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Misty family: Difference between revisions

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**Imported revision 251717492 - Original comment: **
Extensions: hazy (99 & 111), smoky (87 & 99), and ashy (99ef & 111)
 
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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 '''misty family''' [[tempering out|tempers out]] the [[misty comma]] ({{monzo|legend=1| 26 -12 -3 }}, [[ratio]]: 67108864/66430125).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-07 17:51:43 UTC</tt>.<br>
: The original revision id was <tt>251717492</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">[[toc|flat]]


The misty family tempers out the [[misty comma]], 67108864/66430125 = |26 -12 -3&gt;.
== Misty ==
{{Main| Misty }}


=Misty=
Misty features a third-octave period, which represents the [[512/405|ptolemaic diminished fourth (512/405)]]. Four [[4/3|perfect fourths]] (i.e. a minor sixth, [[~]][[128/81]]) minus a period give the interval class of 5. Misty has a [[ploidacot]] of triploid monocot, and equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap| ''n'' {{=}} 3 }}.
Comma: 67108864/66430125 = |26 -12 -3&gt;


POTE generator: ~3/2 = 703.111
[[Subgroup]]: 2.3.5


Map: [&lt;3 0 26|, &lt;0 1 -4|]
[[Comma list]]: 67108864/66430125
EDOs: 12, 51, 63, 75, 87, 99, 285, 384
Badness: 0.1065


==7-limit==
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
Commas: 3136/3125, 5120/5103
: mapping generators: ~512/405, ~3


[[POTE tuning|POTE generator]]: 703.021
[[Optimal tuning]]s:
* [[WE]]: ~512/405 = 399.8595{{c}}, ~3/2 = 702.8645{{c}}
: [[error map]]: {{val| -0.421 +0.488 +0.261 }}
* [[CWE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.1489{{c}}
: error map: {{val| 0.000 +1.194 +1.091 }}


Map: [&lt;3 0 26 56|, &lt;0 1 -4 -10|]
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc }}
Wedgie: &lt;&lt;3 -12 -30 -26 -56 -36||
EDOs: [[12edo|12]], [[87edo|87]], [[99edo|99]]


==11-limit==
[[Badness]] (Sintel): 2.50
Commas: 441/440, 3136/3125, 4375/4356


[[POTE tuning|POTE generator]]: 703.210
=== Overview to extensions ===
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
* Adding 3136/3125 (hemimean) leads to septimal misty.
* Adding 225/224 (marvel) leads to fog.


Map: [&lt;3 0 26 56 77|, &lt;0 1 -4 -10 -14|]
== Septimal misty ==
EDOs: 12, 87, 99, 186
{{Main| Misty }}


==13-limit==
Septimal misty tempers out not only [[3136/3125]] and [[5120/5103]], but [[250047/250000]], mapping [[~]][[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 99 }}, and finds [[7/4]] by [[5/4]] plus six extra fourths octave reduced. [[99edo]] makes for an excellent tuning, and [[87edo]] and [[111edo]] are notable alternatives.
Commas: 196/195, 352/351, 364/363, 635/624


[[POTE tuning|POTE generator]]: 703.303
It extends best to the no-11 no-13 [[19-limit]], where the ~[[16/15]] stands in for ~[[17/16]], and the ~[[25/21]], a fifth minus a period, stands in for ~[[19/16]]. The [[S-expression]]-based comma list for the 2.3.5.7.17-subgroup extension is {([[5120/5103|S8/S9]]), [[2025/2023|S15/S17]], [[256/255|S16]], [[1701/1700|S18/S20]], ([[2500/2499|S50]])}, and for the 2.3.5.7.17.19-subgroup extension, {([[5120/5103|S8/S9]]), [[256/255|S16]], [[324/323|S18]], [[400/399|S20]], [[1445/1444|S17/S19]], ([[2500/2499|S50]])}.  


Map: [&lt;3 0 26 56 77 92|, &lt;0 1 -4 -10 -14 -17|]
=== 7-limit ===
EDOs: 12, 87, 273
[[Subgroup]]: 2.3.5.7


=Mystic=
[[Comma list]]: 3136/3125, 5120/5103
Commas: 325/324, 441/440, 896/891, 1001/1000


[[POTE tuning|POTE generator]]: 703.311
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


Map: [&lt;3 0 26 56 77 -46|, &lt;0 1 -4 -10 -14 12|]
[[Optimal tuning]]s:  
EDOs: 12, 87, 273
* [[WE]]: ~63/50 = 399.8579{{c}}, ~3/2 = 702.7714{{c}}
: [[error map]]: {{val| -0.426 +0.390 +0.611 -0.236 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}
: error map: {{val| 0.000 +1.100 +1.466 +0.623 }}


==17-limit==
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}
Commas: 196/195, 352/351, 364/363, 635/624


[[POTE tuning|POTE generator]]: 703.233
[[Badness]] (Sintel): 0.931


Map: [&lt;3 0 26 56 77 92 -2|, &lt;0 1 -4 -10 -14 -17 3|]
==== 2.3.5.7.17-subgroup ====
EDOs: 12, 87, 186
Subgroup: 2.3.5.7.17


