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Marvel 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 page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-23 22:40:03 UTC</tt>.<br>
: The original revision id was <tt>213446802</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]]


This page discusses some of the temperaments tempering out 225/224, the marvel comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade and mavila.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]


Since (5/4)^3 = 225/224 * 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess 5/4-5/4-9/7 versions of the augmented triad. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo  established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Considered below are tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].  


=Negri=
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Commas: 49/48, 225/224


[[POTE tuning|POTE temperament]]: 125.608
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.


Map: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
Wedgie: &lt;&lt;4 -3 2 -14 -8 13||
EDOs: 9, 10, 19, 86


==11-limit==
== Tritonic ==
Commas: 45/44, 49/48, 56/55
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


[[POTE tuning|POTE temperament]]: 126.474
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.  


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
[[Subgroup]]: 2.3.5.7
EDOs: 9, 10, 19


==13-limit==
[[Comma list]]: 225/224, 50421/50000
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE tuning|POTE generator]]: 126.431
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
[[Optimal tuning]]s:  
EDOs: 9, 10, 19
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}


=Negric=
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Commas: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
[[Badness]] (Sintel): 1.20


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
=== 11-limit ===
EDOs: 9, 19, 85
Subgroup: 2.3.5.7.11
Badness: 0.0306


==13-limit==
Comma list: 121/120, 225/224, 441/440
Commas: 33/32, 49/48, 65/64, 91/90


POTE generator: ~15/14 = 127.039
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Optimal tunings:  
EDOs: 9, 19, 85
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
Badness: 0.0202
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


=Wizard=
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
Commas: 225/224, 118098/117649


[[POTE tuning|POTE temperament]]: 383.256
Badness (Sintel): 0.782


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
==== 13-limit ====
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
Subgroup: 2.3.5.7.11.13
EDOs: 22, 50, 72, 382, 454


==11-limit==
Comma list: 105/104, 121/120, 196/195, 275/273
Commas: 225/224, 385/384, 4000/3993


[[POTE tuning|POTE temperament]]: 383.232
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Optimal tunings:  
EDOs: 22, 50, 72, 166, 310
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE tuning|POTE temperament]]: 383.389
Badness (Sintel): 0.950


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
==== 17-limit ====
EDOs: 22, 50, 72
Subgroup: 2.3.5.7.11.13.17


==17-limit==
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE tuning|POTE temperament]]: 383.381
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Optimal tunings:  
EDOs: 22, 50, 72
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


==19-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE tuning|POTE temperament]]: 383.477
Badness (Sintel): 0.973


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
==== 19-limit ====
EDOs: 22, 50, 72, 194, 460
Subgroup: 2.3.5.7.11.13.17.19


=Tritonic=
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Commas: 225/224, 50421/50000


[[POTE tuning|POTE temperament]]: 580.286
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
Optimal tunings:  
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
EDOs: 29, 31, 60, 91, 122, 213
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 385/384, 27783/27500


[[POTE tuning|POTE temperament]]: 580.389
Badness (Sintel): 1.03


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
==== 23-limit ====
EDOs: 29, 31, 60, 91, 122, 153
Subgroup: 2.3.5.7.11.13.17.19.23


=Septimin=
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
Optimal tunings:  
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
EDOs: 9, 32, 41, 91, 132
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 385/384, 2401/2376


[[POTE tuning|POTE temperament]]: 263.634
Badness (Sintel): 1.04


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
=== Tritoni ===
EDOs: 9, 32, 41, 91
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 225/224, 385/384, 27783/27500
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE tuning|POTE temperament]]: 263.700
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Optimal tunings:  
EDOs: 9, 41, 91
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


=Slender=
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Commas: 225/225, 589824/588245


[[POTE tuning|POTE temperament]]: (~49/48) 38.413
Badness (Sintel): 1.50


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
== Septimin ==
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
EDOs: 31, 94, 125


==11-limit==
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.
Commas: 225/224, 385/384, 1331/1323


