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>2015-07-20 16:15:55 UTC</tt>.<br>
: The original revision id was <tt>555536359</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–limmic 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


</pre></div>
{{Mapping|legend=1| 5 0 -28 | 0 1 5 }}
<h4>Original HTML content:</h4>
: mapping generators: ~59049/51200, ~3
<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;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&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;
* [[WE]]: ~59049/51200 = 240.0389{{c}}, ~3/2 = 701.3235{{c}} (~81/80 = 18.7932{{c}})
Comma: 847288609443/838860800000&lt;br /&gt;
: [[error map]]: {{val| +0.195 -0.437 +0.187 }}
&lt;br /&gt;
* [[CWE]]: ~59049/51200 = 240.0000{{c}}, ~3/2 = 701.2536{{c}} (~81/80 = 18.7464{{c}})
POTE generator: ~3/2 = 701.210&lt;br /&gt;
: error map: {{val| 0.000 -0.701 -0.046 }}
&lt;br /&gt;
 
Map: [&amp;lt;5 0 -28|, &amp;lt;0 1 5|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 60, 65, 190, 255, 575, 830b, 1405b }}
EDOs: 5, 60, 65, 190, 255, 575, 830b, 1405b&lt;br /&gt;
 
Badness: 0.24005&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Badness]] (Sintel): 5.63
 
== Pentacloud ==
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 [[sensamagic comma]] 245/243 in the 7-limit.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 16807/16384
 
{{Mapping|legend=1| 5 0 -28 14 | 0 1 5 0 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~8/7 = 240.2778{{c}}, ~3/2 = 701.3587{{c}} (~81/80 = 19.4746{{c}})
: [[error map]]: {{val| +1.389 +0.793 -0.353 -4.937 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 700.7696{{c}} (~81/80 = 19.2304{{c}})
: error map: {{val| 0.000 -1.185 -2.466 -8.826 }}
 
{{Optimal ET sequence|legend=1| 60, 125d, 185cdd }}
 
[[Badness]] (Sintel): 3.06
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 3087/3025
 
Mapping: {{mapping| 5 0 -28 14 49 | 0 1 5 0 -4 }}
 
Optimal tunings:
* WE: ~8/7 = 240.0919{{c}}, ~3/2 = 701.6455{{c}} (~81/80 = 18.6301{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 701.3981{{c}} (~81/80 = 18.6019{{c}})
 
{{Optimal ET sequence|legend=0| 60, 65 }}
 
Badness (Sintel): 3.08
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 245/243, 3087/3025
 
Mapping: {{mapping| 5 0 -28 14 49 -29 | 0 1 5 0 -4 6 }}
 
Optimal tunings:
* WE: ~8/7 = 240.0470{{c}}, ~3/2 = 701.1333{{c}} (~81/80 = 19.0078{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 701.0105{{c}} (~81/80 = 18.9895{{c}})
 
{{Optimal ET sequence|legend=0| 60, 65 }}
 
Badness (Sintel): 2.79
 
== 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 comma]]) and 5250987/5242880 ([[mitonisma]]).
 
[[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:
* [[WE]]: ~147/128 = 240.0063{{c}}, ~140/81 = 950.6787{{c}} (~1029/1024 = 9.3464{{c}})
: [[error map]]: {{val| +0.031 -0.598 +0.298 +0.608 }}
* [[CWE]]: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6561{{c}} (~1029/1024 = 9.3439{{c}})
: error map: {{val| 0.000 -0.643 +0.247 +0.518 }}
 
{{Optimal ET sequence|legend=1| 125, 130, 255, 385 }}
 
[[Badness]] (Sintel): 2.64
 
=== 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:
* WE: ~147/128 = 240.0080{{c}}, ~140/81 = 950.6658{{c}} (~176/175 = 9.3662{{c}})
* CWE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6365{{c}} (~176/175 = 9.3635{{c}})
 
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }}
 
Badness (Sintel): 1.57
 
==== 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:  
* WE: ~147/128 = 240.0155{{c}}, ~26/15 = 950.7207{{c}} (~176/175 = 9.3413{{c}})
* CWE: ~147/128 = 240.0000{{c}}, ~26/15 = 950.6641{{c}} (~176/175 = 9.3359{{c}})
 
{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}
 
Badness (Sintel): 1.39
 
==== 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:
* WE: ~147/128 = 239.9879{{c}}, ~140/81 = 950.6196{{c}} (~176/175 = 9.3318{{c}})
* CWE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6666{{c}} (~176/175 = 9.3334{{c}})
 
