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Horwell 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]] '''horwell temperaments''', which temper out the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-08 11:39:34 UTC</tt>.<br>
: The original revision id was <tt>396689132</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]]


Horwell temperaments temper out the horwell comma, |-16 1 5 1&gt; = 65625/65536.
Temperaments discussed elsewhere are
* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]
* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Worschmidt]]'' (+126/125) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds|Hemimean clan]]
* ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
* ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[Eris]]'' (+16875/16807) → [[Canopic clan #Eris|Canopic clan]]
* ''[[Soviet ferris wheel]]'' (+{{monzo| -5 -9 -5 11 }}) → [[20th-octave temperaments #Soviet ferris wheel|20th-octave temperaments]]


=Fifthplus=
Considered below are fifthplus, mutt, oquatonic, emkay, kastro, and bezique, in the order of increasing [[badness]].
Commas: 65625/65536, 420175/419904


POTE generator: ~5488/3645 = 708.774
== Fifthplus ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sesesix]].''


Map: [&lt;1 11 -3 20|, &lt;0 -23 13 -42|]
Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].
Wedgie: &lt;&lt;23 -13 42 -74 2 134||
EDOs: 22, 149, 171, 1903c, 2074c, 2245c, 2416c, 2587c, 2758c, 2929c, 3100c, 3271c, 3442c, 3613c
Badness: 0.0258


=Emkay=
[[Subgroup]]: 2.3.5.7
Commas: 65625/65536, 244140625/243045684


POTE generator: ~3125/2268 = 551.775
[[Comma list]]: 65625/65536, 420175/419904


Map: [&lt;1 14 6 -28|, &lt;0 -27 -8 67|]
{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
Wedgie: &lt;&lt;27 8 -67 -50 -182 -178||
: mapping generators: ~2, ~5488/3645
EDOs: 87, 137, 224, 311, 535
Badness: 0.1357


==11-limit==
[[Optimal tuning]]s:
Commas: 3025/3024, 4000/3993, 65625/65536
* [[WE]]: ~2 = 1200.0934{{c}}, ~5488/3645 = 708.8291{{c}}
: [[error map]]: {{val| +0.093 -0.007 -0.158 -0.059 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5488/3645 = 708.7752{{c}}
: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}


POTE generator: ~11/8 = 551.775
{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}


Map: [&lt;1 14 6 -28 3|, &lt;0 -27 -8 67 1|]
[[Badness]] (Sintel): 0.654
EDOs: 87, 137, 224, 311, 535
Badness: 0.0356


==13-limit==
== Mutt ==
Commas: 625/624, 1575/1573, 2080/2079, 2200/2197
{{Main| Mutt }}
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Mutt (5-limit)]].''


POTE generator: ~11/8 = 551.775
Mutt tempers out the [[landscape comma]] in addition to the horwell comma, and may be described as the {{nowrap| 84 & 87 }} temperament.  


Map: [&lt;1 14 6 -28 3 6|, &lt;0 -27 -8 67 1 -5|]
[[Subgroup]]: 2.3.5.7
EDOs: 87, 137, 224, 311, 535
Badness: 0.0179


</pre></div>
[[Comma list]]: 65625/65536, 250047/250000
<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;Horwell temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&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:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Fifthplus"&gt;Fifthplus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Emkay"&gt;Emkay&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
{{Mapping|legend=1| 3 -2 6 20 | 0 7 1 -12 }}
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;br /&gt;
: mapping generators: ~63/50, ~5/4
Horwell temperaments temper out the horwell comma, |-16 1 5 1&amp;gt; = 65625/65536.&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="Fifthplus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Fifthplus&lt;/h1&gt;
* [[WE]]: ~63/50 = 400.0351{{c}}, ~5/4 = 385.9974{{c}} (~126/125 = 14.0377{{c}})
Commas: 65625/65536, 420175/419904&lt;br /&gt;
: [[error map]]: {{val| +0.105 -0.043 -0.105 -0.092 }}
&lt;br /&gt;
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~5/4 = 385.9638{{c}} (~126/125 = 14.0362{{c}})
POTE generator: ~5488/3645 = 708.774&lt;br /&gt;
: error map: {{val| 0.000 -0.208 -0.350 -0.392 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 11 -3 20|, &amp;lt;0 -23 13 -42|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 84, 87, 171 }}
Wedgie: &amp;lt;&amp;lt;23 -13 42 -74 2 134||&lt;br /&gt;
 