[[POTE tuning|POTE temperament]]: 38.387
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
[[Comma list]]: 225/224, 84035/82944
EDOs: 31, 63, 94, 125


=Triton=
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
Commas: 225/224, 1029/1000
: mapping generators: ~2, ~12/7


[[POTE tuning|POTE temperament]]: 568.865
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 19, 135, 289


=Marvo=
[[Badness]] (Sintel): 1.38
Commas: 225/224, 78125000/78121827


[[POTE tuning|POTE temperament]]: (~27/20) 516.694
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Comma list: 225/224, 245/242, 385/384
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
Commas: 225/224, 243/242, 4000/3993


[[POTE tuning|POTE temperament]]: 516.699
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
EDOs: 65, 72, 281, 497</pre></div>
 
<h4>Original HTML content:</h4>
Badness (Sintel): 1.04
<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;Marvel temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:40:&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:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;a href="#Negri"&gt;Negri&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt; | &lt;a href="#Negric"&gt;Negric&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt; | &lt;a href="#Wizard"&gt;Wizard&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt; | &lt;a href="#Tritonic"&gt;Tritonic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt; | &lt;a href="#Septimin"&gt;Septimin&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt; | &lt;a href="#Slender"&gt;Slender&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt; | &lt;a href="#Triton"&gt;Triton&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt; | &lt;a href="#Marvo"&gt;Marvo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;br /&gt;
=== 13-limit ===
This page discusses some of the temperaments tempering out 225/224, the marvel comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade and mavila.&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Since (5/4)^3 = 225/224 * 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess 5/4-5/4-9/7 versions of the augmented triad. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo  established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.&lt;br /&gt;
Comma list: 105/104, 144/143, 196/195, 245/242
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Negri"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Negri&lt;/h1&gt;
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
Commas: 49/48, 225/224&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 125.608&lt;br /&gt;
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
Map: [&amp;lt;1 2 2 3|, &amp;lt;0 -4 3 -2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;4 -3 2 -14 -8 13||&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
EDOs: 9, 10, 19, 86&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.955
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Negri-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 45/44, 49/48, 56/55&lt;br /&gt;
== Merman ==
&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 126.474&lt;br /&gt;
 
&lt;br /&gt;
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
Map: [&amp;lt;1 2 2 3 4|, &amp;lt;0 -4 3 -2 -5|]&lt;br /&gt;
 
EDOs: 9, 10, 19&lt;br /&gt;
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Negri-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
[[Subgroup]]: 2.3.5.7
Commas: 45/44, 49/48, 56/55, 78/77&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 225/224, 2500000/2470629
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 126.431&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
Map: [&amp;lt;1 2 2 3 4 4|, &amp;lt;0 -4 3 -2 -5 -3|]&lt;br /&gt;
: mapping generators: ~2, ~10/7
EDOs: 9, 10, 19&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Negric"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Negric&lt;/h1&gt;
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
Commas: 33/32, 49/48, 77/75&lt;br /&gt;
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
POTE generator: ~15/14 = 127.039&lt;br /&gt;
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 3 3|, &amp;lt;0 -4 3 -2 4|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
EDOs: 9, 19, 85&lt;br /&gt;
 
Badness: 0.0306&lt;br /&gt;
[[Badness]] (Sintel): 1.39
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Negric-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h2&gt;
=== 11-limit ===
Commas: 33/32, 49/48, 65/64, 91/90&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
POTE generator: ~15/14 = 127.039&lt;br /&gt;
Comma list: 225/224, 441/440, 1344/1331
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 3 3 4|, &amp;lt;0 -4 3 -2 4 -3|]&lt;br /&gt;
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
EDOs: 9, 19, 85&lt;br /&gt;
 
