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Misty family: 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 '''misty family''' tempers out the [[misty comma]], 67108864/66430125 = {{monzo| 26 -12 -3 }}, equating the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]]. It features a third-octave period, which is represented by [[512/405]] and acts as the interval separating ~[[128/81]] from ~[[5/4]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-07 17:43:32 UTC</tt>.<br>
: The original revision id was <tt>251715262</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;.
The second comma of the [[Normal lists #Normal interval list|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.


=Misty=
== Misty ==
Comma: 67108864/66430125 = |26 -12 -3&gt;
{{Main| Misty }}


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


{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}


=Misty=
: mapping generators: ~512/405, ~3
Commas: 3136/3125, 5120/5103


[[POTE tuning|POTE generator]]: 703.021
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111


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 }}
Wedgie: &lt;&lt;3 -12 -30 -26 -56 -36||
EDOs: [[12edo|12]], [[87edo|87]], [[99edo|99]]


==11-limit==
[[Badness]]: 0.106540
Commas: 441/440, 3136/3125, 4375/4356


[[POTE tuning|POTE generator]]: 703.210
== Septimal misty ==
{{Main| Misty }}


Map: [&lt;3 0 26 56 77|, &lt;0 1 -4 -10 -14|]
[[Subgroup]]: 2.3.5.7
EDOs: 12, 87, 99, 186


==13-limit==
[[Comma list]]: 3136/3125, 5120/5103
Commas: 196/195, 352/351, 364/363, 635/624


[[POTE tuning|POTE generator]]: 703.303
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


Map: [&lt;3 0 26 56 77 92|, &lt;0 1 -4 -10 -14 -17|]
: mapping generators: ~63/50, ~3
EDOs: 12, 87, 273


=Mystic=
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448
Commas: 325/324, 441/440, 896/891, 1001/1000


[[POTE tuning|POTE generator]]: 703.311
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}


Map: [&lt;3 0 26 56 77 -46|, &lt;0 1 -4 -10 -14 12|]
[[Badness]]: 0.036802
EDOs: 12, 87, 273


==17-limit==
=== 2.3.5.7.17 subgroup ===
Commas: 196/195, 352/351, 364/363, 635/624
The [[S-expression]]-based comma list for this 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]])}.


[[POTE tuning|POTE generator]]: 703.233
Subgroup: 2.3.5.7.17


Map: [&lt;3 0 26 56 77 92 -2|, &lt;0 1 -4 -10 -14 -17 3|]
Comma list: 256/255, 1701/1700, 2025/2023
EDOs: 12, 87, 186


==19-limit==
Sval mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324


[[POTE tuning|POTE generator]]: 703.253
: sval 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 tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963
EDOs: 12, 87, 186


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


POTE generator: ~11/8 = 551.527
Badness: 0.0186


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
Sval 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|]
: sval mapping generators: ~24/19, ~3
EDOs: 15, 24, 87, 111, 198
Badness: 0.0229


Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}


Badness: 0.0132


=== Undecimal misty ===
Subgroup: 2.3.5.7.11


Comma list: 441/440, 3136/3125, 4375/4356


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


Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* POTE: ~44/35 = 1\3, ~3/2 = 703.210


</pre></div>
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
<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;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;a href="#Misty"&gt;Misty&lt;/a&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;
Badness: 0.035932
&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;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
 
Comma: 67108864/66430125 = |26 -12 -3&amp;gt;&lt;br /&gt;
Comma list: 196/195, 352/351, 364/363, 625/624
&lt;br /&gt;
 
POTE generator: ~3/2 = 703.111&lt;br /&gt;
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26|, &amp;lt;0 1 -4|]&lt;br /&gt;
Optimal tunings:
EDOs: 12, 51, 63, 75, 87, 99, 285, 384&lt;br /&gt;
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
Badness: 0.1065&lt;br /&gt;
* POTE: ~44/35 = 1\3, ~3/2 = 703.303
&lt;br /&gt;
 
&lt;br /&gt;
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Misty"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Misty&lt;/h1&gt;
 
