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Quince 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 '''quince clan''' [[Tempering out|tempers out]] the [[quince comma]] ({{monzo|legend=1| -15 0 -2 7 }}, [[ratio]]: 823543/819200).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-21 15:31:43 UTC</tt>.<br>
: The original revision id was <tt>338028770</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]]


This tempers out quince, the no-threes comma |-15 0 -2 7&gt; = 823543/819200. Quince temperaments include miracle, quincy, octowerck and birds as well as the ones listed below.
== Mercy ==
Mercy is the no-3 version of [[miracle]]. Two generators make [[~]][[8/7]], five generators make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph.  


=Essence=
[[Subgroup]]: 2.5.7
Essence is of interest because of the abundant supply of essentially tempered chords.


Commas: 321489/320000, 823543/820125
[[Comma list]]: 823543/819200


POTE generator: ~243/196 = 372.597
{{Mapping|legend=2| 1 3 3 | 0 -7 -2 }}


Map: [&lt;2 10 27 23|, &lt;0 -11 -36 -28|]
{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
Wedgie: &lt;&lt;22 72 56 63 27 -72||
: mapping generators: ~2, ~343/320
EDOs: 58, 190, 248, 438d, 686d
Badness: 0.1736


==11-limit==
[[Optimal tuning]]s:
Commas: 441/440, 8019/8000, 456533/455625
* [[WE]]: ~2 = 1200.2169{{c}}, ~343/320 = 116.3116{{c}}
: [[error map]]: {{val| +0.217 +0.156 -0.798 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
: error map: {{val| 0.000 -0.117 -1.341 }}


POTE generator: ~243/196 = 372.599
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}


Map: [&lt;2 10 27 23 33|, &lt;0 -11 -36 -28 -42|]
[[Badness]] (Sintel): 0.428
EDOs: 58, 190, 248, 438d
Badness: 0.0464


==13-limit==
=== Overview to extensions ===
Commas: 441/440, 729/728, 847/845, 1001/1000
Temperaments discussed elsewhere are:
* ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Slendi]]'' (+4000/3969) → [[Octagar temperaments #Slendi|Octagar temperaments]]
* ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]
* ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]
* ''[[Octowerck]]'' (+321489/320000 or 420175/419904) → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* [[Cotoneum]] (+10976/10935) → [[Garischismic clan #Cotoneum|Garischismic clan]]


POTE generator: ~243/196 = 372.602
Discussed below is countermiracle.  


Map: [&lt;2 10 27 23 33 31|, &lt;0 -11 -36 -28 -42 -38|]
== Countermiracle ==
EDOs: 58, 190, 248, 438d
The countermiracle temperament tempers out the porwell comma ([[6144/6125]]), the trimyna comma ([[50421/50000]]) and the quince comma ([[823543/819200]]). It can be described as the {{nowrap| 31 & 176 }} temperament, and finds the [[3/2|perfect fifth]] -25 generators away (normalized for 2.5.7) as opposed to miracle's +6 generators. It has the [[ploidacot]] signature of 22-sheared 25-cot (or omega-heptaseph).  
Badness: 0.0261


</pre></div>
Like miracle, it is naturally an [[11-limit]] temperament with the generator representing [[77/72]], but here the generator does not represent [[15/14]] or [[16/15]]. [[176edo]] makes for a recommendable tuning for both the 7- and 11-limit.
<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;Quince clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&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:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Essence"&gt;Essence&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
Countermiracle extends less naturally to the [[13-limit|13-]] and [[17-limit]], with countermiraculous (31 & 145), countermanna (145 & 176), and counterbenediction (176 & 207) being possible candidates, each favoring a tuning range shown by the [[temperament merging|equal-temperament joins]].
&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&gt;
 
This tempers out quince, the no-threes comma |-15 0 -2 7&amp;gt; = 823543/819200. Quince temperaments include miracle, quincy, octowerck and birds as well as the ones listed below.&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Essence"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Essence&lt;/h1&gt;
[[Comma list]]: 6144/6125, 50421/50000
Essence is of interest because of the abundant supply of essentially tempered chords.&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 -21 -4 1 | 0 25 7 2 }}
Commas: 321489/320000, 823543/820125&lt;br /&gt;
: mapping generators: ~2, ~625/336
&lt;br /&gt;
 
