Misty family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 251687440 - Original comment: **
 
Extensions: hazy (99 & 111), smoky (87 & 99), and ashy (99ef & 111)
 
(45 intermediate revisions by 13 users not shown)
Line 1: Line 1:
<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 16:17:46 UTC</tt>.<br>
: The original revision id was <tt>251687440</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">


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


POTE generator: ~3/2 = 703.111
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 }}.  


Map: [&lt;3 0 26|, &lt;0 1 -4|]
[[Subgroup]]: 2.3.5
EDOs: 12, 51, 63, 75, 87, 99, 285, 384
Badness: 0.1065


[[Comma list]]: 67108864/66430125


=Misty=
{{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}}
</pre></div>
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 702.9837{{c}}
<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;br /&gt;
{{Optimal ET sequence|legend=0| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
&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): 0.688
Comma: 67108864/66430125 = |26 -12 -3&amp;gt;&lt;br /&gt;
 
&lt;br /&gt;
==== 2.3.5.7.17.19 subgroup ====
POTE generator: ~3/2 = 703.111&lt;br /&gt;
Subgroup: 2.3.5.7.17.19
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26|, &amp;lt;0 1 -4|]&lt;br /&gt;
Comma list: 256/255, 324/323, 400/399, 476/475
EDOs: 12, 51, 63, 75, 87, 99, 285, 384&lt;br /&gt;
 
Badness: 0.1065&lt;br /&gt;
Subgroup-val mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}
&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}
Commas: 3136/3125, 5120/5103&lt;br /&gt;
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 702.9418{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.021&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, 87, 99, 210gh, 309cdghh }}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56|, &amp;lt;0 1 -4 -10|]&lt;br /&gt;
Badness (Sintel): 0.637
Wedgie: &amp;lt;&amp;lt;3 -12 -30 -26 -56 -36||&lt;br /&gt;
 
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;
=== Murky ===
&lt;br /&gt;
Murky (87 & 99ef) tempers out [[441/440]] and [[896/891]], which joins misty with [[parapyth]]/[[pele]].
&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;
This extension used to be known as ''undecimal misty'', but was decanonicalized in 2026 in favor of the no-11 no-13 temperament.
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.210&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56 77|, &amp;lt;0 1 -4 -10 -14|]&lt;br /&gt;
Comma list: 441/440, 896/891, 3136/3125
EDOs: 12, 87, 99, 186&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}
&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;
Optimal tunings:
&lt;br /&gt;
* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.303&lt;br /&gt;
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 26 56 77 92|, &amp;lt;0 1 -4 -10 -14 -17|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, …, 87, 99e, 186e }}
EDOs: 12, 87, 273&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.19
&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;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.311&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 196/195, 352/351, 364/363, 625/624
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;
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
&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;
Optimal tunings:  
Commas: 196/195, 352/351, 364/363, 635/624&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;: 703.233&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, …, 87, 186ef }}
Map: [&amp;lt;3 0 26 56 77 92 -2|, &amp;lt;0 1 -4 -10 -14 -17 3|]&lt;br /&gt;
 
EDOs: 12, 87, 186&lt;br /&gt;
Badness (Sintel): 1.15
&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;
===== 17-limit =====
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.253&lt;br /&gt;
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
&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;
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
EDOs: 12, 87, 186&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Optimal tunings:  
* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efg }}
 
Badness (Sintel): 1.22
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
 
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}
 
Optimal tunings:  
* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.2216{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efgh }}
 
Badness (Sintel): 1.21
 
=== Hazy ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1375/1372, 5120/5103
 
Mapping: {{mapping| 3 0 26 56 96 | 0 1 -4 -10 -18 }}
 
Optimal tunings:
* WE: ~63/50 = 399.8097{{c}}, ~3/2 = 702.4039{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7731{{c}}
 
{{Optimal ET sequence|legend=0| 12, 87e, 99, 111 }}
 
Badness (Sintel): 1.84
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 847/845, 1375/1372
 
Mapping: {{mapping| 3 0 26 56 96 111 | 0 1 -4 -10 -18 -21 }}
 
Optimal tunings:
* WE: ~63/50 = 399.8089{{c}}, ~3/2 = 702.3993{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7661{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 87ef, 99, 111 }}
 
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]]

Latest revision as of 06:22, 12 February 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The misty family tempers out the misty comma (monzo: [26 -12 -3⟩, ratio: 67108864/66430125).

