Orwellismic temperaments: 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>
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[orwellisma]] ({{monzo|legend=1| 6 3 -1 -3 }}, [[ratio]]: 1728/1715).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-19 09:46:41 UTC</tt>.<br>
: The original revision id was <tt>303109366</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]]
These temper out |6 3 -1 -3&gt; = 1728/1715, the orwellisma.


=Secant=
Temperaments discussed elsewhere are:
Commas: 1728/1715, 177147/175000
* [[Beep]] (+21/20 or 27/25) → [[Bug family #Beep|Bug family]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* [[Superpyth]] (+64/63) → [[Archytas clan #Superpyth|Archytas clan]]
* [[Mothra]] (+81/80) → [[Meantone family #Mothra|Meantone family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Secund]]'' (+405/392 or 525/512) → [[Greenwoodmic temperaments #Secund|Greenwoodmic temperaments]]
* ''[[Quartonic]]'' (+4000/3969) → [[Quartonic family]]
* [[Buzzard]] (+5120/5103) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* ''[[Trisensory]]'' (+78732/78125) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trimabila]]'' (+268435456/263671875) → [[Mabila family #Trimabila|Mabila family]]


POTE generator: ~10/9 = 185.885
Considered below are sentinel, secant, diesic, infraorwell, pentaorwell, triskaidekic, and philips.  


Map: [&lt;2 1 0 5|, &lt;0 7 15 2|]
== Sentinel ==
Wedgie: &lt;&lt;14 30 4 15 -33 -75|
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sentinel]].''
EDOs: 26, 58, 84, 142, 368cd
Badness: 0.0953


==11-limit==
Sentinel tempers out [[3645/3584]] and may be described as the {{nowrap| 14 & 17 }} temperament. It is related to [[squares]], but the mappings differ for the [[5/1|5th harmonic]]. Like squares, it splits the [[6/1|6th harmonic]] into four subminor sixths of 11/7~14/9 (or splits a perfect eleventh into four supermajor thirds of 9/7~14/11), and uses it for a generator; its [[ploidacot]] is beta-tetracot. However, the 5th harmonic is found by -15 generator steps instead of +16 as in squares.
Commas: 441/440, 1344/1331, 1728/1715


POTE genertor: ~10/9 = 185.962
As one might expect, [[31edo]] is a good tuning for this temperament, in which case it is identical to squares. Among the alternatives are [[48edo]] and [[79edo]], both in their [[patent val]]s.  


Map: [&lt;2 1 0 5 6|, &lt;0 7 15 2 3|]
[[Subgroup]]: 2.3.5.7
EDOs: 26, 58, 142e, 200cde
Badness: 0.0464


==13-limit==
[[Comma list]]: 1728/1715, 3645/3584
Commas: 144/143, 351/350, 364/363, 441/440


POTE generator: ~10/9 = 185.955
{{Mapping|legend=1| 1 -1 12 -3 | 0 4 -15 9 }}
: mapping generators: ~2, ~14/9


Map: [&lt;2 1 0 5 6 4|, &lt;0 7 15 2 3 11|]
[[Optimal tuning]]s:  
EDOs: 26, 58, 84, 142ef
* [[WE]]: ~2 = 1200.8626{{c}}, ~14/9 = 774.9609{{c}}
Badness: 0.0250
: [[error map]]: {{val| +0.863 -2.974 -0.376 +3.234 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.4021{{c}}
: error map: {{val| 0.000 -4.347 -2.345 +0.793 }}


=Infraorwell=
{{Optimal ET sequence|legend=1| 14, 17, 31, 110, 141, 172b }}
Commas: 1728/1715, 28672/28125


POTE generator: ~7/6 = 269.032
[[Badness]] (Sintel): 2.37


Map: [&lt;1 14 0 16|, &lt;0 -16 3 -17|]
=== 11-limit ===
Wedgie: &lt;&lt;16 -3 17 -42 -18 48||
Subgroup: 2.3.5.7.11
EDOs: 9, 49, 58, 165cd, 223bcd
Badness: 0.1171


