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Lehmerismic 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-3 temperament|rank-3]] [[regular temperament|temperaments]] that [[temper out]] the lehmerisma, [[3025/3024]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-20 20:30:02 UTC</tt>.<br>
: The original revision id was <tt>277502570</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|flat]]


These are rank three temperaments tempering out 3025/3024, but we can began by looking at the rank four temperament.
Temperaments discussed elsewhere are:
* ''[[Oxpecker]]'' (+121/120 or 126/125) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Manwe]]'' (+176/175) → [[Valinorsmic clan #Manwe|Valinorsmic clan]]
* ''[[Spectacle]]'' (+225/224 or 243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Portent]]'' (+385/384 or 441/440) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Indra]]'' (+540/539) → [[Swetismic temperaments #Indra|Swetismic temperaments]]
* ''[[Tolerant]]'' (+896/891) → [[Pentacircle clan #Tolerant|Pentacircle clan]]
* ''[[Freya]]'' (+2401/2400) → [[Breed family #Freya|Breed family]]
* ''[[Triglav]]'' (+3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
* ''[[Thor]]'' (+4375/4374) → [[Ragismic family #Thor|Ragismic family]]
* ''[[Kapo]]'' (+5120/5103) → [[Hemifamity family #Kapo|Hemifamity family]]
* ''[[Tyr]]'' (+102487/102400) → [[Landscape family #Tyr|Landscape family]]


Comma: 3025/3024
Considered below are skadi, ganesha, hanuman, lux, and galaxy, in the order of increasing [[badness]]. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].


Map: [&lt;1 0 0 0 2|, &lt;0 1 0 1 2|, &lt;0 0 1 0 -1|, &lt;0 0 0 2 1|]
== Skadi ==
Generators: 2, 3, 5, 55/36
[[Subgroup]]: 2.3.5.7.11
EDOs: 5, 10, 15, 26, 31, 41, 72, 118, 224, 270, 342, 612, 836, 1106, 1448, 2554


=Tyr=
[[Comma list]]: 3025/3024, 703125/702464
Commas: 3025/3024, 102487/102400


Map: [&lt;3 0 0 4 8|, &lt;0 2 0 -4 1|, &lt;0 0 1 2 0|]
{{Mapping|legend=1| 1 0 -1 -6 0 | 0 1 1 3 2 | 0 0 6 14 1 }}
Generators: 63/50, 400/231, 5
: mapping generators: ~2, ~3, ~11/9
EDOs: 15, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566
Badness: 0.000259


=Skadi=
[[Optimal tuning]]s:
Commas: 3025/3024, 703125/702464
* [[WE]]: ~2 = 1200.0399{{c}}, ~3/2 = 701.9113{{c}}, ~11/9 = 347.3783{{c}}
: [[error map]]: {{val| +0.040 -0.004 -0.132 +0.085 -0.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9144{{c}}, ~11/9 = 347.3654{{c}}
: error map: {{val| 0.000 -0.041 -0.207 +0.033 -0.124 }}


Map: [&lt;1 0 5 8 1|, &lt;0 1 1 3 2|, &lt;0 0 -6 -14 -1|]
{{Optimal ET sequence|legend=1| 24d, 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}
Generators: 2, 3, 18/11
EDOs: 31, 97, 121, 152, 311, 342, 836, 1178, 2014
Badness: 0.000349


=Ganesha=
[[Badness]] (Sintel): 0.419
Commas: 3025/3024, 5632/5625


Map: [&lt;1 0 0 -22 -9|, &lt;0 1 0 1 2|, &lt;0 0 1 10 4|]
== Ganesha ==
Generators: 2, 3, 5
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Quasiorwellismic]].''
EDOs: 31, 87, 118, 152, 239, 270, 962, 1232, 1502
Badness: 0.000390


=Hanuman=
[[Subgroup]]: 2.3.5.7.11
Commas: 4000/3993, 3025/3024


Map: [&lt;1 2 0 -8 1|, &lt;0 -3 0 11 1|, &lt;0 0 1 4 1|]
[[Comma list]]: 3025/3024, 5632/5625
Generators: 2, 11/10, 5
EDOs: 15, 57, 72, 152, 224, 311, 463, 535, 998, 1533ce, 2307ce, 2531cde, 2842bce
Badness: 0.000500


==13-limit==
{{Mapping|legend=1| 1 0 0 -22 -9 | 0 1 0 1 2 | 0 0 1 10 4 }}
Commas: 625/624, 1575/1573, 2080/2079