==19-limit==
Comma list: 256/255, 1701/1700, 2025/2023
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324


[[POTE tuning|POTE generator]]: 703.253
Subgroup-val mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
: mapping generators: ~34/27, ~3


Map: [&lt;3 0 26 56 77 92 -2 8|, &lt;0 1 -4 -10 -14 -17 3 1|]
Optimal tunings:  
EDOs: 12, 87, 186
* WE: ~34/27 = 399.8023{{c}}, ~3/2 = 702.6087{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 702.9837{{c}}


=Hemimist=
{{Optimal ET sequence|legend=0| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
Commas: 3136/3125, 5120/5103, 3025/3024


POTE generator: ~11/8 = 551.527
Badness (Sintel): 0.688


Map: [&lt;3 0 26 56 8|, &lt;0 2 -8 -20 1|]
==== 2.3.5.7.17.19 subgroup ====
EDOs: 15, 24, 87, 111, 198
Subgroup: 2.3.5.7.17.19
Badness: 0.0474


==13-limit==
Comma list: 256/255, 324/323, 400/399, 476/475
Commas: 352/351, 676/675, 847/845, 3025/3024


POTE generator: ~11/8 = 551.525
Subgroup-val mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}


Map: [&lt;3 0 26 56 8 23|, &lt;0 2 -8 -20 1 -5|]
Optimal tunings:  
EDOs: 15, 24, 87, 111, 198
* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}
Badness: 0.0229
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 702.9418{{c}}


{{Optimal ET sequence|legend=0| 12, 87, 99, 210gh, 309cdghh }}


Badness (Sintel): 0.637


=== Murky ===
Murky (87 & 99ef) tempers out [[441/440]] and [[896/891]], which joins misty with [[parapyth]]/[[pele]].


This extension used to be known as ''undecimal misty'', but was decanonicalized in 2026 in favor of the no-11 no-13 temperament.


Subgroup: 2.3.5.7.11


Comma list: 441/440, 896/891, 3136/3125


Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


</pre></div>
Optimal tunings:
<h4>Original HTML content:</h4>
* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
<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;Misty family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:18:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;a href="#Misty"&gt;Misty&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Mystic"&gt;Mystic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Hemimist"&gt;Hemimist&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}
&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;br /&gt;
 
The misty family tempers out the &lt;a class="wiki_link" href="/misty%20comma"&gt;misty comma&lt;/a&gt;, 67108864/66430125 = |26 -12 -3&amp;gt;.&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, …, 87, 99e, 186e }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Misty"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Misty&lt;/h1&gt;
Badness (Sintel): 1.19
Comma: 67108864/66430125 = |26 -12 -3&amp;gt;&lt;br /&gt;
 
&lt;br /&gt;
==== 13-limit ====
POTE generator: ~3/2 = 703.111&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26|, &amp;lt;0 1 -4|]&lt;br /&gt;
Comma list: 196/195, 352/351, 364/363, 625/624
EDOs: 12, 51, 63, 75, 87, 99, 285, 384&lt;br /&gt;
 
Badness: 0.1065&lt;br /&gt;
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Misty-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
Optimal tunings:
Commas: 3136/3125, 5120/5103&lt;br /&gt;
* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}
&lt;br /&gt;
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3258{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.021&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, …, 87, 186ef }}
Map: [&amp;lt;3 0 26 56|, &amp;lt;0 1 -4 -10|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;3 -12 -30 -26 -56 -36||&lt;br /&gt;
Badness (Sintel): 1.15
EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
===== 17-limit =====
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Misty-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13.17
Commas: 441/440, 3136/3125, 4375/4356&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.210&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
Map: [&amp;lt;3 0 26 56 77|, &amp;lt;0 1 -4 -10 -14|]&lt;br /&gt;
 
EDOs: 12, 87, 99, 186&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Misty-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{c}}
Commas: 196/195, 352/351, 364/363, 635/624&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efg }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.303&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.22
Map: [&amp;lt;3 0 26 56 77 92|, &amp;lt;0 1 -4 -10 -14 -17|]&lt;br /&gt;
 
EDOs: 12, 87, 273&lt;br /&gt;
===== 19-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Mystic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Mystic&lt;/h1&gt;
 
Commas: 325/324, 441/440, 896/891, 1001/1000&lt;br /&gt;
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.311&lt;br /&gt;
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56 77 -46|, &amp;lt;0 1 -4 -10 -14 12|]&lt;br /&gt;
Optimal tunings:  
EDOs: 12, 87, 273&lt;br /&gt;
* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}
&lt;br /&gt;
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.2216{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Mystic-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;17-limit&lt;/h2&gt;
 