==11-limit==
Comma list: 99/98, 243/242, 385/384
Commas: 176/175, 540/539, 1344/1331


POTE generator: ~7/6 = 269.036
Mapping: {{mapping| 1 -1 12 -3 -3 | 0 4 -15 9 10 }}


Map: [&lt;1 14 0 16 12|, &lt;0 -16 3 -17 -11|]
Optimal tunings:  
EDOs: 9, 49, 58, 165cde
* WE: ~2 = 1201.0893{{c}}, ~11/7 = 775.1534{{c}}
Badness: 0.0407
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4564{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 14, 17, 31, 79, 110e, 141e }}
Commas: 144/143, 176/175, 196/195, 364/363


POTE generator: ~7/6 = 269.021
Badness (Sintel): 1.31


Map: [&lt;1 14 0 16 12 20|, &lt;0 -16 3 -17 -11 -21|]
=== 13-limit ===
EDOs: 9, 49f, 58
Subgroup: 2.3.5.7.11.13
Badness: 0.0237


Comma list: 99/98, 105/104, 144/143, 243/242


Mapping: {{mapping| 1 -1 12 -3 -3 5 | 0 4 -15 9 10 -2 }}


Optimal tunings:
* WE: ~2 = 1200.4045{{c}}, ~11/7 = 774.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.5000{{c}}


{{Optimal ET sequence|legend=0| 14, 17, 31 }}


</pre></div>
Badness (Sintel): 1.55
<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;Orwellismic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Secant"&gt;Secant&lt;/a&gt;&lt;/div&gt;
== Secant ==
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Secant-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Secant, the {{nowrap| 26 & 58 }} temperament, is generated by a slightly sharp ~10/9, seven of which plus a semi-octave period give the [[3/1|3rd harmonic]]. Its ploidacot is diploid alpha-heptacot. [[58edo]], [[84edo]] and [[142edo]] all make for excellent tunings.
&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Secant-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Infraorwell"&gt;Infraorwell&lt;/a&gt;&lt;/div&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Infraorwell-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Infraorwell-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
[[Comma list]]: 1728/1715, 177147/175000
&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;These temper out |6 3 -1 -3&amp;gt; = 1728/1715, the orwellisma.&lt;br /&gt;
{{Mapping|legend=1| 2 1 0 5 | 0 7 15 2 }}
&lt;br /&gt;
: mapping generators: ~567/400, ~10/9
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Secant"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Secant&lt;/h1&gt;
 
Commas: 1728/1715, 177147/175000&lt;br /&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~567/400 = 599.8218{{c}}, ~10/9 = 185.8303{{c}}
POTE generator: ~10/9 = 185.885&lt;br /&gt;
: [[error map]]: {{val| -0.356 -1.321 +1.141 +1.944 }}
&lt;br /&gt;
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~10/9 = 185.8618{{c}}
Map: [&amp;lt;2 1 0 5|, &amp;lt;0 7 15 2|]&lt;br /&gt;
: error map: {{val| 0.000 -0.922 +1.613 +2.898 }}
Wedgie: &amp;lt;&amp;lt;14 30 4 15 -33 -75|&lt;br /&gt;
 
EDOs: 26, 58, 84, 142, 368cd&lt;br /&gt;
{{Optimal ET sequence|legend=1| 26, 58, 84, 142, 368cd }}
Badness: 0.0953&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.41
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Secant-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 441/440, 1344/1331, 1728/1715&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE genertor: ~10/9 = 185.962&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 441/440, 1344/1331, 1728/1715
Map: [&amp;lt;2 1 0 5 6|, &amp;lt;0 7 15 2 3|]&lt;br /&gt;
 
EDOs: 26, 58, 142e, 200cde&lt;br /&gt;
Mapping: {{mapping| 2 1 0 5 6 | 0 7 15 2 3 }}
Badness: 0.0464&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Secant-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
* WE: ~99/70 = 599.5785{{c}}, ~10/9 = 185.8332{{c}}
Commas: 144/143, 351/350, 364/363, 441/440&lt;br /&gt;
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 185.9140{{c}}
&lt;br /&gt;
 
