Wesley 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 '''wesley family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] 78125/73728, the [[wesley comma]]. The wesley comma is unchanged in [[Wesley Woolhouse]]'s 7/26-comma meantone – it is an [[eigenmonzo|eigenmonzo (i.e. unchanged-interval)]], which [[Gene Ward Smith]] has talked about<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12793.html Yahoo! Tuning Group | ''26et and meantone projections'']</ref>.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-30 02:51:40 UTC</tt>.<br>
: The original revision id was <tt>327181930</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">This tempers out the wesley comma, 78125/73728. The wesley comma is unchanged in Wesley Woolhouse's 7/26-comma meantone--it is an eigenmonzo, in other words. More information on that can be found [[http://tech.groups.yahoo.com/group/tuning-math/message/12793|here]].


=Wesley=
== Wesley ==
Comma: 78125/73728
The generator of wesley is one half of a [[~]][[5/2]], around 786 [[cent]]s, seven of which after [[octave reduction]] reach the [[3/2|perfect fifth]]. The [[ploidacot]] signature of this temperament is delta-heptacot. A good tuning for the generator may be found between [[26edo|17\26]] and [[29edo|19\29]].


POTE generator: ~125/96 = 414.509
[[Subgroup]]: 2.3.5


Map: [&lt;1 4 3|, &lt;0 -7 -2|]
[[Comma list]]: 78125/73728
EDOs: 23, 26, 29, 55c
Badness: 0.2477


==7-limit==
{{Mapping|legend=1| 1 -3 1 | 0 7 2 }}
Commas: 405/392, 875/864
: mapping generators: ~2, ~192/125


POTE generator: ~9/7 = 415.519
[[Optimal tuning]]s:  
* [[WE]] ~2 = 1202.9225{{c}}, ~192/125 = 787.4042{{c}}
: [[error map]]: {{val| +2.922 +1.107 -8.583 }}
* [[CWE]] ~2 = 1200.0000{{c}}, ~192/125 = 785.7808{{c}}
: error map: {{val| 0.000 -1.490 -14.752 }}


Map: [&lt;1 4 3 8|, &lt;0 -7 -2 -15|]
{{Optimal ET sequence|legend=1| 3, …, 23, 26, 29, 55c }}
Wedgie: &lt;&lt;7 2 15 -13 4 29||
EDOs: 26
Badness: 0.0953


==11-limit==
[[Badness]] (Sintel): 5.81
Commas: 45/44, 99/98, 875/864


POTE generator: ~9/7 = 415.769
== Septimal wesley ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 4 3 8 9|, &lt;0 -7 -2 -15 -16|]
[[Comma list]]: 405/392, 875/864
EDOs: 23de, 26, 75bcd
Badness: 0.0491


==13-limit==
{{Mapping|legend=1| 1 -3 1 -7 | 0 7 2 15 }}
Commas: 45/44, 78/77, 99/98, 325/324


POTE generator: ~9/7 = 415.645
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.6434{{c}}, ~14/9 = 786.8631{{c}}
: [[error map]]: {{val| +3.634 -4.844 -8.944 +8.617 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 784.7754{{c}}
: error map: {{val| 0.000 -8.527 -16.763 +2.805 }}


Map: [&lt;1 4 3 8 9 12|, &lt;0 -7 -2 -15 -16 -24|]
{{Optimal ET sequence|legend=1| 3d, …, 23d, 26 }}
EDOs: 26
Badness: 0.0384


==Snipes==
[[Badness]] (Sintel): 2.41
Commas: 225/224, 6125/5832


POTE generator: ~35/27 = 413.513
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 4 3 9|, &lt;0 -7 -2 -18|]
Comma list: 45/44, 99/98, 875/864
Wedgie: &lt;&lt;7 2 18 -13 9 36||
EDOs: 29, 90cd, 119cd
Badness: 0.1179