Commas: 3136/3125, 5120/5103&lt;br /&gt;
Badness: 0.027820
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.021&lt;br /&gt;
===== 17-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
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;
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
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;
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
&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;
 
Commas: 441/440, 3136/3125, 4375/4356&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.210&lt;br /&gt;
* POTE: ~34/27 = 1\3, ~3/2 = 703.233
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56 77|, &amp;lt;0 1 -4 -10 -14|]&lt;br /&gt;
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}
EDOs: 12, 87, 99, 186&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.023908
&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;
 
Commas: 196/195, 352/351, 364/363, 635/624&lt;br /&gt;
===== 19-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.303&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
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;
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}
&lt;br /&gt;
 
&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;
Optimal tunings:
Commas: 325/324, 441/440, 896/891, 1001/1000&lt;br /&gt;
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
&lt;br /&gt;
* POTE: ~34/27 = 1\3, ~3/2 = 703.253
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.311&lt;br /&gt;
 
&lt;br /&gt;
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}
Map: [&amp;lt;3 0 26 56 77 -46|, &amp;lt;0 1 -4 -10 -14 12|]&lt;br /&gt;
 
EDOs: 12, 87, 273&lt;br /&gt;
Badness: 0.019878
&lt;br /&gt;
 
&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;
==== Mystic ====
Commas: 196/195, 352/351, 364/363, 635/624&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.233&lt;br /&gt;
Comma list: 325/324, 441/440, 896/891, 1001/1000
&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;
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
EDOs: 12, 87, 186&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&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;
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324&lt;br /&gt;
* POTE: ~44/35 = 1\3, ~3/2 = 703.311
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.253&lt;br /&gt;
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56 77 92 -2 8|, &amp;lt;0 1 -4 -10 -14 -17 3 1|]&lt;br /&gt;
Badness: 0.036872
EDOs: 12, 87, 186&lt;br /&gt;
 
&lt;br /&gt;
=== Semimist ===
&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;
Subgroup: 2.3.5.7.11
Commas: 3136/3125, 5120/5103, 3025/3024&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 3136/3125, 5120/5103, 9801/9800
POTE generator: ~11/8 = 551.527&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
Map: [&amp;lt;3 0 26 56 8|, &amp;lt;0 2 -8 -20 1|]&lt;br /&gt;
 
EDOs: 15, 24, 87, 111, 198&lt;br /&gt;
: mapping generators: ~55/49, ~3
Badness: 0.0474&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&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;
* CTE: ~55/49 = 1\6, ~3/2 = 703.0864
Commas: 352/351, 676/675, 847/845, 3025/3024&lt;br /&gt;
* POTE: ~55/49 = 1\6, ~3/2 = 702.9764
&lt;br /&gt;
 
POTE generator: ~11/8 = 551.525&lt;br /&gt;
Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56 8 23|, &amp;lt;0 2 -8 -20 1 -5|]&lt;br /&gt;
Badness: 0.068509
EDOs: 15, 24, 87, 111, 198&lt;br /&gt;
 
Badness: 0.0229&lt;/body&gt;&lt;/html&gt;</pre></div>
==== 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:
* CTE: ~55/49 = 1\6, ~3/2 = 703.1045
* POTE: ~55/49 = 1\6, ~3/2 = 703.0087
 
Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}
 
Badness: 0.044073
 
=== 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 tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721
 
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
 
Badness: 0.047439
 
==== 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 tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810
 
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
 
Badness: 0.022923
 
== Fog ==
The fog temperament (12 &amp; 63) tempers out the marvel comma, [[225/224]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250/151263
 
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
 
: mapping generators: ~80/63, ~3
 
Optimal tunings:
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* POTE: ~80/63 = 1\3, ~3/2 = 704.118
 
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}
 
[[Badness]]: 0.086866
 
=== 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 }}
 
: mapping generators: ~44/35, ~3
 
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* POTE: ~44/35 = 1\3, ~3/2 = 704.411
 
Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
 
Badness: 0.051348
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Misty_family"