POTE generator: ~10/9 = 185.955&lt;br /&gt;
{{Optimal ET sequence|legend=0| 26, 58, 142e, 200cdee }}
&lt;br /&gt;
 
Map: [&amp;lt;2 1 0 5 6 4|, &amp;lt;0 7 15 2 3 11|]&lt;br /&gt;
Badness (Sintel): 1.53
EDOs: 26, 58, 84, 142ef&lt;br /&gt;
 
Badness: 0.0250&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Infraorwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Infraorwell&lt;/h1&gt;
 
Commas: 1728/1715, 28672/28125&lt;br /&gt;
Comma list: 144/143, 351/350, 364/363, 441/440
&lt;br /&gt;
 
POTE generator: ~7/6 = 269.032&lt;br /&gt;
Mapping: {{mapping| 2 1 0 5 6 4 | 0 7 15 2 3 11 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 14 0 16|, &amp;lt;0 -16 3 -17|]&lt;br /&gt;
Optimal tunings:  
Wedgie: &amp;lt;&amp;lt;16 -3 17 -42 -18 48||&lt;br /&gt;
* WE: ~55/39 = 599.5432{{c}}, ~10/9 = 185.8136{{c}}
EDOs: 9, 49, 58, 165cd, 223bcd&lt;br /&gt;
* CWE: ~55/39 = 600.0000{{c}}, ~10/9 = 185.9012{{c}}
Badness: 0.1171&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 26, 58, 84, 142ef }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Infraorwell-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
 
Commas: 176/175, 540/539, 1344/1331&lt;br /&gt;
Badness (Sintel): 1.03
&lt;br /&gt;
 
POTE generator: ~7/6 = 269.036&lt;br /&gt;
=== 17-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
Map: [&amp;lt;1 14 0 16 12|, &amp;lt;0 -16 3 -17 -11|]&lt;br /&gt;
 
EDOs: 9, 49, 58, 165cde&lt;br /&gt;
Comma list: 144/143, 170/169, 221/220, 351/350, 441/440
Badness: 0.0407&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 2 1 0 5 6 4 6 | 0 7 15 2 3 11 7 }}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Infraorwell-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
 
Commas: 144/143, 176/175, 196/195, 364/363&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~17/12 = 599.7117{{c}}, ~10/9 = 185.8270{{c}}
POTE generator: ~7/6 = 269.021&lt;br /&gt;
* CWE: ~17/12 = 600.0000{{c}}, ~10/9 = 185.8834{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 14 0 16 12 20|, &amp;lt;0 -16 3 -17 -11 -21|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 26, 58, 84 }}
EDOs: 9, 49f, 58&lt;br /&gt;
 
Badness: 0.0237&lt;/body&gt;&lt;/html&gt;</pre></div>
Badness (Sintel): 1.10
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 144/143, 153/152, 170/169, 210/209, 221/220, 400/399
 
Mapping: {{mapping| 2 1 0 5 6 4 6 2 | 0 7 15 2 3 11 7 21 }}
 
Optimal tunings:  
* WE: ~17/12 = 599.7288{{c}}, ~10/9 = 185.7738{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/9 = 185.8279{{c}}
 
{{Optimal ET sequence|legend=0| 26, 58h, 84 }}
 
Badness (Sintel): 1.13
 
== Diesic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Diesic]].''
 
