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Werckismic temperaments: Difference between revisions

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Naming of the history temp (reference thread #96550)
 
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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-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out]] the [[werckisma]] ([[ratio]]: 441/440, {{monzo|legend=1| -3 2 -1 2 -1 }}). For the rank-4 werckismic temperament, see [[Catalog of rank-4 temperaments #Werckismic (441/440)]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-13 03:43:30 UTC</tt>.<br>
: The original revision id was <tt>187637547</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">These are rank three temperaments tempering out 441/440, but we can began by looking at the rank four temperament.


Minimax tuning
Temperaments discussed elsewhere are:
[|1 0 0 0 0&gt;, |1/2 2/3 1/6 -1/3 1/6&gt;, |0 0 1 0 0&gt;,
* ''[[Festival]]'' (+45/44 or 50/49) → [[Jubilismic family #Festival|Jubilismic family]]
|1 -2/3 1/3 1/3 1/3&gt;, |0 0 0 0 1&gt;]
* ''[[Vulcan]]'' (+56/55 or 64/63) → [[Archytas family #Vulcan|Archytas family]]
Eigenmonzos: 2, 5, 11, 9/7
* ''[[Euterpe]]'' (+81/80 or 99/98) → [[Didymus rank-3 family #Euterpe|Didymus rank-3 family]]
* ''[[Nakika]]'' (+100/99 or 245/242) → [[Octagar family #Nakika|Octagar family]]
* ''[[Aphrodite]]'' (+121/120) → [[Biyatismic clan #Aphrodite|Biyatismic clan]]
* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[Prodigy]]'' (+225/224) → [[Marvel family #Prodigy|Marvel family]]
* [[Jove]] (+243/242) → [[Breed family #Jove|Breed family]]
* ''[[Sensawer]]'' (+245/243) → [[Sensamagic family #Sensawer|Sensamagic family]]
* [[Portent]] (+385/384) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Demeter]]'' (+686/675) → [[Sengic family #Demeter|Sengic family]]
* ''[[Pele]]'' (+896/891) → [[Hemifamity family #Pele|Hemifamity family]]
* ''[[Tiberius]]'' (+2048/2025) → [[Diaschismic rank-3 family #Tiberius|Diaschismic rank-3 family]]
* ''[[Belobog]]'' (+3136/3125) → [[Hemimean family #Belobog|Hemimean family]]
* ''[[Landy]]'' (+4375/4356) → [[Landscape family #Landy|Landscape family]]
* ''[[Turan]]'' (+10976/10935) → [[Hemimage family #Turan|Hemimage family]]
* ''[[Snape]]'' (+19683/19600) → [[Cataharry family #Snape|Cataharry family]]


Map: [&lt;1 0 0 0 -3|, &lt;0 1 0 0 2|, &lt;0 0 1 0 -1|, &lt;0 0 0 1 2|]
Considered below are history, varuna, and trimyna.


==Babar==
== History ==
Commas: 441/400, 4000/3993
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #History]].''


Minimax tuning
History, named by [[Graham Breed]] some time before 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96550.html#96556 Yahoo! Tuning Group | ''Madagascar<nowiki>[</nowiki>19<nowiki>]</nowiki>'']</ref>, splits the perfect fourth into six, each for [[21/20]][[~]][[22/21]]. A recommendable tuning is given by [[289edo]].
[|1 0 0 0 0&gt;, |4/17 12/17 0 -6/17 6/17&gt;,
|-27/17 4/17 0 -2/17 19/17&gt;,  
|8/17 -10/17 0 5/17 12/17&gt;, |0 0 0 0 1&gt;]
Eigenmonzos: 2, 11, 9/7


Map: [&lt;1 2 0 0 1|, &lt;0 -6 0 7 2|, &lt;0 0 1 1 1|]
[[Subgroup]]: 2.3.5.7.11
Generators: 2, 21/20, 5


===13-limit===
[[Comma list]]: 441/440, 4000/3993
Commas: 364/363, 441/440, 1001/1000


Minimax tuning
{{Mapping|legend=1| 1 2 0 0 1 | 0 -6 0 7 2 | 0 0 1 1 1 }}
[|1 0 0 0 0 0&gt;, |5/4 0 0 -3/8 0 3/8&gt;, |0 0 1 0 0 0&gt;,
: mapping generators: ~2, ~21/20, ~5
|7/8 0 1 7/16 0 -7/16&gt;, |5/4 0 1 1/8 0 -1/8&gt;,
|7/8 0 1 -9/16 0 9/16&gt;]
Eigenmonzos: 2, 5, 13/7