POTE generator: ~243/196 = 372.597&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1199.7944{{c}}, ~625/336 = 1083.8973{{c}} (~343/320 = 115.8970{{c}})
Map: [&amp;lt;2 10 27 23|, &amp;lt;0 -11 -36 -28|]&lt;br /&gt;
: [[error map]]: {{val| -0.206 -0.203 +1.790 -1.237 }}
Wedgie: &amp;lt;&amp;lt;22 72 56 63 27 -72||&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/336 = 1084.0787{{c}} (~343/320 = 115.9213{{c}})
EDOs: 58, 190, 248, 438d, 686d&lt;br /&gt;
: error map: {{val| 0.000 +0.013 +2.237 -0.669 }}
Badness: 0.1736&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 114, 145, 176 }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Essence-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 441/440, 8019/8000, 456533/455625&lt;br /&gt;
[[Badness]] (Sintel): 2.59
&lt;br /&gt;
 
POTE generator: ~243/196 = 372.599&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;2 10 27 23 33|, &amp;lt;0 -11 -36 -28 -42|]&lt;br /&gt;
 
EDOs: 58, 190, 248, 438d&lt;br /&gt;
Comma list: 441/440, 3388/3375, 6144/6125
Badness: 0.0464&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -21 -4 1 -39 | 0 25 7 2 47 }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Essence-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
 
Commas: 441/440, 729/728, 847/845, 1001/1000&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.7942{{c}}, ~144/77 = 1083.8983{{c}} (~77/72 = 115.8959{{c}})
POTE generator: ~243/196 = 372.602&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0798{{c}} (~77/72 = 115.9202{{c}})
&lt;br /&gt;
 
Map: [&amp;lt;2 10 27 23 33 31|, &amp;lt;0 -11 -36 -28 -42 -38|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}
EDOs: 58, 190, 248, 438d&lt;br /&gt;
 
Badness: 0.0261&lt;/body&gt;&lt;/html&gt;</pre></div>
Badness (Sintel): 1.29
 
==== Countermiraculous ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 1001/1000, 6144/6125
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}
 
Optimal tunings:  
* WE: ~2 = 1199.4465{{c}}, ~144/77 = 1083.6196{{c}} (~77/72 = 115.8268{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1148{{c}} (~77/72 = 115.8852{{c}})
 
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}
 
Badness (Sintel): 1.62
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}
 
Optimal tunings:  
* WE: ~2 = 1199.4013{{c}}, ~144/77 = 1083.5835{{c}} (~77/72 = 115.8178{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1219{{c}} (~77/72 = 115.8781{{c}})
 
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}
 
Badness (Sintel): 1.50
 
==== Counterbenediction ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 3146/3125, 3584/3575
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}
 
Optimal tunings:
* WE: ~2 = 1199.9460{{c}}, ~144/77 = 1084.0178{{c}} (~77/72 = 115.9283{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0663{{c}} (~77/72 = 115.9337{{c}})
 
{{Optimal ET sequence|legend=0| 31, 145f, 176, 207 }}
 
Badness (Sintel): 1.88
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
 
Mapping: {{mapping| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}
 
Optimal tunings:
* WE: ~2 = 1199.9950{{c}}, ~144/77 = 1084.0564{{c}} (~77/72 = 115.9386{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0609{{c}} (~77/72 = 115.9391{{c}})
 
{{Optimal ET sequence|legend=0| 31, 176, 207 }}
 
Badness (Sintel): 1.85
 
==== Countermanna ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
 
Mapping: {{mapping| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}
 
Optimal tunings:
* WE: ~2 = 1199.6687{{c}}, ~144/77 = 1083.8108{{c}} (~77/72 = 115.8578{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1065{{c}} (~77/72 = 115.8935{{c}})
 
{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}
 
Badness (Sintel): 2.21
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
 
Mapping: {{mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
 
Optimal tunings:
* WE: ~2 = 1199.6458{{c}}, ~144/77 = 1083.7968{{c}} (~77/72 = 115.8490{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1132{{c}} (~77/72 = 115.8868{{c}})
 
{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}
 
Badness (Sintel): 2.08
 
== Subgroup extensions ==
=== Mercy (2.5.7.13) ===
This extension may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].
 
Subgroup: 2.5.7.13
 
Comma list: 343/338, 640/637
 
Subgroup-val mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
 
Badness (Sintel): 0.546
 
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
 
Comma list: 170/169, 224/221, 343/338
 
Subgroup-val mapping: {{mapping| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
 
Optimal tunings:  
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}
 
Badness (Sintel): 0.451
 
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
 
Comma list: 170/169, 224/221, 343/338, 476/475
 
Subgroup-val mapping: {{mapping| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
 
Optimal tunings:
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
 
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}
 
Badness (Sintel): 0.759
 
[[Category:Quince clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Quince_clan"