Diesic is generated by a diesis-sized interval of ~36/35, hence the name. It may be described as the {{nowrap| 31 & 32c }} temperament. It is related to [[slender]], both sharing the ploidacot of omega-13-cot, but the mappings differ for the [[5/1|5th harmonic]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 5103/5000
 
{{Mapping|legend=1| 1 2 3 3 | 0 -13 -21 -6 }}
: mapping generators: ~2, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1293{{c}}, ~36/35 = 38.5709{{c}}
: [[error map]]: {{val| +0.129 -3.118 +4.086 +0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 38.5603{{c}}
: error map: {{val| 0.000 -3.239 +3.920 -0.188 }}
 
{{Optimal ET sequence|legend=1| 1c, 30bc, 31, 156c }}
 
[[Badness]] (Sintel): 2.72
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 891/875
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -13 -21 -6 -17 }}
 
Optimal tunings:
* WE: ~2 = 1200.4923{{c}}, ~36/35 = 38.5806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.5397{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}
 
Badness (Sintel): 1.46
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 121/120, 343/338, 441/440
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -13 -21 -6 -17 -9 }}
 
Optimal tunings:  
* WE: ~2 = 1199.3005{{c}}, ~36/35 = 38.4213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.4764{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}
 
Badness (Sintel): 1.57
 
== Infraorwell ==
Infraorwell may be described as the {{nowrap| 49 & 58 }} temperament. It is generated by a ~7/6, less sharp than the generator of [[orwell]] but still sharp of just, so that three generators make a ~8/5, but seven generators does not give the [[3/1|3rd harmonic]]. Instead, sixteen generators give the [[12/1|12th harmonic]]; the ploidacot for this temperament is therefore gamma-16-cot. [[58edo]] makes for a recommendable tuning, though [[49edo]] and [[67edo]] are among the possibilities.  
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 28672/28125
 
{{Mapping|legend=1| 1 -2 3 -1 | 0 16 -3 17 }}
: mapping generators: ~2, ~7/6
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.4649{{c}}, ~7/6 = 268.6878{{c}}
: [[error map]]: {{val| -1.535 +0.120 +3.018 +0.401 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 268.9844{{c}}
: error map: {{val| 0.000 +1.796 +6.733 +3.909 }}
 
{{Optimal ET sequence|legend=1| 9, 40bd, 49, 58, 165cd, 223bccd, 281bcccdd }}
 
[[Badness]] (Sintel): 2.96
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 1344/1331
 
Mapping: {{mapping| 1 -2 3 -1 1 | 0 16 -3 17 11 }}
 
Optimal tunings:
* WE: ~2 = 1198.2928{{c}}, ~7/6 = 268.6532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9825{{c}}
 
{{Optimal ET sequence|legend=0| 9, 40bde, 49, 58, 165cdee, 223bccdeee, 281bcccddeee }}
 
Badness (Sintel): 1.35
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 196/195, 364/363
 
Mapping: {{mapping| 1 -2 3 -1 1 -1 | 0 16 -3 17 11 21 }}
 
Optimal tunings:  
* WE: ~2 = 1198.2216{{c}}, ~7/6 = 268.6220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9616{{c}}
 
{{Optimal ET sequence|legend=0| 9, 40bdef, 49f, 58, 223bccdeeefff, 281bcccddeeeffff }}
 
Badness (Sintel): 0.979
 
== Pentaorwell ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Counterpental]].''
 
Named by [[User:Flirora|+merlan #flirora]] in 2021, pentaorwell tempers out 179200/177147 and is the {{nowrap| 75 & 80 }} temperament. Its ploidacot is pentaploid monocot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 179200/177147
 
{{Mapping|legend=1| 5 0 -36 22 | 0 1 6 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~280/243 = 239.7268{{c}}, ~3/2 = 704.1103{{c}} (~64/63 = 15.0700{{c}})
: [[error map]]: {{val| -1.366 +0.789 -0.013 +2.419 }}
* [[CWE]]: ~280/243 = 240.0000{{c}}, ~3/2 = 704.7723{{c}} (~64/63 = 15.2277{{c}})
: error map: {{val| 0.000 +2.817 +2.320 +6.402 }}
 
{{Optimal ET sequence|legend=1| 5, 75, 80 }}
 
[[Badness]] (Sintel): 3.76
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 896/891, 1728/1715, 2200/2187
 
Mapping: {{mapping| 5 0 -36 22 57 | 0 1 6 -1 -5 }}
 
Optimal tunings:
* WE: ~55/48 = 239.7357{{c}}, ~3/2 = 704.1025{{c}} (~99/98 = 15.1045{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.8440{{c}} (~99/98 = 15.1560{{c}})
 