Map: [&lt;1 2 0 0 1 2|, &lt;0 -6 0 7 2 -9|, &lt;0 0 1 1 1 1|]
[[Optimal tuning]]s:  
Generators: 2, 21/20, 5</pre></div>
* [[WE]]: ~2 = 1200.0906{{c}}, ~21/20 = 83.0793{{c}}, ~5/4 = 386.1759{{c}}
<h4>Original HTML content:</h4>
: [[error map]]: {{val| +0.091 -0.250 +0.043 -0.914 +1.288 }}
<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;Werckismic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These are rank three temperaments tempering out 441/440, but we can began by looking at the rank four temperament.&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0654{{c}}, ~5/4 = 386.2710{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -0.348 -0.043 -1.097 +1.084 }}
Minimax tuning&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |1/2 2/3 1/6 -1/3 1/6&amp;gt;, |0 0 1 0 0&amp;gt;, &lt;br /&gt;
[[Minimax tuning]]:
|1 -2/3 1/3 1/3 1/3&amp;gt;, |0 0 0 0 1&amp;gt;]&lt;br /&gt;
* [[11-odd-limit]]
Eigenmonzos: 2, 5, 11, 9/7&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/17 12/17 0 -6/17 6/17 }}, {{monzo| -27/17 4/17 0 -2/17 19/17 }}, {{monzo| 8/17 -10/17 0 5/17 12/17 }}, {{monzo| 0 0 0 0 1 }}]
&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11
Map: [&amp;lt;1 0 0 0 -3|, &amp;lt;0 1 0 0 2|, &amp;lt;0 0 1 0 -1|, &amp;lt;0 0 0 1 2|]&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491 }}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Babar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Babar&lt;/h2&gt;
 
Commas: 441/400, 4000/3993&lt;br /&gt;
[[Badness]] (Sintel): 0.725
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
=== 13-limit ===
[|1 0 0 0 0&amp;gt;, |4/17 12/17 0 -6/17 6/17&amp;gt;, &lt;br /&gt;
Subgroup: 2.3.5.7.11.13
|-27/17 4/17 0 -2/17 19/17&amp;gt;, &lt;br /&gt;
 
|8/17 -10/17 0 5/17 12/17&amp;gt;, |0 0 0 0 1&amp;gt;]&lt;br /&gt;
Comma list: 364/363, 441/440, 676/675
Eigenmonzos: 2, 11, 9/7&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 0 0 1 2 | 0 -6 0 7 2 -9 | 0 0 1 1 1 1 }}
Map: [&amp;lt;1 2 0 0 1|, &amp;lt;0 -6 0 7 2|, &amp;lt;0 0 1 1 1|]&lt;br /&gt;
 
Generators: 2, 21/20, 5&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}, ~5/4 = 386.6582{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Babar-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit&lt;/h3&gt;
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}, ~5/4 = 386.6802{{c}}
Commas: 364/363, 441/440, 1001/1000&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:
Minimax tuning&lt;br /&gt;
* 13-odd-limit
[|1 0 0 0 0 0&amp;gt;, |5/4 0 0 -3/8 0 3/8&amp;gt;, |0 0 1 0 0 0&amp;gt;, &lt;br /&gt;
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 0 0 -3/8 0 3/8 }}, {{monzo| 0 0 1 0 0 0 }}, {{monzo| 7/8 0 1 7/16 0 -7/16 }}, {{monzo| 5/4 0 1 1/8 0 -1/8 }}, {{monzo| 7/8 0 1 -9/16 0 9/16 }}]
|7/8 0 1 7/16 0 -7/16&amp;gt;, |5/4 0 1 1/8 0 -1/8&amp;gt;, &lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.5.13/7
|7/8 0 1 -9/16 0 9/16&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 5, 13/7&lt;br /&gt;
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 0 0 1 2|, &amp;lt;0 -6 0 7 2 -9|, &amp;lt;0 0 1 1 1 1|]&lt;br /&gt;
Badness (Sintel): 0.505
Generators: 2, 21/20, 5&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Complexity spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
 
== Varuna ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Varuna]].''
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 8019/8000
 
{{Mapping|legend=1| 2 0 0 9 12 | 0 1 0 -4 -6 | 0 0 1 2 3 }}
: mapping generators: ~99/70, ~3, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.1002{{c}}, ~3/2 = 701.3812{{c}}, ~5/4 = 386.3126{{c}}
: [[error map]]: {{val| +0.200 -0.373 +0.400 -0.823 +0.536 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.3457{{c}}, ~5/4 = 386.5123{{c}}
: error map: {{val| 0.000 -0.609 +0.199 -1.184 +0.144 }}
 
{{Optimal ET sequence|legend=1| 12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320 }}
 
[[Badness]] (Sintel): 0.503
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 441/440
 
Mapping: {{mapping| 2 0 0 9 12 11 | 0 1 0 -4 -6 -7 | 0 0 1 2 3 4 }}
 
Optimal tunings:  
* WE: ~55/39 = 600.0052{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 387.1861{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.4418{{c}}, ~5/4 = 387.1949{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 14cf, 26, 46, 58, 72, 130 }}
 
Badness (Sintel): 0.595
 
== Trimyna ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Trimyna]].''
 