Misty

Main article: Misty

Misty features a third-octave period, which represents the ptolemaic diminished fourth (512/405). Four 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 schismas, making it a member of the schismic–Pythagorean equivalence continuum with n = 3.

Subgroup: 2.3.5

Comma list: 67108864/66430125

Mapping: [⟨3 0 26], ⟨0 1 -4]]

mapping generators: ~512/405, ~3

Optimal tunings:

  • WE: ~512/405 = 399.8595 ¢, ~3/2 = 702.8645 ¢
error map: ⟨-0.421 +0.488 +0.261]
  • CWE: ~512/405 = 400.0000 ¢, ~3/2 = 703.1489 ¢
error map: ⟨0.000 +1.194 +1.091]

Optimal ET sequence: 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc

Badness (Sintel): 2.50

Overview to extensions

The second comma of the 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.

Septimal misty

Main article: Misty

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 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.

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 {(S8/S9), S15/S17, S16, S18/S20, (S50)}, and for the 2.3.5.7.17.19-subgroup extension, {(S8/S9), S16, S18, S20, S17/S19, (S50)}.

7-limit

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]

Optimal tunings:

  • WE: ~63/50 = 399.8579 ¢, ~3/2 = 702.7714 ¢
error map: ⟨-0.426 +0.390 +0.611 -0.236]
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.0551 ¢
error map: ⟨0.000 +1.100 +1.466 +0.623]

Optimal ET sequence: 12, …, 87, 99

Badness (Sintel): 0.931

2.3.5.7.17-subgroup

Subgroup: 2.3.5.7.17

Comma list: 256/255, 1701/1700, 2025/2023

Subgroup-val mapping: [⟨3 0 26 56 -2], ⟨0 1 -4 -10 3]]

mapping generators: ~34/27, ~3

Optimal tunings:

  • WE: ~34/27 = 399.8023 ¢, ~3/2 = 702.6087 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~3/2 = 702.9837 ¢

Optimal ET sequence: 12, 87, 99, 309cdg, 408cdgg, 507cdggg

Badness (Sintel): 0.688

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 256/255, 324/323, 400/399, 476/475

Subgroup-val mapping: [⟨3 0 26 56 -2 8], ⟨0 1 -4 -10 3 1]]

Optimal tunings:

  • WE: ~24/19 = 399.7556 ¢, ~3/2 = 702.4862 ¢
  • CWE: ~24/19 = 400.0000 ¢, ~3/2 = 702.9418 ¢

Optimal ET sequence: 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: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]

Optimal tunings:

  • WE: ~44/35 = 399.8727 ¢, ~3/2 = 702.9867 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.2385 ¢

Optimal ET sequence: 12, …, 87, 99e, 186e

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 625/624

Mapping: [⟨3 0 26 56 77 92], ⟨0 1 -4 -10 -14 -17]]

Optimal tunings:

  • WE: ~44/35 = 399.8888 ¢, ~3/2 = 703.1080 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.3258 ¢

Optimal ET sequence: 12f, …, 87, 186ef

Badness (Sintel): 1.15

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2], ⟨0 1 -4 -10 -14 -17 3]]

Optimal tunings:

  • WE: ~34/27 = 399.8046 ¢, ~3/2 = 702.8894 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~3/2 = 703.2597 ¢

Optimal ET sequence: 12f, 87, 99ef, 186efg

Badness (Sintel): 1.22

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2 8], ⟨0 1 -4 -10 -14 -17 3 1]]

Optimal tunings:

  • WE: ~24/19 = 399.7530 ¢, ~3/2 = 702.7601 ¢
  • CWE: ~24/19 = 400.0000 ¢, ~3/2 = 703.2216 ¢

Optimal ET sequence: 12f, 87, 99ef, 186efgh

Badness (Sintel): 1.21

Hazy

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 5120/5103

Mapping: [⟨3 0 26 56 96], ⟨0 1 -4 -10 -18]]