{{Optimal ET sequence|legend=0| 5, 75e, 80, 315bcdddee }}
 
Badness (Sintel): 2.37
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 364/363, 1728/1715
 
Mapping: {{mapping| 5 0 -36 22 57 82 | 0 1 6 -1 -5 -8 }}
 
Optimal tunings:
* WE: ~55/48 = 239.7711{{c}}, ~3/2 = 704.0302{{c}} (~99/98 = 15.2830{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.7112{{c}} (~99/98 = 15.2888{{c}})
 
{{Optimal ET sequence|legend=0| 5f, 75e, 80, 155de }}
 
Badness (Sintel): 2.30
 
== Triskaidekic ==
: ''For the 5-limit version, see [[13th-octave temperaments #Triskaidekic]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 1875/1792
 
{{Mapping|legend=1| 13 0 30 16 | 0 1 0 1 }}
: mapping generators: ~15/14, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 92.5254{{c}}, ~3/2 = 695.7801{{c}}
: [[error map]]: {{val| +2.831 -3.344 -10.551 +10.192 }}
* [[CWE]]: ~15/14 = 92.3077{{c}}, ~3/2 = 695.8319{{c}}
: error map: {{val| 0.000 -6.123 -17.083 +3.929 }}
 
{{Optimal ET sequence|legend=1| 13d, 26 }}
 
[[Badness]] (Sintel): 5.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 125/121, 385/384
 
Mapping: {{mapping| 13 0 30 16 45 | 0 1 0 1 0 }}
 
Optimal tunings:
* WE: ~15/14 = 92.4364{{c}}, ~3/2 = 697.2675{{c}}
* CWE: ~15/14 = 92.3077{{c}}, ~3/2 = 697.0548{{c}}
 
{{Optimal ET sequence|legend=0| 13d, 26 }}
 
Badness (Sintel): 3.27
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 99/98, 125/121, 1200/1183
 
Mapping: {{mapping| 13 0 30 16 45 48 | 0 1 0 1 0 0 }}
 
Optimal tunings:
* WE: ~15/14 = 92.4535{{c}}, ~3/2 = 696.9785{{c}}
* CWE: ~15/14 = 92.3077{{c}}, ~3/2 = 696.5711{{c}}
 
{{Optimal ET sequence|legend=0| 13d, 26 }}
 
Badness (Sintel): 2.45
 
== Phillips ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Phillips]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 6561/6272
 
{{Mapping|legend=1| 2 0 33 -7 | 0 1 -9 4 }}
: mapping generators: ~81/56, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~81/56 = 601.3944{{c}}, ~3/2 = 692.7807{{c}}
: [[error map]]: {{val| +2.789 -6.385 -0.424 +3.692 }}
* [[CWE]]: ~81/56 = 600.0000{{c}}, ~3/2 = 691.1141{{c}}
: error map: {{val| 0.000 -10.841 -6.340 -4.370 }}
 
{{Optimal ET sequence|legend=1| 14, 26, 66b, 92bc }}
 
[[Badness]] (Sintel): 5.79
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 385/384, 729/704
 
Mapping: {{mapping| 2 0 33 -7 -12 | 0 1 -9 4 6 }}
 
Optimal tunings:
* WE: ~63/44 = 601.2349{{c}}, ~3/2 = 692.4623{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 691.0383{{c}}
 
{{Optimal ET sequence|legend=0| 14, 26, 40, 66b }}
 
Badness (Sintel): 3.17
 
[[Category:Temperament collections]]
[[Category:Orwellismic temperaments| ]] <!-- main article -->
[[Category:Orwellismic| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 22:34, 21 January 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.

This is a collection of rank-2 temperaments that temper out the orwellisma (monzo: [6 3 -1 -3⟩, ratio: 1728/1715).

Temperaments discussed elsewhere are:

Considered below are sentinel, secant, diesic, infraorwell, pentaorwell, triskaidekic, and philips.

