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Quintile 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 '''quintile family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quintile comma]] ({{monzo|legend=1| -28 25 -5 }}, [[ratio]]: 847 288 609 443 / 838 860 800 000).  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-07-21 11:36:14 UTC</tt>.<br>
: The original revision id was <tt>555566759</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">The pental family tempers out the pental comma, 847288609443/838860800000 = |-28 25 -5&gt;.


=Pental=
== Quintile ==
Comma: 847288609443/838860800000
Quintile reaches the interval class of 5 by five [[3/2|perfect fifths]] (i.e. a major seventh) plus two periods of 1/5-octave; this two-period interval represents a grave fourth of [[320/243]], that is, a [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Quintile is a member of the [[syntonic–diatonic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of five [[81/80|syntonic commas]].


POTE generator: ~3/2 = 701.210
The temperament was first introduced by [[Mike Battaglia]] in 2011 along with other temperaments in the continuum mentioned above<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_99315.html#99323 Yahoo! Tuning Group | ''Some new 5-limit microtemperaments'']</ref>. It did not get named until 2012, when [[Petr Pařízek]] called it ''pental''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104270.html Yahoo! Tuning Group | ''2D temperaments, part II -- new 5-limit temperaments'']</ref>. In 2024, the community has decided to rename it for fear of confusion with the more common usage of the term ''pental'' to refer to the [[5-limit]].  


Map: [&lt;5 0 -28|, &lt;0 1 5|]
[[Subgroup]]: 2.3.5
EDOs: 5, 60, 65, 190, 255, 575, 830b, 1405b
Badness: 0.24005


[[Comma list]]: 847288609443/838860800000


=Decal=
{{Mapping|legend=1| 5 0 -28 | 0 1 5 }}
Commas: 235298/234375, 321489/320000


POTE generator: ~3/2 = 701.303
: mapping generators: ~59049/51200, ~3


Map: [&lt;10 0 -56 -67|, &lt;0 1 5 6|]
[[Optimal tuning]]s:
EDOs: 60, 130, 320, 450, 770d
* [[CTE]]: ~59049/51200 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
Badness: 0.1049
: [[error map]]: {{val| 0.000 -0.638 +0.274 }}
* [[POTE]]: ~59049/51200 = 240.000, ~3/2 = 701.210 (~81/80 = 18.790)
: error map: {{val| 0.000 -0.745 -0.265 }}


==11-limit==
{{Optimal ET sequence|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}
Commas: 441/440, 8019/8000, 234375/234256


POTE generator: ~3/2 = 701.240
[[Badness]] (Smith): 0.240050


Map: [&lt;10 0 -56 -67 -108|, &lt;0 1 5 6 9|]
== Pentacloud ==
EDOs: 130, 190, 320
Pentacloud can be described as the 5 & 60 temperament. It identifies the period as ~8/7, tempering out the [[cloudy comma]] 16807/16384 and the [[245/243|sensamagic comma]] 245/243 in the 7-limit.
Badness: 0.0406


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 441/440, 729/728, 1001/1000, 4225/4224


POTE generator: ~3/2 = 701.252
[[Comma list]]: 245/243, 16807/16384


Map: [&lt;10 0 -56 -67 -108 37|, &lt;0 1 5 6 9 0|]
{{Mapping|legend=1| 5 0 -28 14 | 0 1 5 0 }}
EDOs: 130, 190, 320
 
Badness: 0.0239</pre></div>
[[Optimal tuning]]s:
<h4>Original HTML content:</h4>
* [[CTE]]: ~8/7 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
<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;Pental family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The pental family tempers out the pental comma, 847288609443/838860800000 = |-28 25 -5&amp;gt;.&lt;br /&gt;
: [[error map]]: {{val| 0.000 -0.638 +0.274 -8.826 }}
&lt;br /&gt;
* [[POTE]]: ~8/7 = 240.000, ~3/2 = 700.548 (~81/80 = 19.452)
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Pental"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Pental&lt;/h1&gt;
: error map: {{val| 0.000 -1.407 -3.574 -8.826 }}
Comma: 847288609443/838860800000&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5, 60, 65, 125d, 185cdd }}
POTE generator: ~3/2 = 701.210&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Smith): 0.120942
Map: [&amp;lt;5 0 -28|, &amp;lt;0 1 5|]&lt;br /&gt;
 
EDOs: 5, 60, 65, 190, 255, 575, 830b, 1405b&lt;br /&gt;
=== 11-limit ===
Badness: 0.24005&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 245/243, 385/384, 3087/3025
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Decal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Decal&lt;/h1&gt;
 
Commas: 235298/234375, 321489/320000&lt;br /&gt;
Mapping: {{mapping| 5 0 -28 14 49 | 0 1 5 0 -4 }}
&lt;br /&gt;
 
POTE generator: ~3/2 = 701.303&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* CTE: ~8/7 = 240.000, ~3/2 = 701.496 (~81/80 = 18.304)
Map: [&amp;lt;10 0 -56 -67|, &amp;lt;0 1 5 6|]&lt;br /&gt;
* POTE: ~8/7 = 240.000, ~3/2 = 701.377 (~81/80 = 18.623)
EDOs: 60, 130, 320, 450, 770d&lt;br /&gt;
 