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}
 
Badness (Sintel): 1.70
 
== Decile ==
Decile (formerly ''decal'') can be described as 130 & 190 and tempers out the [[triwellisma]] 235298/234375 in the 7-limit, as well as the [[varunisma]] 321489/320000, 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:
* [[WE]]: ~15/14 = 120.0171{{c}}, ~3/2 = 701.4027{{c}} (~81/80 = 18.6999{{c}})
: [[error map]]: {{val| +0.171 -0.381 +0.597 -0.529 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 701.3381{{c}} (~81/80 = 18.6619{{c}})
: error map: {{val| 0.000 -0.617 +0.377 -0.797 }}
 
{{Optimal ET sequence|legend=1| 60, 130, 320, 450, 770d }}
 
[[Badness]] (Sintel): 2.65
 
=== 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:
* WE: ~15/14 = 120.0208{{c}}, ~3/2 = 701.3614{{c}} (~99/98 = 18.7632{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 701.2776{{c}} (~99/98 = 18.7224{{c}})
 
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}
 
Badness (Sintel): 1.34
 
=== 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:
* WE: ~15/14 = 120.0182{{c}}, ~3/2 = 701.3582{{c}} (~91/90 = 18.7510{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 701.2754{{c}} (~91/90 = 18.7246{{c}})
 
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}
 
Badness (Sintel): 0.990
 
== References ==
 
[[Category:Quintile family| ]] <!-- main article -->
[[Category:Quintile| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 06:42, 4 September 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 quintile family of temperaments tempers out the quintile comma (monzo: [-28 25 -5⟩, ratio: 847 288 609 443 / 838 860 800 000).

Quintile

Quintile reaches the interval class of 5 by five 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 perfect fourth minus a syntonic comma. Quintile is a member of the syntonic–limmic equivalence continuum with n = 5, so it equates a Pythagorean limma with a stack of five syntonic commas.

The temperament was first introduced by Mike Battaglia in 2011 along with other temperaments in the continuum mentioned above[1]. It did not get named until 2012, when Petr Pařízek called it pental[2]. 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.

Subgroup: 2.3.5

Comma list: 847288609443/838860800000

Mapping: [⟨5 0 -28], ⟨0 1 5]]

mapping generators: ~59049/51200, ~3

Optimal tunings:

  • WE: ~59049/51200 = 240.0389 ¢, ~3/2 = 701.3235 ¢ (~81/80 = 18.7932 ¢)
error map: ⟨+0.195 -0.437 +0.187]
  • CWE: ~59049/51200 = 240.0000 ¢, ~3/2 = 701.2536 ¢ (~81/80 = 18.7464 ¢)
error map: ⟨0.000 -0.701 -0.046]

Optimal ET sequence: 60, 65, 190, 255, 575, 830b, 1405b

Badness (Sintel): 5.63

Pentacloud

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 sensamagic comma 245/243 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 245/243, 16807/16384

Mapping: [⟨5 0 -28 14], ⟨0 1 5 0]]

Optimal tunings:

  • WE: ~8/7 = 240.2778 ¢, ~3/2 = 701.3587 ¢ (~81/80 = 19.4746 ¢)
error map: ⟨+1.389 +0.793 -0.353 -4.937]
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 700.7696 ¢ (~81/80 = 19.2304 ¢)
error map: ⟨0.000 -1.185 -2.466 -8.826]

Optimal ET sequence: 60, 125d, 185cdd

Badness (Sintel): 3.06

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3087/3025

Mapping: [⟨5 0 -28 14 49], ⟨0 1 5 0 -4]]

Optimal tunings:

  • WE: ~8/7 = 240.0919 ¢, ~3/2 = 701.6455 ¢ (~81/80 = 18.6301 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 701.3981 ¢ (~81/80 = 18.6019 ¢)

Optimal ET sequence: 60, 65

Badness (Sintel): 3.08

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/243, 3087/3025

Mapping: [⟨5 0 -28 14 49 -29], ⟨0 1 5 0 -4 6]]

Optimal tunings:

  • WE: ~8/7 = 240.0470 ¢, ~3/2 = 701.1333 ¢ (~81/80 = 19.0078 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 701.0105 ¢ (~81/80 = 18.9895 ¢)

Optimal ET sequence: 60, 65

Badness (Sintel): 2.79

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 comma) and 5250987/5242880 (mitonisma).