Sentinel

For the 5-limit version, see Miscellaneous 5-limit temperaments #Sentinel.

Sentinel tempers out 3645/3584 and may be described as the 14 & 17 temperament. It is related to squares, but the mappings differ for the 5th harmonic. Like squares, it splits the 6th harmonic into four subminor sixths of 11/7~14/9 (or splits a perfect eleventh into four supermajor thirds of 9/7~14/11), and uses it for a generator; its ploidacot is beta-tetracot. However, the 5th harmonic is found by -15 generator steps instead of +16 as in squares.

As one might expect, 31edo is a good tuning for this temperament, in which case it is identical to squares. Among the alternatives are 48edo and 79edo, both in their patent vals.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3645/3584

Mapping: [⟨1 -1 12 -3], ⟨0 4 -15 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1200.8626 ¢, ~14/9 = 774.9609 ¢
error map: ⟨+0.863 -2.974 -0.376 +3.234]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.4021 ¢
error map: ⟨0.000 -4.347 -2.345 +0.793]

Optimal ET sequence: 14, 17, 31, 110, 141, 172b

Badness (Sintel): 2.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 243/242, 385/384

Mapping: [⟨1 -1 12 -3 -3], ⟨0 4 -15 9 10]]

Optimal tunings:

  • WE: ~2 = 1201.0893 ¢, ~11/7 = 775.1534 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.4564 ¢

Optimal ET sequence: 14, 17, 31, 79, 110e, 141e

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 105/104, 144/143, 243/242

Mapping: [⟨1 -1 12 -3 -3 5], ⟨0 4 -15 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1200.4045 ¢, ~11/7 = 774.7596 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.5000 ¢

Optimal ET sequence: 14, 17, 31

Badness (Sintel): 1.55

Secant

Secant, the 26 & 58 temperament, is generated by a slightly sharp ~10/9, seven of which plus a semi-octave period give the 3rd harmonic. Its ploidacot is diploid alpha-heptacot. 58edo, 84edo and 142edo all make for excellent tunings.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 177147/175000

Mapping: [⟨2 1 0 5], ⟨0 7 15 2]]

mapping generators: ~567/400, ~10/9

Optimal tunings:

  • WE: ~567/400 = 599.8218 ¢, ~10/9 = 185.8303 ¢
error map: ⟨-0.356 -1.321 +1.141 +1.944]
  • CWE: ~567/400 = 600.0000 ¢, ~10/9 = 185.8618 ¢
error map: ⟨0.000 -0.922 +1.613 +2.898]

Optimal ET sequence: 26, 58, 84, 142, 368cd

Badness (Sintel): 2.41

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 1728/1715

Mapping: [⟨2 1 0 5 6], ⟨0 7 15 2 3]]

Optimal tunings:

  • WE: ~99/70 = 599.5785 ¢, ~10/9 = 185.8332 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~10/9 = 185.9140 ¢

Optimal ET sequence: 26, 58, 142e, 200cdee

Badness (Sintel): 1.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 351/350, 364/363, 441/440

Mapping: [⟨2 1 0 5 6 4], ⟨0 7 15 2 3 11]]

Optimal tunings:

  • WE: ~55/39 = 599.5432 ¢, ~10/9 = 185.8136 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~10/9 = 185.9012 ¢

Optimal ET sequence: 26, 58, 84, 142ef

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 144/143, 170/169, 221/220, 351/350, 441/440

Mapping: [⟨2 1 0 5 6 4 6], ⟨0 7 15 2 3 11 7]]

Optimal tunings:

  • WE: ~17/12 = 599.7117 ¢, ~10/9 = 185.8270 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~10/9 = 185.8834 ¢

Optimal ET sequence: 26, 58, 84

Badness (Sintel): 1.10

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 144/143, 153/152, 170/169, 210/209, 221/220, 400/399

Mapping: [⟨2 1 0 5 6 4 6 2], ⟨0 7 15 2 3 11 7 21]]

Optimal tunings:

  • WE: ~17/12 = 599.7288 ¢, ~10/9 = 185.7738 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~10/9 = 185.8279 ¢

Optimal ET sequence: 26, 58h, 84

Badness (Sintel): 1.13

Diesic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Diesic.