Badness: 0.1049&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5, 60, 65 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Decal-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Badness (Smith): 0.093248
Commas: 441/440, 8019/8000, 234375/234256&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
POTE generator: ~3/2 = 701.240&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;10 0 -56 -67 -108|, &amp;lt;0 1 5 6 9|]&lt;br /&gt;
Comma list: 105/104, 144/143, 245/243, 3087/3025
EDOs: 130, 190, 320&lt;br /&gt;
 
Badness: 0.0406&lt;br /&gt;
Mapping: {{mapping| 5 0 -28 14 49 -29 | 0 1 5 0 -4 6 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Decal-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Optimal tunings:  
Commas: 441/440, 729/728, 1001/1000, 4225/4224&lt;br /&gt;
* CTE: ~8/7 = 240.000, ~3/2 = 701.085 (~81/80 = 18.915)
&lt;br /&gt;
* POTE: ~8/7 = 240.000, ~3/2 = 700.996 (~81/80 = 19.004)
POTE generator: ~3/2 = 701.252&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5, 60, 65, 125de, 190ddef }}
Map: [&amp;lt;10 0 -56 -67 -108 37|, &amp;lt;0 1 5 6 9 0|]&lt;br /&gt;
 
EDOs: 130, 190, 320&lt;br /&gt;
Badness (Smith): 0.067549
Badness: 0.0239&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Hemiquintile ==
Hemiquintile (formerly ''hemipental'') can be described as 125 & 130 and tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 ([[hewuermera temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 ([[mitonismic temperaments|mitonisma]], laquadzo-agu).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 19683/19600, 589824/588245
 
{{Mapping|legend=1| 5 0 -28 18 | 0 2 10 -1 }}
 
: mapping generators: ~147/128, ~140/81
 
[[Optimal tuning]]s:
* [[CTE]]: ~147/128 = 240.0000, ~140/81 = 950.6620 (~1029/1024 = 9.3380)
: [[error map]]: {{val| 0.000 -0.6311 +0.3059 +0.5121 }}
* [[POTE]]: ~147/128 = 240.0000, ~140/81 = 950.6536 (~1029/1024 = 9.3464)
: error map: {{val| 0.000 -0.6473 +0.2249 +0.5202 }}
 
{{Optimal ET sequence|legend=1| 125, 130, 255, 385 }}
 
[[Badness]] (Smith): 0.104163
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 8019/8000, 180224/180075
 
Mapping: {{mapping| 5 0 -28 18 -54 | 0 2 10 -1 18 }}
 
Optimal tunings:
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6430 (~176/175 = 9.3570)
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6341 (~176/175 = 9.3659)
 
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }}
 
Badness (Smith): 0.047624
 
==== Hemiquintilis ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 676/675, 124215/123904
 
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}
 
Optimal tunings:
* CTE: ~147/128 = 240.0000, ~26/15 = 950.6775 (~176/175 = 9.3225)
* POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407)
 
{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}
 
Badness (Smith): 0.033542
 
==== Hemiquint ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
 
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}
 
Optimal tunings:  
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393)
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323)
 
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}
 
Badness (Smith): 0.041043
 
== Decile ==
Decile (formerly ''decal'') can be described as 130 & 190 and tempers out the [[varunisma]] 321489/320000 in the 7-limit, as well as the [[triwellisma]] 235298/234375, the [[breeze comma]] 2460375/2458624, and the [[linus comma]] {{monzo| 11 -10 -10 10 }}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 235298/234375, 321489/320000
 
{{Mapping|legend=1| 10 0 -56 -67 | 0 1 5 6 }}
 
: mapping generators: ~15/14, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~15/14 = 120.000, ~3/2 = 701.390 (~81/80 = 18.610)
: [[error map]]: {{val| 0.000 -0.565 +0.639 -0.483 }}
* [[POTE]]: ~15/14 = 120.000, ~3/2 = 701.303 (~81/80 = 18.697)
: error map: {{val| 0.000 -0.652 +0.200 -1.009 }}
 
{{Optimal ET sequence|legend=1| 60, 130, 320, 450, 770d }}
 
[[Badness]] (Smith): 0.104859
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 8019/8000, 234375/234256
 
Mapping: {{mapping| 10 0 -56 -67 -108 | 0 1 5 6 9 }}
 
Optimal tunings:
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~99/98 = 18.664)
* POTE: ~15/14 = 120.000, ~3/2 = 701.240 (~99/98 = 18.760)
 
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}
 
Badness (Smith): 0.040633
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 729/728, 1001/1000, 4225/4224
 
Mapping: {{mapping| 10 0 -56 -67 -108 37 | 0 1 5 6 9 0 }}
 
Optimal tunings:
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~91/90 = 18.664)
* POTE: ~15/14 = 120.000, ~3/2 = 701.252 (~91/90 = 18.748)
 
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}
 
Badness (Smith): 0.023948
 
== Notes ==
 
[[Category:Temperament families]]
[[Category:Quintile family| ]] <!-- main article -->
[[Category:Quintile| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Quintile_family"