Trimyna can be described as the {{nowrap| 27e & 29 & 31 }} temperament. All three equal temperaments have a fairly accurate tritone for ~10/7, and trimyna takes it as one of its generators, five of which get us to the [[6/1|6th]] [[harmonic]]. The name ''trimyna'' is a contraction of ''[[tritonic]]'' and ''[[myna]]''.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 3388/3375
 
{{Mapping|legend=1| 1 -1 0 1 -3 | 0 5 0 -1 8 | 0 0 1 1 1 }}
: mapping generators: ~2, ~10/7, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0909{{c}}, ~10/7 = 620.4267{{c}}, ~5/4 = 387.4364{{c}}
: [[error map]]: {{val| +0.091 +0.088 +1.304 -1.544 -0.559 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3901{{c}}, ~5/4 = 387.5065{{c}}
: error map: {{val| 0.000 -0.004 +1.193 -1.710 -0.691 }}
 
{{Optimal ET sequence|legend=1| 27e, 31, 58, 87, 118, 294, 412d }}
 
[[Badness]] (Sintel): 0.711
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 1001/1000
 
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 5 0 -1 8 -7 | 0 0 1 1 1 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.6799{{c}}, ~10/7 = 620.4057{{c}}, ~5/4 = 387.7593{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 620.5510{{c}}, ~5/4 = 387.5132{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }}
 
Badness (Sintel): 0.761
 
==== Trimy ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 676/675, 847/845
 
Mapping: {{mapping| 1 -6 0 2 -11 -10 | 0 10 0 -2 16 15 | 0 0 1 1 1 1 }}
: mapping generators: ~2, ~22/13, ~5
 
Optimal tunings:  
* WE: ~2 = 1200.0924{{c}}, ~22/13 = 910.2591{{c}}, ~5/4 = 387.4264{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 910.1939{{c}}, ~5/4 = 387.4946{{c}}
 
{{Optimal ET sequence|legend=0| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }}
 
Badness (Sintel): 1.38
 
== Seminaiad ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Seminaiad]].''
 
Named by [[User:Xenllium|Xenllium]] in 2026, seminaiad is a weak extension of [[No-threes subgroup temperaments #No-threes naiad|no-threes naiad]]. It splits ~13/5 in two to represent ~21/13.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 1127357/1125000
 
{{Mapping|legend=1| 1 5 0 -2 3 | 0 -11 0 8 -6 | 0 0 1 1 1 }}
: mapping generators: ~2, ~150/121, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.203{{c}}, ~150/121 = 372.665{{c}}, ~5/4 = 386.499{{c}}
: [[error map]]: {{val| +0.203 -0.257 +0.592 -1.004 +0.206 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~150/121 = 372.591{{c}}, ~5/4 = 386.702{{c}}
: error map: {{val| 0.000 -0.453 +0.388 -1.399 -0.161 }}
 
{{Optimal ET sequence|legend=1| 16, 29, 45e, 58, 87, 103, 145, 161, 190 }}
 
[[Badness]] (Sintel): 2.582
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 847/845, 1001/1000
 
Mapping: {{mapping| 1 5 0 -2 3 2 | 0 -11 0 8 -6 -2 | 0 0 1 1 1 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.161{{c}}, ~26/21 = 372.652{{c}}, ~5/4 = 386.381{{c}}
* CWE: ~2 = 1200.000{{c}}, ~26/21 = 372.593{{c}}, ~5/4 = 386.546{{c}}
 
{{Optimal ET sequence|legend=0| 16, 29, 45ef, 58, 87, 103, 145, 161, 190 }}
 
Badness (Sintel): 0.944
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 170/169, 221/220, 441/440, 847/845
 
Mapping: {{mapping| 1 5 0 -2 3 2 3 | 0 -11 0 8 -6 -2 -4 | 0 0 1 1 1 1 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.375{{c}}, ~21/17 = 372.711{{c}}, ~5/4 = 386.747{{c}}
* CWE: ~2 = 1200.000{{c}}, ~21/17 = 372.573{{c}}, ~5/4 = 387.138{{c}}
 
{{Optimal ET sequence|legend=0| 16, 29g, 45efg, 58, 87g }}
 
Badness (Sintel): 1.748
 
==== Seminaiadia ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 441/440, 833/832, 847/845, 1001/1000
 
Mapping: {{mapping| 1 5 0 -2 3 2 12 | 0 -11 0 8 -6 -2 -18 | 0 0 1 1 1 1 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.925{{c}}, ~26/21 = 372.566{{c}}, ~5/4 = 387.117{{c}}
* CWE: ~2 = 1200.000{{c}}, ~26/21 = 372.594{{c}}, ~5/4 = 387.080{{c}}
 
{{Optimal ET sequence|legend=0| 16, 29, 45ef, 58, 87, 103, 145, 161, 190 }}
 
Badness (Sintel): 1.284
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Werckismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Varuna]]
[[Category:Trimyna]]
Retrieved from "https://en.xen.wiki/w/Werckismic_temperaments"