Diesic is generated by a diesis-sized interval of ~36/35, hence the name. It may be described as the 31 & 32c temperament. It is related to slender, both sharing the ploidacot of omega-13-cot, but the mappings differ for the 5th harmonic.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 5103/5000

Mapping: [⟨1 2 3 3], ⟨0 -13 -21 -6]]

mapping generators: ~2, ~36/35

Optimal tunings:

  • WE: ~2 = 1200.1293 ¢, ~36/35 = 38.5709 ¢
error map: ⟨+0.129 -3.118 +4.086 +0.137]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 38.5603 ¢
error map: ⟨0.000 -3.239 +3.920 -0.188]

Optimal ET sequence: 1c, 30bc, 31, 156c

Badness (Sintel): 2.72

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 891/875

Mapping: [⟨1 2 3 3 4], ⟨0 -13 -21 -6 -17]]

Optimal tunings:

  • WE: ~2 = 1200.4923 ¢, ~36/35 = 38.5806 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 38.5397 ¢

Optimal ET sequence: 1ce, 30bce, 31

Badness (Sintel): 1.46

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 343/338, 441/440

Mapping: [⟨1 2 3 3 4 4], ⟨0 -13 -21 -6 -17 -9]]

Optimal tunings:

  • WE: ~2 = 1199.3005 ¢, ~36/35 = 38.4213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 38.4764 ¢

Optimal ET sequence: 1ce, 30bce, 31

Badness (Sintel): 1.57

Infraorwell

Infraorwell may be described as the 49 & 58 temperament. It is generated by a ~7/6, less sharp than the generator of orwell but still sharp of just, so that three generators make a ~8/5, but seven generators does not give the 3rd harmonic. Instead, sixteen generators give the 12th harmonic; the ploidacot for this temperament is therefore gamma-16-cot. 58edo makes for a recommendable tuning, though 49edo and 67edo are among the possibilities.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 28672/28125

Mapping: [⟨1 -2 3 -1], ⟨0 16 -3 17]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • WE: ~2 = 1198.4649 ¢, ~7/6 = 268.6878 ¢
error map: ⟨-1.535 +0.120 +3.018 +0.401]
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 268.9844 ¢
error map: ⟨0.000 +1.796 +6.733 +3.909]

Optimal ET sequence: 9, 40bd, 49, 58, 165cd, 223bccd, 281bcccdd

Badness (Sintel): 2.96

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 1344/1331

Mapping: [⟨1 -2 3 -1 1], ⟨0 16 -3 17 11]]

Optimal tunings:

  • WE: ~2 = 1198.2928 ¢, ~7/6 = 268.6532 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 268.9825 ¢

Optimal ET sequence: 9, 40bde, 49, 58, 165cdee, 223bccdeee, 281bcccddeee

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 364/363

Mapping: [⟨1 -2 3 -1 1 -1], ⟨0 16 -3 17 11 21]]

Optimal tunings:

  • WE: ~2 = 1198.2216 ¢, ~7/6 = 268.6220 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 268.9616 ¢

Optimal ET sequence: 9, 40bdef, 49f, 58, 223bccdeeefff, 281bcccddeeeffff

Badness (Sintel): 0.979

Pentaorwell

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Counterpental.