</pre></div>
Mapping: {{mapping| 1 -3 1 -7 -7 | 0 7 2 15 16 }}
<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;Wesley family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This tempers out the wesley comma, 78125/73728. The wesley comma is unchanged in Wesley Woolhouse's 7/26-comma meantone--it is an eigenmonzo, in other words. More information on that can be found &lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/12793" rel="nofollow"&gt;here&lt;/a&gt;.&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1203.6405{{c}}, ~11/7 = 786.6101{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Wesley"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Wesley&lt;/h1&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 784.5242{{c}}
Comma: 78125/73728&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 3de, …, 23de, 26 }}
POTE generator: ~125/96 = 414.509&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.62
Map: [&amp;lt;1 4 3|, &amp;lt;0 -7 -2|]&lt;br /&gt;
 
EDOs: 23, 26, 29, 55c&lt;br /&gt;
=== 13-limit ===
Badness: 0.2477&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Wesley-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
Comma list: 45/44, 78/77, 99/98, 325/324
Commas: 405/392, 875/864&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -3 1 -7 -7 -12 | 0 7 2 15 16 24 }}
POTE generator: ~9/7 = 415.519&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 4 3 8|, &amp;lt;0 -7 -2 -15|]&lt;br /&gt;
* WE: ~2 = 1200.0000{{c}}, ~11/7 = 786.5973{{c}}
Wedgie: &amp;lt;&amp;lt;7 2 15 -13 4 29||&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 784.5923{{c}}
EDOs: 26&lt;br /&gt;
 
Badness: 0.0953&lt;br /&gt;
{{Optimal ET sequence|legend=0| 3def, 23deff, 26 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Wesley-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Badness (Sintel): 1.59
Commas: 45/44, 99/98, 875/864&lt;br /&gt;
 
&lt;br /&gt;
== Snipes ==
POTE generator: ~9/7 = 415.769&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 4 3 8 9|, &amp;lt;0 -7 -2 -15 -16|]&lt;br /&gt;
[[Comma list]]: 225/224, 6125/5832
EDOs: 23de, 26, 75bcd&lt;br /&gt;
 
Badness: 0.0491&lt;br /&gt;
{{Mapping|legend=1| 1 -3 1 -9 | 0 7 2 18 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Wesley-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
[[Optimal tuning]]s:
Commas: 45/44, 78/77, 99/98, 325/324&lt;br /&gt;
* [[WE]]: ~2 = 1201.9268{{c}}, ~54/35 = 787.7496{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +1.927 +6.511 -8.888 -6.675 }}
POTE generator: ~9/7 = 415.645&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/35 = 786.6189{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +4.377 -13.076 -9.686 }}
Map: [&amp;lt;1 4 3 8 9 12|, &amp;lt;0 -7 -2 -15 -16 -24|]&lt;br /&gt;
 
EDOs: 26&lt;br /&gt;
{{Optimal ET sequence|legend=1| 3d, …, 26d, 29 }}
Badness: 0.0384&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.98
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Wesley-Snipes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Snipes&lt;/h2&gt;
 
Commas: 225/224, 6125/5832&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~35/27 = 413.513&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 225/224, 245/242
Map: [&amp;lt;1 4 3 9|, &amp;lt;0 -7 -2 -18|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;7 2 18 -13 9 36||&lt;br /&gt;
Mapping: {{mapping| 1 -3 1 -9 -9 | 0 7 2 18 19 }}
EDOs: 29, 90cd, 119cd&lt;br /&gt;
 
Badness: 0.1179&lt;/body&gt;&lt;/html&gt;</pre></div>
Optimal tunings:  
* WE: ~2 = 1201.9201{{c}}, ~11/7 = 787.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 786.6402{{c}}
 
{{Optimal ET sequence|legend=0| 3de, …, 26de, 29 }}
 
Badness (Sintel): 1.79
 
== Roman ==
Roman tempers out 525/512, the [[avicennma]]. The 7-limit version has also been proposed as ''crusher''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 525/512, 3125/3024
 
{{Mapping|legend=1| 1 -3 1 10 | 0 7 2 -11 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.0258{{c}}, ~63/40 = 787.4289{{c}}
: [[error map]]: {{val| +3.026 +0.970 -8.430 -0.286 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 785.4717{{c}}
: error map: {{val| 0.000 -3.653 -15.370 -9.0145 }}
 