Badness: 0.0202&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Wizard"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Wizard&lt;/h1&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
Commas: 225/224, 118098/117649&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.256&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.20
Map: [&amp;lt;2 1 5 2|, &amp;lt;0 6 -1 10|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;12 -2 20 -31 -2 52||&lt;br /&gt;
==== 13-limit ====
EDOs: 22, 50, 72, 382, 454&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Wizard-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
Comma list: 144/143, 225/224, 364/363, 441/440
Commas: 225/224, 385/384, 4000/3993&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.232&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;2 1 5 2 8|, &amp;lt;0 6 -1 10 -3|]&lt;br /&gt;
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
EDOs: 22, 50, 72, 166, 310&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Wizard-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
Commas: 225/224, 351/350, 364/363, 385/384&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.14
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.389&lt;br /&gt;
 
&lt;br /&gt;
=== Mermaid ===
Map: [&amp;lt;2 1 5 2 8 11|, &amp;lt;0 6 -1 10 -3 -10|]&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 22, 50, 72&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 225/224, 385/384, 532400/531441
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Wizard-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;17-limit&lt;/h2&gt;
 
Commas: 221/220, 273/272, 289/288, 351/350, 375/374&lt;br /&gt;
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.381&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
Map: [&amp;lt;2 1 5 2 8 11 6|, &amp;lt;0 6 -1 10 -3 -10 6|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
EDOs: 22, 50, 72&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Wizard-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;19-limit&lt;/h2&gt;
 
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342&lt;br /&gt;
Badness (Sintel): 1.46
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.477&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;2 1 5 2 8 11 6 2|, &amp;lt;0 6 -1 10 -3 -10 6 18|]&lt;br /&gt;
 
EDOs: 22, 50, 72, 194, 460&lt;br /&gt;
Comma list: 225/224, 325/324, 385/384, 10648/10647
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Tritonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Tritonic&lt;/h1&gt;
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
Commas: 225/224, 50421/50000&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 580.286&lt;br /&gt;
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
Map: [&amp;lt;1 4 -3 -3|, &amp;lt;0 -5 11 12|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;5 -11 -12 -29 -33 3||&lt;br /&gt;
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
EDOs: 29, 31, 60, 91, 122, 213&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.47
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Tritonic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
 
Commas: 225/224, 385/384, 27783/27500&lt;br /&gt;
== Slender ==
&lt;br /&gt;
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 580.389&lt;br /&gt;
 
&lt;br /&gt;
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
Map: [&amp;lt;1 4 -3 -3 17|, &amp;lt;0 -5 11 12 -28|]&lt;br /&gt;
 
EDOs: 29, 31, 60, 91, 122, 153&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Septimin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Septimin&lt;/h1&gt;
[[Comma list]]: 225/224, 589824/588245
Commas: 225/224, 84035/82944&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 263.632&lt;br /&gt;
: mapping generators: ~2, ~49/48
&lt;br /&gt;
 
Map: [&amp;lt;1 4 1 5|, &amp;lt;0 -11 6 -10|]&lt;br /&gt;
[[Optimal tuning]]s:  
Wedgie: &amp;lt;&amp;lt;11 -6 10 -35 -15 40||&lt;br /&gt;
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
EDOs: 9, 32, 41, 91, 132&lt;br /&gt;
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Septimin-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;11-limit&lt;/h2&gt;
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}
Commas: 225/224, 385/384, 2401/2376&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 263.634&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.44
Map: [&amp;lt;1 4 1 5 5|, &amp;lt;0 -11 6 -10 -7|]&lt;br /&gt;
 
EDOs: 9, 32, 41, 91&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Septimin-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;13-limit&lt;/h2&gt;
 
Commas: 105/104, 144/143, 196/195, 245/242&lt;br /&gt;
Comma list: 225/224, 385/384, 1331/1323
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 263.700&lt;br /&gt;
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 4 1 5 5 7|, &amp;lt;0 -11 6 -10 -7 -15|]&lt;br /&gt;
Optimal tunings:  
EDOs: 9, 41, 91&lt;br /&gt;
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Slender"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Slender&lt;/h1&gt;
 