Named by +merlan #flirora in 2021, pentaorwell tempers out 179200/177147 and is the 75 & 80 temperament. Its ploidacot is pentaploid monocot.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 179200/177147

Mapping: [⟨5 0 -36 22], ⟨0 1 6 -1]]

Optimal tunings:

  • WE: ~280/243 = 239.7268 ¢, ~3/2 = 704.1103 ¢ (~64/63 = 15.0700 ¢)
error map: ⟨-1.366 +0.789 -0.013 +2.419]
  • CWE: ~280/243 = 240.0000 ¢, ~3/2 = 704.7723 ¢ (~64/63 = 15.2277 ¢)
error map: ⟨0.000 +2.817 +2.320 +6.402]

Optimal ET sequence: 5, 75, 80

Badness (Sintel): 3.76

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 1728/1715, 2200/2187

Mapping: [⟨5 0 -36 22 57], ⟨0 1 6 -1 -5]]

Optimal tunings:

  • WE: ~55/48 = 239.7357 ¢, ~3/2 = 704.1025 ¢ (~99/98 = 15.1045 ¢)
  • CWE: ~55/48 = 240.0000 ¢, ~3/2 = 704.8440 ¢ (~99/98 = 15.1560 ¢)

Optimal ET sequence: 5, 75e, 80, 315bcdddee

Badness (Sintel): 2.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 1728/1715

Mapping: [⟨5 0 -36 22 57 82], ⟨0 1 6 -1 -5 -8]]

Optimal tunings:

  • WE: ~55/48 = 239.7711 ¢, ~3/2 = 704.0302 ¢ (~99/98 = 15.2830 ¢)
  • CWE: ~55/48 = 240.0000 ¢, ~3/2 = 704.7112 ¢ (~99/98 = 15.2888 ¢)

Optimal ET sequence: 5f, 75e, 80, 155de

Badness (Sintel): 2.30

Triskaidekic

For the 5-limit version, see 13th-octave temperaments #Triskaidekic.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 1875/1792

Mapping: [⟨13 0 30 16], ⟨0 1 0 1]]

mapping generators: ~15/14, ~3

Optimal tunings:

  • WE: ~15/14 = 92.5254 ¢, ~3/2 = 695.7801 ¢
error map: ⟨+2.831 -3.344 -10.551 +10.192]
  • CWE: ~15/14 = 92.3077 ¢, ~3/2 = 695.8319 ¢
error map: ⟨0.000 -6.123 -17.083 +3.929]

Optimal ET sequence: 13d, 26

Badness (Sintel): 5.54

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 125/121, 385/384

Mapping: [⟨13 0 30 16 45], ⟨0 1 0 1 0]]

Optimal tunings:

  • WE: ~15/14 = 92.4364 ¢, ~3/2 = 697.2675 ¢
  • CWE: ~15/14 = 92.3077 ¢, ~3/2 = 697.0548 ¢

Optimal ET sequence: 13d, 26

Badness (Sintel): 3.27

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 125/121, 1200/1183

Mapping: [⟨13 0 30 16 45 48], ⟨0 1 0 1 0 0]]

Optimal tunings:

  • WE: ~15/14 = 92.4535 ¢, ~3/2 = 696.9785 ¢
  • CWE: ~15/14 = 92.3077 ¢, ~3/2 = 696.5711 ¢

Optimal ET sequence: 13d, 26

Badness (Sintel): 2.45

Phillips

For the 5-limit version, see Miscellaneous 5-limit temperaments #Phillips.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 6561/6272

Mapping: [⟨2 0 33 -7], ⟨0 1 -9 4]]

mapping generators: ~81/56, ~3

Optimal tunings:

  • WE: ~81/56 = 601.3944 ¢, ~3/2 = 692.7807 ¢
error map: ⟨+2.789 -6.385 -0.424 +3.692]
  • CWE: ~81/56 = 600.0000 ¢, ~3/2 = 691.1141 ¢
error map: ⟨0.000 -10.841 -6.340 -4.370]

Optimal ET sequence: 14, 26, 66b, 92bc

Badness (Sintel): 5.79

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 729/704

Mapping: [⟨2 0 33 -7 -12], ⟨0 1 -9 4 6]]

Optimal tunings:

  • WE: ~63/44 = 601.2349 ¢, ~3/2 = 692.4623 ¢
  • CWE: ~63/44 = 600.0000 ¢, ~3/2 = 691.0383 ¢

Optimal ET sequence: 14, 26, 40, 66b

Badness (Sintel): 3.17

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