Subgroup: 2.3.5.7

Comma list: 19683/19600, 589824/588245

Mapping: [⟨5 0 -28 18], ⟨0 2 10 -1]]

mapping generators: ~147/128, ~140/81

Optimal tunings:

  • WE: ~147/128 = 240.0063 ¢, ~140/81 = 950.6787 ¢ (~1029/1024 = 9.3464 ¢)
error map: ⟨+0.031 -0.598 +0.298 +0.608]
  • CWE: ~147/128 = 240.0000 ¢, ~140/81 = 950.6561 ¢ (~1029/1024 = 9.3439 ¢)
error map: ⟨0.000 -0.643 +0.247 +0.518]

Optimal ET sequence: 125, 130, 255, 385

Badness (Sintel): 2.64

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000, 180224/180075

Mapping: [⟨5 0 -28 18 -54], ⟨0 2 10 -1 18]]

Optimal tunings:

  • WE: ~147/128 = 240.0080 ¢, ~140/81 = 950.6658 ¢ (~176/175 = 9.3662 ¢)
  • CWE: ~147/128 = 240.0000 ¢, ~140/81 = 950.6365 ¢ (~176/175 = 9.3635 ¢)

Optimal ET sequence: 125, 130, 255, 385, 640

Badness (Sintel): 1.57

Hemiquintilis

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 124215/123904

Mapping: [⟨5 0 -28 18 -54 34], ⟨0 2 10 -1 18 13]]

Optimal tunings:

  • WE: ~147/128 = 240.0155 ¢, ~26/15 = 950.7207 ¢ (~176/175 = 9.3413 ¢)
  • CWE: ~147/128 = 240.0000 ¢, ~26/15 = 950.6641 ¢ (~176/175 = 9.3359 ¢)

Optimal ET sequence: 125f, 130, 255f, 385f

Badness (Sintel): 1.39

Hemiquint

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000

Mapping: [⟨5 0 -28 18 -54 34], ⟨0 2 10 -1 18 -13]]

Optimal tunings:

  • WE: ~147/128 = 239.9879 ¢, ~140/81 = 950.6196 ¢ (~176/175 = 9.3318 ¢)
  • CWE: ~147/128 = 240.0000 ¢, ~140/81 = 950.6666 ¢ (~176/175 = 9.3334 ¢)

Optimal ET sequence: 125, 130, 255, 385, 515

Badness (Sintel): 1.70

Decile

Decile (formerly decal) can be described as 130 & 190 and tempers out the triwellisma 235298/234375 in the 7-limit, as well as the varunisma 321489/320000, the breeze comma 2460375/2458624, and the linus comma [11 -10 -10 10⟩.

Subgroup: 2.3.5.7

Comma list: 235298/234375, 321489/320000

Mapping: [⟨10 0 -56 -67], ⟨0 1 5 6]]

mapping generators: ~15/14, ~3

Optimal tunings:

  • WE: ~15/14 = 120.0171 ¢, ~3/2 = 701.4027 ¢ (~81/80 = 18.6999 ¢)
error map: ⟨+0.171 -0.381 +0.597 -0.529]
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 701.3381 ¢ (~81/80 = 18.6619 ¢)
error map: ⟨0.000 -0.617 +0.377 -0.797]

Optimal ET sequence: 60, 130, 320, 450, 770d

Badness (Sintel): 2.65

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 234375/234256

Mapping: [⟨10 0 -56 -67 -108], ⟨0 1 5 6 9]]

Optimal tunings:

  • WE: ~15/14 = 120.0208 ¢, ~3/2 = 701.3614 ¢ (~99/98 = 18.7632 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 701.2776 ¢ (~99/98 = 18.7224 ¢)

Optimal ET sequence: 60e, 130, 190, 320

Badness (Sintel): 1.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 4225/4224

Mapping: [⟨10 0 -56 -67 -108 37], ⟨0 1 5 6 9 0]]

Optimal tunings:

  • WE: ~15/14 = 120.0182 ¢, ~3/2 = 701.3582 ¢ (~91/90 = 18.7510 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 701.2754 ¢ (~91/90 = 18.7246 ¢)

Optimal ET sequence: 60e, 130, 190, 320

Badness (Sintel): 0.990

References

  1. ↑ Yahoo! Tuning Group | Some new 5-limit microtemperaments
  2. ↑ Yahoo! Tuning Group | 2D temperaments, part II -- new 5-limit temperaments
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