Optimal tunings:

  • WE: ~63/50 = 399.8097 ¢, ~3/2 = 702.4039 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.7731 ¢

Optimal ET sequence: 12, 87e, 99, 111

Badness (Sintel): 1.84

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 1375/1372

Mapping: [⟨3 0 26 56 96 111], ⟨0 1 -4 -10 -18 -21]]

Optimal tunings:

  • WE: ~63/50 = 399.8089 ¢, ~3/2 = 702.3993 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.7661 ¢

Optimal ET sequence: 12f, 87ef, 99, 111

Badness (Sintel): 1.54

Smoky

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3136/3125

Mapping: [⟨3 0 26 56 -61], ⟨0 1 -4 -10 15]]

Optimal tunings:

  • WE: ~63/50 = 400.0072 ¢, ~3/2 = 702.3068 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.2941 ¢

Optimal ET sequence: 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: [⟨3 0 26 56 -61 -46], ⟨0 1 -4 -10 15 12]]

Optimal tunings:

  • WE: ~63/50 = 400.0184 ¢, ~3/2 = 703.3420 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.3101 ¢

Optimal ET sequence: 12e, 87, 186, 273

Badness (Sintel): 1.63

Ashy

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56 -80], ⟨0 1 -4 -10 19]]

Optimal tunings:

  • WE: ~63/50 = 399.7776 ¢, ~3/2 = 702.4764 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.8652 ¢

Optimal ET sequence: 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: [⟨3 0 26 56 -80 -65], ⟨0 1 -4 -10 19 16]]

Optimal tunings:

  • WE: ~63/50 = 399.7470 ¢, ~3/2 = 702.3782 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.8135 ¢

Optimal ET sequence: 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: [⟨6 0 52 112 173], ⟨0 1 -4 -10 -16]]

mapping generators: ~55/49, ~3

Optimal tunings:

  • WE: ~55/49 = 199.9260 ¢, ~3/2 = 702.7162 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0062 ¢

Optimal ET sequence: 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: [⟨6 0 52 112 173 203], ⟨0 1 -4 -10 -16 -19]]

Optimal tunings:

  • WE: ~55/49 = 199.9293 ¢, ~3/2 = 702.7602 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0343 ¢

Optimal ET sequence: 12f, 174ef, 186, 198

Badness (Sintel): 1.82

Hemimist

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]

mapping generators: ~63/50, ~400/231

Optimal tunings:

  • WE: ~63/50 = 399.8922 ¢, ~400/231 = 951.2704 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~400/231 = 951.5363 ¢

Optimal ET sequence: 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: [⟨3 0 26 56 8 23], ⟨0 2 -8 -20 1 -5]]

Optimal tunings:

  • WE: ~63/50 = 399.8818 ¢, ~26/15 = 951.2442 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~26/15 = 951.5358 ¢

Optimal ET sequence: 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 12 & 63.

Subgroup: 2.3.5.7

Comma list: 225/224, 156250/151263

Mapping: [⟨3 0 26 37], ⟨0 1 -4 -6]]

Optimal tunings:

  • WE: ~80/63 = 399.7918 ¢, ~3/2 = 703.7511 ¢
error map: ⟨-0.624 +1.172 -4.232 +4.712]
  • CWE: ~80/63 = 400.0000 ¢, ~3/2 = 704.1865 ¢
error map: ⟨0.000 +2.232 -3.060 +6.055]

Optimal ET sequence: 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: [⟨3 0 26 37 58], ⟨0 1 -4 -6 -10]]

Optimal tunings:

  • WE: ~44/35 = 399.8078 ¢, ~3/2 = 704.0728 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 704.4697 ¢

Optimal ET sequence: 12, 39ce, 51, 63, 138de

Badness (Sintel): 1.70

Retrieved from "https://en.xen.wiki/index.php?title=Misty_family&oldid=223840"