Map: [&lt;1 2 0 -8 1 -6|, &lt;0 -3 0 11 1 3|, &lt;0 0 1 4 1 4|]
[[Optimal tuning]]s:  
Generators: 2, 11/10, 5
* [[WE]]: ~2 = 1199.9176{{c}}, ~3/2 = 702.1769{{c}}, ~5/4 = 386.6642{{c}}
EDOs: 15, 72, 87, 137, 159, 224, 311, 535, 918c, 1142c, 1453c
: [[error map]]: {{val| -0.082 +0.139 +0.186 +0.075 -0.390 }}
Badness: 0.000653</pre></div>
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1836{{c}}, ~5/4 = 386.6860{{c}}
<h4>Original HTML content:</h4>
: error map: {{val| 0.000 +0.229 +0.372 +0.218 -0.207 }}
<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;Lehmerismic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Tyr"&gt;Tyr&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Skadi"&gt;Skadi&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#Ganesha"&gt;Ganesha&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Hanuman"&gt;Hanuman&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}
These are rank three temperaments tempering out 3025/3024, but we can began by looking at the rank four temperament.&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 0.469
Comma: 3025/3024&lt;br /&gt;
 
&lt;br /&gt;
== Hanuman ==
Map: [&amp;lt;1 0 0 0 2|, &amp;lt;0 1 0 1 2|, &amp;lt;0 0 1 0 -1|, &amp;lt;0 0 0 2 1|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Generators: 2, 3, 5, 55/36&lt;br /&gt;
 
EDOs: 5, 10, 15, 26, 31, 41, 72, 118, 224, 270, 342, 612, 836, 1106, 1448, 2554&lt;br /&gt;
[[Comma list]]: 3025/3024, 4000/3993
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Tyr"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Tyr&lt;/h1&gt;
{{Mapping|legend=1| 1 2 0 -8 1 | 0 -3 0 11 1 | 0 0 1 4 1 }}
Commas: 3025/3024, 102487/102400&lt;br /&gt;
: mapping generators: ~2, ~11/10, ~5
&lt;br /&gt;
 
Map: [&amp;lt;3 0 0 4 8|, &amp;lt;0 2 0 -4 1|, &amp;lt;0 0 1 2 0|]&lt;br /&gt;
[[Optimal tuning]]s:  
Generators: 63/50, 400/231, 5&lt;br /&gt;
* [[WE]]: ~2 = 1199.9539{{c}}, ~11/10 = 165.9317{{c}}, ~5/4 = 385.9220{{c}}
EDOs: 15, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566&lt;br /&gt;
: [[error map]]: {{val| -0.046 +0.158 -0.484 +0.111 +0.398 }}
Badness: 0.000259&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/10 = 165.9501{{c}}, ~5/4 = 385.8882{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +0.195 -0.426 +0.178 +0.520 }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Skadi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Skadi&lt;/h1&gt;
 
Commas: 3025/3024, 703125/702464&lt;br /&gt;
{{Optimal ET sequence|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 5 8 1|, &amp;lt;0 1 1 3 2|, &amp;lt;0 0 -6 -14 -1|]&lt;br /&gt;
[[Badness]] (Sintel): 0.600
Generators: 2, 3, 18/11&lt;br /&gt;
 
EDOs: 31, 97, 121, 152, 311, 342, 836, 1178, 2014&lt;br /&gt;
=== 13-limit ===
Badness: 0.000349&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Ganesha"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Ganesha&lt;/h1&gt;
Comma list: 625/624, 1575/1573, 2080/2079
Commas: 3025/3024, 5632/5625&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}
Map: [&amp;lt;1 0 0 -22 -9|, &amp;lt;0 1 0 1 2|, &amp;lt;0 0 1 10 4|]&lt;br /&gt;
 
Generators: 2, 3, 5&lt;br /&gt;
Optimal tunings:
EDOs: 31, 87, 118, 152, 239, 270, 962, 1232, 1502&lt;br /&gt;
* WE: ~2 = 1200.0188{{c}}, ~11/10 = 166.0060{{c}}, ~5/4 = 385.6865{{c}}
Badness: 0.000390&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0007{{c}}, ~5/4 = 385.6918{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Hanuman"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Hanuman&lt;/h1&gt;
{{Optimal ET sequence|legend=0| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
Commas: 4000/3993, 3025/3024&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.610
Map: [&amp;lt;1 2 0 -8 1|, &amp;lt;0 -3 0 11 1|, &amp;lt;0 0 1 4 1|]&lt;br /&gt;
 