Commas: 196/195, 352/351, 364/363, 635/624&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efgh }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.233&lt;br /&gt;
Badness (Sintel): 1.21
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56 77 92 -2|, &amp;lt;0 1 -4 -10 -14 -17 3|]&lt;br /&gt;
=== Hazy ===
EDOs: 12, 87, 186&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Mystic-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;19-limit&lt;/h2&gt;
Comma list: 176/175, 1375/1372, 5120/5103
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 0 26 56 96 | 0 1 -4 -10 -18 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.253&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;3 0 26 56 77 92 -2 8|, &amp;lt;0 1 -4 -10 -14 -17 3 1|]&lt;br /&gt;
* WE: ~63/50 = 399.8097{{c}}, ~3/2 = 702.4039{{c}}
EDOs: 12, 87, 186&lt;br /&gt;
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7731{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Hemimist"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Hemimist&lt;/h1&gt;
{{Optimal ET sequence|legend=0| 12, 87e, 99, 111 }}
Commas: 3136/3125, 5120/5103, 3025/3024&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.84
POTE generator: ~11/8 = 551.527&lt;br /&gt;
 
&lt;br /&gt;
==== 13-limit ====
Map: [&amp;lt;3 0 26 56 8|, &amp;lt;0 2 -8 -20 1|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 15, 24, 87, 111, 198&lt;br /&gt;
 
Badness: 0.0474&lt;br /&gt;
Comma list: 176/175, 351/350, 847/845, 1375/1372
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Hemimist-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit&lt;/h2&gt;
Mapping: {{mapping| 3 0 26 56 96 111 | 0 1 -4 -10 -18 -21 }}
Commas: 352/351, 676/675, 847/845, 3025/3024&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~11/8 = 551.525&lt;br /&gt;
* WE: ~63/50 = 399.8089{{c}}, ~3/2 = 702.3993{{c}}
&lt;br /&gt;
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7661{{c}}
Map: [&amp;lt;3 0 26 56 8 23|, &amp;lt;0 2 -8 -20 1 -5|]&lt;br /&gt;
 
EDOs: 15, 24, 87, 111, 198&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, 87ef, 99, 111 }}
Badness: 0.0229&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Badness (Sintel): 1.54
 
=== Smoky ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 2200/2187, 3136/3125
 
Mapping: {{mapping| 3 0 26 56 -61 | 0 1 -4 -10 15 }}
 
Optimal tunings:
* WE: ~63/50 = 400.0072{{c}}, ~3/2 = 702.3068{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 703.2941{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 87, 186, 273 }}
 
Badness (Sintel): 2.25
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 3136/3125
 
Mapping: {{mapping| 3 0 26 56 -61 -46 | 0 1 -4 -10 15 12 }}
 
Optimal tunings:
* WE: ~63/50 = 400.0184{{c}}, ~3/2 = 703.3420{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 703.3101{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 87, 186, 273 }}
 
Badness (Sintel): 1.63
 
=== Ashy ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 3136/3125, 5120/5103
 
Mapping: {{mapping| 3 0 26 56 -80 | 0 1 -4 -10 19 }}
 
Optimal tunings:
* WE: ~63/50 = 399.7776{{c}}, ~3/2 = 702.4764{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.8652{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 99e, 210e }}
 
Badness (Sintel): 1.89
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 3136/3125
 
Mapping: {{mapping| 3 0 26 56 -80 -65 | 0 1 -4 -10 19 16 }}
 
Optimal tunings:  
* WE: ~63/50 = 399.7470{{c}}, ~3/2 = 702.3782{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.8135{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 99ef, 111, 210ef, 321cdeff, 531ccddeeffff }}
 
Badness (Sintel): 1.72
 
=== Semimist ===
Subgroup: 2.3.5.7.11
 
Comma list: 3136/3125, 5120/5103, 9801/9800
 
Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
: mapping generators: ~55/49, ~3
 
Optimal tunings:
* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0062{{c}}
 
{{Optimal ET sequence|legend=0| 12, 174e, 186, 198, 408cde, 606bcde }}
 
Badness (Sintel): 2.26
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 1716/1715, 3136/3125
 
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0343{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 174ef, 186, 198 }}
 
Badness (Sintel): 1.82
 
=== Hemimist ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 3136/3125, 5120/5103
 
Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}
: mapping generators: ~63/50, ~400/231
 
Optimal tunings:  
* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5363{{c}}
 
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}
 
Badness (Sintel): 1.57
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 3025/3024
 
Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}
 
Optimal tunings:
* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5358{{c}}
 
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}
 
Badness (Sintel): 0.947
 
== Fog ==
Named by [[Xenllium]] in 2021, the fog temperament tempers out the marvel comma, [[225/224]], and may be described as {{nowrap| 12 & 63 }}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250/151263
 
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~80/63 = 399.7918{{c}}, ~3/2 = 703.7511{{c}}
: [[error map]]: {{val| -0.624 +1.172 -4.232 +4.712 }}
* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}
: error map: {{val| 0.000 +2.232 -3.060 +6.055 }}
 
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}
 
[[Badness]] (Sintel): 2.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 2420/2401
 
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
 
Optimal tunings:
* WE: ~44/35 = 399.8078{{c}}, ~3/2 = 704.0728{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.4697{{c}}
 
{{Optimal ET sequence|legend=0| 12, 39ce, 51, 63, 138de }}
 
Badness (Sintel): 1.70
 
[[Category:Temperament families]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Misty_family"