Commas: 225/225, 589824/588245&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: (~49/48) 38.413&lt;br /&gt;
Badness (Sintel): 0.838
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 3|, &amp;lt;0 -13 10 -6|]&lt;br /&gt;
=== 13-limit ===
Wedgie: &amp;lt;&amp;lt;13 -10 6 -46 -27 42||&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 31, 94, 125&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 225/224, 275/273, 385/384, 1331/1323
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Slender-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;11-limit&lt;/h2&gt;
 
Commas: 225/224, 385/384, 1331/1323&lt;br /&gt;
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 38.387&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
Map: [&amp;lt;1 2 2 3 4|, &amp;lt;0 -13 10 -6 -17|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
EDOs: 31, 63, 94, 125&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Triton"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Triton&lt;/h1&gt;
 
Commas: 225/224, 1029/1000&lt;br /&gt;
Badness (Sintel): 1.07
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 568.865&lt;br /&gt;
== Triton ==
&lt;br /&gt;
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
Map: [&amp;lt;1 0 6 7|, &amp;lt;0 3 -7 -8|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;3 -7 -8 -18 -21 1||&lt;br /&gt;
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.  
EDOs: 19, 135, 289&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;a name="Marvo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Marvo&lt;/h1&gt;
 
Commas: 225/224, 78125000/78121827&lt;br /&gt;
[[Comma list]]: 225/224, 1029/1000
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: (~27/20) 516.694&lt;br /&gt;
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
&lt;br /&gt;
: mapping generators: ~2, ~10/7
Map: [&amp;lt;1 5 12 29|, &amp;lt;0 -6 -17 -46|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;6, 17, 46, 13, 56 59||&lt;br /&gt;
[[Optimal tuning]]s:
EDOs: 65, 72&lt;br /&gt;
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Marvo-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;11-limit&lt;/h2&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
Commas: 225/224, 243/242, 4000/3993&lt;br /&gt;
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 516.699&lt;br /&gt;
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
&lt;br /&gt;
 
Map: [&amp;lt;1 5 12 29 12|, &amp;lt;0 -6 -17 -46 -15|]&lt;br /&gt;
[[Badness]] (Sintel): 1.50
EDOs: 65, 72, 281, 497&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 56/55, 1029/1000
 
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
 
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}
 
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
 
[[Badness]] (Sintel): 2.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4000/3993
 
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}
 
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.958
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 441/440
 
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.887
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
 
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
 
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.760
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
 
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
 
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.895
 
== Decic ==
{{Main| Decic }}
 
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].
 
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 16807/16384
 
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3
 
[[Optimal tuning]]s:  
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}
 
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
 
[[Badness]] (Sintel): 2.26
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 3087/3025
 
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}
 
Optimal tunings:  
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 2.11
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 2200/2197
 
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
 
Optimal tunings:  
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 1.52
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
 
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
 
Optimal tunings:  
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})
 
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
 
Badness (Sintel): 1.28
 
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.
 
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 1617/1600, 2401/2376
 
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}
 
Optimal tunings:  
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})
 
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}
 
Badness (Sintel): 1.98
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 1001/1000, 1188/1183
 
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
 
Optimal tunings:
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})
 
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}
 
Badness (Sintel): 1.57
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
 
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
 
Optimal tunings:
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})
 
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}
 
Badness (Sintel): 1.33
 
=== Prodecic ===
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.
 
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 5929/5832
 
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}
 
Optimal tunings:  
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})
 
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
 
Badness (Sintel): 2.20
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 275/273, 5929/5832
 
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
 
Optimal tunings:  
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})
 
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
 
Badness (Sintel): 1.73
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
 
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
 
Optimal tunings:  
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})
 
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
 
Badness (Sintel): 1.41
 
== Enneaportent ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 40353607/40310784
 
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912
 
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}
 
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
 
[[Badness]] (Sintel): 2.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 12005/11979
 
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}
 
Optimal tunings:
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})
 
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
 
Badness (Sintel): 1.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 1716/1715
 