Generators: 2, 11/10, 5&lt;br /&gt;
== Lux ==
EDOs: 15, 57, 72, 152, 224, 311, 463, 535, 998, 1533ce, 2307ce, 2531cde, 2842bce&lt;br /&gt;
The last generator of lux, represented by [[55/48]], exceeds [[8/7]] by [[385/384]], which is equated with a number of important [[superparticular ratio]]s in the [[13-limit]]: [[325/324]], [[352/351]], [[364/363]], and [[441/440]].
Badness: 0.000500&lt;br /&gt;
 
&lt;br /&gt;
This ultra-efficient full 13-limit temperament was first considered by [[Flora Canou]] in 2021. After a few attempts to come up with a memorable name, ''lux'', suggested by [[Godtone]], who associated certain intervals of 13 with light, was adopted.
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Hanuman-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h2&gt;
 
Commas: 625/624, 1575/1573, 2080/2079&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 2 0 -8 1 -6|, &amp;lt;0 -3 0 11 1 3|, &amp;lt;0 0 1 4 1 4|]&lt;br /&gt;
[[Comma list]]: 3025/3024, 131072/130977
Generators: 2, 11/10, 5&lt;br /&gt;
 
EDOs: 15, 72, 87, 137, 159, 224, 311, 535, 918c, 1142c, 1453c&lt;br /&gt;
{{Mapping|legend=1| 1 0 -5 4 9 | 0 1 4 -1 -3 | 0 0 5 2 -4 }}
Badness: 0.000653&lt;/body&gt;&lt;/html&gt;</pre></div>
: mapping generators: ~2, ~3, ~55/48
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9605{{c}}, ~3/2 = 702.0886{{c}}, ~55/48 = 235.5706{{c}}
: [[error map]]: {{val| -0.039 +0.094 -0.067 +0.108 -0.103 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1120{{c}}, ~55/48 = 235.5763{{c}}
: error map: {{val| 0.000 +0.157 +0.016 +0.215 +0.041 }}
 
{{Optimal ET sequence|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}
 
[[Badness]] (Sintel): 0.611
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}
 
Optimal tunings:
* WE: ~2 = 1199.9727{{c}}, ~3/2 = 702.0946{{c}}, ~55/48 = 235.5597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1109{{c}}, ~55/48 = 235.5649{{c}}
 
{{Optimal ET sequence|legend=0| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
 
Badness (Sintel): 0.337
 
== Galaxy ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 20614528/20588575
 
{{Mapping|legend=1| 1 0 -4 10 11 | 0 1 3 -3 -3 | 0 0 9 -14 -16 }}
: mapping generators: ~2, ~3, ~1936/1715
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9912{{c}}, ~3/2 = 701.9805{{c}}, ~1936/1715 = 208.9296{{c}}
: [[error map]]: {{val| -0.009 +0.017 +0.003 +0.156 -0.204 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9856{{c}}, ~1936/1715 = 208.9333{{c}}
: error map: {{val| 0.000 +0.023 +0.020 +0.174 -0.184 }}
 
{{Optimal ET sequence|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}
 
[[Badness]] (Sintel): 1.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 3025/3024, 4225/4224
 
Mapping: {{mapping| 1 0 -4 10 11 13 | 0 1 3 -3 -3 -4 | 0 0 9 -14 -16 -17 }}
 
Optimal tunings:
* WE: ~2 = 1200.0131{{c}}, ~3/2 = 701.9445{{c}}, ~44/39 = 208.9426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9368{{c}}, ~44/39 = 208.9373{{c}}
 
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}
 
Badness (Sintel): 0.433
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 936/935, 1225/1224, 4225/4224
 
Mapping: {{mapping| 1 0 -4 10 11 13 9 | 0 1 3 -3 -3 -4 -2 | 0 0 9 -14 -16 -17 -10 }}
 
Optimal tunings:
* WE: ~2 = 1199.9094{{c}}, ~3/2 = 702.0614{{c}}, ~44/39 = 208.8807{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0462{{c}}, ~44/39 = 208.9167{{c}}
 
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}
 
Badness (Sintel): 0.888
 
[[Category:Temperament collections]]
[[Category:Lehmerismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Lehmerismic_temperaments"