{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
 
[[Badness]] (Sintel): 2.87
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/242, 525/512
 
Mapping: {{mapping| 1 -3 1 10 10 | 0 7 2 -11 -10 }}
 
Optimal tunings:
* WE: ~2 = 1202.7762{{c}}, ~11/7 = 787.3465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 785.5130{{c}}
 
{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55c }}
 
Badness (Sintel): 1.75
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 65/64, 100/99, 105/104, 245/242
 
Mapping: {{mapping| 1 -3 1 10 10 5 | 0 7 2 -11 -10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1202.8327{{c}}, ~11/7 = 787.3820{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 785.5085{{c}}
 
{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55cf }}
 
Badness (Sintel): 1.24
 
== Dubbla ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 78125/73728
 
{{Mapping|legend=1| 2 1 4 5 | 0 7 2 2 }}
: mapping generators: ~7/5, ~192/175
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.8055{{c}}, ~192/175 = 185.9882{{c}}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~192/175 = 185.8808{{c}}
 
{{Optimal ET sequence|legend=1| 6, 20b, 26, 58c }}
 
[[Badness]] (Sintel): 4.60
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 125/121, 1344/1331
 
Mapping: {{mapping| 2 1 4 5 6 | 0 7 2 2 3 }}
 
Optimal tunings:
* WE: ~7/5 = 600.3702{{c}}, ~11/10 = 185.9936{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~11/10 = 185.9408{{c}}
 
{{Optimal ET sequence|legend=0| 6, 20b, 26 }}
 
Badness (Sintel): 2.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 105/104, 125/121, 144/143
 
Mapping: {{mapping| 2 1 4 5 6 4 | 0 7 2 2 3 11 }}
 
Optimal tunings:
* WE: ~7/5 = 600.3563{{c}}, ~11/10 = 185.8121{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~11/10 = 185.7623{{c}}
 
{{Optimal ET sequence|legend=0| 6f, 20bff, 26 }}
 
Badness (Sintel): 2.05
 
== References ==
 
[[Category:Wesley family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 12:25, 14 August 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 wesley family of temperaments tempers out 78125/73728, the wesley comma. The wesley comma is unchanged in Wesley Woolhouse's 7/26-comma meantone – it is an eigenmonzo (i.e. unchanged-interval), which Gene Ward Smith has talked about[1].

Wesley

The generator of wesley is one half of a ~5/2, around 786 cents, seven of which after octave reduction reach the perfect fifth. The ploidacot signature of this temperament is delta-heptacot. A good tuning for the generator may be found between 17\26 and 19\29.

Subgroup: 2.3.5

Comma list: 78125/73728

Mapping: [⟨1 -3 1], ⟨0 7 2]]

mapping generators: ~2, ~192/125

Optimal tunings:

  • WE ~2 = 1202.9225 ¢, ~192/125 = 787.4042 ¢
error map: ⟨+2.922 +1.107 -8.583]
  • CWE ~2 = 1200.0000 ¢, ~192/125 = 785.7808 ¢
error map: ⟨0.000 -1.490 -14.752]

Optimal ET sequence: 3, …, 23, 26, 29, 55c

Badness (Sintel): 5.81

Septimal wesley

Subgroup: 2.3.5.7

Comma list: 405/392, 875/864

Mapping: [⟨1 -3 1 -7], ⟨0 7 2 15]]

Optimal tunings:

  • WE: ~2 = 1203.6434 ¢, ~14/9 = 786.8631 ¢
error map: ⟨+3.634 -4.844 -8.944 +8.617]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 784.7754 ¢
error map: ⟨0.000 -8.527 -16.763 +2.805]

Optimal ET sequence: 3d, …, 23d, 26

Badness (Sintel): 2.41

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 875/864

Mapping: [⟨1 -3 1 -7 -7], ⟨0 7 2 15 16]]

Optimal tunings:

  • WE: ~2 = 1203.6405 ¢, ~11/7 = 786.6101 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 784.5242 ¢