EDOs: 22, 149, 171, 1903c, 2074c, 2245c, 2416c, 2587c, 2758c, 2929c, 3100c, 3271c, 3442c, 3613c&lt;br /&gt;
[[Badness]] (Sintel): 0.719
Badness: 0.0258&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Emkay"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Emkay&lt;/h1&gt;
Subgroup: 2.3.5.7.11
Commas: 65625/65536, 244140625/243045684&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 441/440, 4375/4356, 16384/16335
POTE generator: ~3125/2268 = 551.775&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 -2 6 20 21 | 0 7 1 -12 -11 }}
Map: [&amp;lt;1 14 6 -28|, &amp;lt;0 -27 -8 67|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;27 8 -67 -50 -182 -178||&lt;br /&gt;
Optimal tunings:  
EDOs: 87, 137, 224, 311, 535&lt;br /&gt;
* WE: ~44/35 = 399.9783{{c}}, ~5/4 = 385.9993{{c}} (~126/125 = 13.9790{{c}})
Badness: 0.1357&lt;br /&gt;
* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0208{{c}} (~126/125 = 13.9792{{c}})
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Emkay-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 84, 87, 171, 258 }}
Commas: 3025/3024, 4000/3993, 65625/65536&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.93
POTE generator: ~11/8 = 551.775&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 14 6 -28 3|, &amp;lt;0 -27 -8 67 1|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 87, 137, 224, 311, 535&lt;br /&gt;
 
Badness: 0.0356&lt;br /&gt;
Comma list: 364/363, 441/440, 625/624, 2200/2197
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Emkay-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Mapping: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}
Commas: 625/624, 1575/1573, 2080/2079, 2200/2197&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~11/8 = 551.775&lt;br /&gt;
* WE: ~44/35 = 399.9610{{c}}, ~5/4 = 385.9842{{c}} (~126/125 = 13.9768{{c}})
&lt;br /&gt;
* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0231{{c}} (~126/125 = 13.9769{{c}})
Map: [&amp;lt;1 14 6 -28 3 6|, &amp;lt;0 -27 -8 67 1 -5|]&lt;br /&gt;
 
EDOs: 87, 137, 224, 311, 535&lt;br /&gt;
{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429ef }}
Badness: 0.0179&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Badness (Sintel): 1.20
 
== Oquatonic ==
: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''
 
Oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the [[dimcomp comma]] (390625/388962). In this temperament, the [[5/4]] major third is mapped to 9\28.  
 
The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref name="petr's long post"/>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 390625/388962
 
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
: mapping generators: ~128/125, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~128/125 = 42.8570{{c}}, ~3/2 = 702.1112{{c}}
: [[error map]]: {{val| -0.004 +0.152 -0.609 +0.477 }}
* [[CWE]]: ~128/125 = 42.8571{{c}}, ~3/2 = 702.1132{{c}}
: error map: {{val| 0.000 +0.158 -0.599 +0.489 }}
 
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
 
[[Badness]] (Sintel): 2.23
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 6250/6237, 65625/65536
 
Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
 
Optimal tunings:
* WE: ~128/125 = 42.8577{{c}}, ~3/2 = 702.0275{{c}}
* CWE: ~128/125 = 42.8571{{c}}, ~3/2 = 702.0174{{c}}
 
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
 
Badness (Sintel): 1.58
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
 
Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
 
Optimal tunings:
* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}
 
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
 
Badness (Sintel): 0.908
 
== Emkay ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Emka]].''
 