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
 
Optimal tunings:
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})
 
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
 
Badness (Sintel): 0.922
 
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 781250000/771895089
 
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}
 
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
 
[[Badness]] (Sintel): 2.44
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 236328125/234365481
 
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
 
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}
 
Badness (Sintel): 2.96
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 831875/830466
 
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
 
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
 
Badness (Sintel): 2.16
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
 
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
 
Badness (Sintel): 1.96
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
 
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
 
Badness (Sintel): 1.71
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
 
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
 
Optimal tunings:
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
 
Badness (Sintel): 1.57
 
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
 
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
 
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
 
Badness (Sintel): 1.50
 
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
 
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
 
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
 
Badness (Sintel): 1.53
 
=== Gracecord ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 109375/107811
 
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
 
Optimal tunings:
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
 
Badness (Sintel): 2.21
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 364/363, 441/440, 6125/6084
 
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
 
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
 
Badness (Sintel): 1.83
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
 
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
 
Badness (Sintel): 1.87
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
 
{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
 
Badness (Sintel): 1.68
 
== Alphorn ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 5764801/5668704
 
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}
 
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
 
[[Badness]] (Sintel): 3.27
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 12250/11979
 
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}
 
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}
 
Badness (Sintel): 2.43
 
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 4194304/4117715
 
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}
 
{{Optimal ET sequence|legend=1| 21, 32, 53 }}
 
[[Badness]] (Sintel): 3.57
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1310720/1294139
 
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}
 
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}
 
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
 
Badness (Sintel): 2.82
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 176/175, 640/637, 847/845
 
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
 
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}
 
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
 
Badness (Sintel): 1.88
 
=== Musneb ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 66550/64827
 
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
 
Optimal tunings:
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}
 
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}
 
Badness (Sintel): 2.89
 
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 125000000/121060821
 
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}
 
{{Optimal ET sequence|legend=1| 51, 53 }}
 
[[Badness]] (Sintel): 3.64
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 22000/21609
 
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
 
Optimal tunings:
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}
 
{{Optimal ET sequence|legend=0| 51, 53 }}
 
Badness (Sintel): 2.46
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
 
Optimal tunings:
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}
 
{{Optimal ET sequence|legend=0| 51f, 53 }}
 
Badness (Sintel): 1.96
 
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 823543/800000
 
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}
 
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
 
[[Badness]] (Sintel): 3.67
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 245/242, 1617/1600
 
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}
 
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
 
Badness (Sintel): 2.00
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 245/242, 1001/1000
 
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
 
Badness (Sintel): 1.43
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
 
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
 
Optimal tunings:
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}
 
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
 
Badness (Sintel): 1.26
 
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 9805926501/9765625000
 
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}
 
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
 
[[Badness]] (Sintel): 3.81
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 43923/43750
 
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
 
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}
 
Badness (Sintel): 1.74
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
 
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.35
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
 
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
 
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}
 
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
 
Badness (Sintel): 1.17
 
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
 
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 122880/117649
 
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}
 
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
 
[[Badness]] (Sintel): 4.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 352/343
 
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}
 
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}
 
{{Optimal ET sequence|legend=0| 22, 55d }}
 
Badness (Sintel): 2.46
 
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 5971968/5764801
 
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32
 
[[Optimal tuning]]s:
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}
 
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
 
[[Badness]] (Sintel): 4.53
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 65536/65219
 
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}
 
Optimal tunings:
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}
 
{{Optimal ET sequence|legend=0| 22, 74, 96 }}
 
Badness (Sintel): 2.26
 
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''
 
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 14495514624/13841287201
 
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715
 
[[Optimal tuning]]s:
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}
 
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
 
[[Badness]] (Sintel): 10.9
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 3840/3773, 12005/11979
 
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
Optimal tunings:
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}
 
{{Optimal ET sequence|legend=0| 21, 54, 75e }}
 
Badness (Sintel): 5.74
 
== References ==
 
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"