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 }}
 
Badness (Sintel): 1.43
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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:
* 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 }}
 
Badness (Sintel): 0.873
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197
 
Mapping: {{mapping| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}
 
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 }}
 
Badness (Sintel): 0.948
 
== Browser ==
{{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 }}
: mapping generators: ~2, ~90/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9313{{c}}, ~90/49 = 1048.5414{{c}}
: [[error map]]: {{val| -0.069 -1.013 +0.592 +1.264 }}
* [[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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 78732/78125
 
Mapping: {{mapping| 1 -29 -37 -47 -28 | 0 35 45 57 36 }}
 
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5981{{c}}
 
{{Optimal ET sequence|legend=0| 103, 214 }}
 
Badness (Sintel): 1.91
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5984{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.19
 
=== 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 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}
 
Optimal tunings:
* WE: ~2 = 1199.9191{{c}}, ~11/6 = 1048.5324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6014{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.04
 
=== 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 -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
 
== Grazer ==
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 }}
: mapping generators: ~2, ~49/45
 
[[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 }}
 
[[Badness]] (Sintel): 5.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 218750/216513
 
Mapping: {{mapping| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}
 
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 }}
 
Badness (Sintel): 2.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 540/539, 2200/2197
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}
 
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 }}
 
Badness (Sintel): 1.50
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}
 
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 }}
 
Badness (Sintel): 1.29
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
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:
* 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 }}
 
Badness (Sintel): 1.37
 
[[Category:Temperament clans]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Canopic_clan"