[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]
 
Emkay may be described as the {{nowrap| 87 & 224 }} temperament. It tempers out the same 5-limit comma as the [[emka]] (37 & 50), but with the horwell comma (65625/65536) rather than the hemimean comma (3136/3125) tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 244140625/243045684
 
{{Mapping|legend=1| 1 -13 -2 39 | 0 27 8 -67 }}
: mapping generators: ~2, ~4536/3125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0279{{c}}, ~4536/3125 = 648.2405{{c}}
: [[error map]]: {{val| +0.028 +0.177 -0.445 +0.146 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4536/3125 = 648.2254{{c}}
: error map: {{val| 0.000 +0.133 -0.510 +0.069 }}
 
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
 
[[Badness]] (Sintel): 3.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 65625/65536
 
Mapping: {{mapping| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9958{{c}}, ~16/11 = 648.2231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2254{{c}}
 
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}
 
Badness (Sintel): 1.18
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197
 
Mapping: {{mapping| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.9694{{c}}, ~16/11 = 648.2085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2251{{c}}
 
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}
 
Badness (Sintel): 0.738
 
== Kastro ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Astro]].''
 
Kastro may be described as the {{nowrap| 109 & 118 }} temperament, named by [[Petr Pařízek]] in 2011 as a variation of ''astro''<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 117649/116640
 
{{Mapping|legend=1| 1 -26 13 -23 | 0 31 -12 29 }}
: mapping generators: ~2, ~6272/3375
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1529{{c}}, ~6272/3375 = 1067.9515{{c}}
: [[error map]]: {{val| +0.153 +0.567 +0.256 -1.749 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6272/3375 = 1067.8174{{c}}
: error map: {{val| 0.000 +0.384 -0.122 -2.122 }}
 
{{Optimal ET sequence|legend=1| 109, 118, 345d, 463d, 581dd }}
 
[[Badness]] (Sintel): 4.64
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 3388/3375, 12005/11979
 
Mapping: {{mapping| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}
 
Optimal tunings:  
* WE: ~2 = 1200.2427{{c}}, ~224/121 = 1068.0296{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/121 = 1067.8166{{c}}
 
{{Optimal ET sequence|legend=0| 109, 118, 345de, 463de, 581dde }}
 
Badness (Sintel): 1.74
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 364/363, 385/384, 3388/3375
 
Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}
 
Optimal tunings:
* WE: ~2 = 1200.4303{{c}}, ~13/7 = 1068.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1067.8267{{c}}
 
{{Optimal ET sequence|legend=0| 109, 118f, 227f }}
 
Badness (Sintel): 1.93
 
== Bezique ==
Bezique splits the octave into 32 equal parts and reaches 3/2, 8/5 and 11/8 in just one generator with the 64-tone mos. A notable edo tuning overshadowed by [[224edo]] is [[320edo]]. Bezique was named by [[Eliora]] in 2023 for the fact that the card game of bezique is played with two packs of 32 cards.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 847288609443/843308032000
 
{{Mapping|legend=1| 32 0 125 -113 | 0 1 -1 4 }}
: mapping generators: ~100352/98415, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~100352/98415 = 37.5038{{c}}, ~3/2 = 701.6058{{c}}
: [[error map]]: {{val| +0.120 -0.229 -0.071 +0.154 }}
* [[CWE]]: ~100352/98415 = 37.5000{{c}}, ~3/2 = 701.5544{{c}}
: error map: {{val| 0.000 -0.401 -0.368 -0.108 }}
 
{{Optimal ET sequence|legend=1| 96d, 224, 544, 768, 1312, 2080bc }}
 
[[Badness]] (Sintel): 6.82
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 46656/46585, 65625/65536
 
Mapping: {{mapping| 32 0 125 -113 60 | 0 1 -1 4 1 }}
 
Optimal tunings:
* WE: ~45/44 = 37.5025{{c}}, ~3/2 = 701.5912{{c}}
* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5566{{c}}
 
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768 }}
 
Badness (Sintel): 2.25
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655
 
Mapping: {{mapping| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
 
Optimal tunings:
* WE: ~45/44 = 37.5021{{c}}, ~3/2 = 701.5769{{c}}
* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5490{{c}}
 
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768, 1312 }}
 
Badness (Sintel): 1.23
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Horwell_temperaments"