Optimal ET sequence: 3de, …, 23de, 26

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 99/98, 325/324

Mapping: [⟨1 -3 1 -7 -7 -12], ⟨0 7 2 15 16 24]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~11/7 = 786.5973 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 784.5923 ¢

Optimal ET sequence: 3def, 23deff, 26

Badness (Sintel): 1.59

Snipes

Subgroup: 2.3.5.7

Comma list: 225/224, 6125/5832

Mapping: [⟨1 -3 1 -9], ⟨0 7 2 18]]

Optimal tunings:

  • WE: ~2 = 1201.9268 ¢, ~54/35 = 787.7496 ¢
error map: ⟨+1.927 +6.511 -8.888 -6.675]
  • CWE: ~2 = 1200.0000 ¢, ~54/35 = 786.6189 ¢
error map: ⟨0.000 +4.377 -13.076 -9.686]

Optimal ET sequence: 3d, …, 26d, 29

Badness (Sintel): 2.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 225/224, 245/242

Mapping: [⟨1 -3 1 -9 -9], ⟨0 7 2 18 19]]

Optimal tunings:

  • WE: ~2 = 1201.9201 ¢, ~11/7 = 787.7690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 786.6402 ¢

Optimal ET sequence: 3de, …, 26de, 29

Badness (Sintel): 1.79

Roman

Roman tempers out 525/512, the avicennma. The 7-limit version has also been proposed as crusher.

Subgroup: 2.3.5.7

Comma list: 525/512, 3125/3024

Mapping: [⟨1 -3 1 10], ⟨0 7 2 -11]]

Optimal tunings:

  • WE: ~2 = 1203.0258 ¢, ~63/40 = 787.4289 ¢
error map: ⟨+3.026 +0.970 -8.430 -0.286]
  • CWE: ~2 = 1200.0000 ¢, ~63/40 = 785.4717 ¢
error map: ⟨0.000 -3.653 -15.370 -9.0145]

Optimal ET sequence: 3, 23, 26, 29, 55c

Badness (Sintel): 2.87

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 525/512

Mapping: [⟨1 -3 1 10 10], ⟨0 7 2 -11 -10]]

Optimal tunings:

  • WE: ~2 = 1202.7762 ¢, ~11/7 = 787.3465 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 785.5130 ¢

Optimal ET sequence: 3, 23, 26, 29, 55c

Badness (Sintel): 1.75

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 100/99, 105/104, 245/242

Mapping: [⟨1 -3 1 10 10 5], ⟨0 7 2 -11 -10 -2]]

Optimal tunings:

  • WE: ~2 = 1202.8327 ¢, ~11/7 = 787.3820 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 785.5085 ¢

Optimal ET sequence: 3, 23, 26, 29, 55cf

Badness (Sintel): 1.24

Dubbla

Subgroup: 2.3.5.7

Comma list: 50/49, 78125/73728

Mapping: [⟨2 1 4 5], ⟨0 7 2 2]]

mapping generators: ~7/5, ~192/175

Optimal tunings:

  • WE: ~7/5 = 600.8055 ¢, ~192/175 = 185.9882 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~192/175 = 185.8808 ¢

Optimal ET sequence: 6, 20b, 26, 58c

Badness (Sintel): 4.60

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 125/121, 1344/1331

Mapping: [⟨2 1 4 5 6], ⟨0 7 2 2 3]]

Optimal tunings:

  • WE: ~7/5 = 600.3702 ¢, ~11/10 = 185.9936 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~11/10 = 185.9408 ¢

Optimal ET sequence: 6, 20b, 26

Badness (Sintel): 2.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 105/104, 125/121, 144/143

Mapping: [⟨2 1 4 5 6 4], ⟨0 7 2 2 3 11]]

Optimal tunings:

  • WE: ~7/5 = 600.3563 ¢, ~11/10 = 185.8121 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~11/10 = 185.7623 ¢

Optimal ET sequence: 6f, 20bff, 26

Badness (Sintel): 2.05

References

  1. ↑ Yahoo! Tuning Group | 26et and meantone projections
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