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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]]s tempering out [[126/125]], the starling comma or septimal semicomma.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-16 17:41:55 UTC</tt>.<br>
: The original revision id was <tt>202527642</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">This page discusses some of the temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. 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 three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.


==Myna temperament==
Temperaments discussed in families and clans are:
In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&amp;31 temperament, or in terms of its wedgie &lt;&lt;10 9 7 -9 -17 -9||. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]


[[Comma|Commas]]: 126/125, 1728/1715
Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].


7 and 9 limit minimax
Since {{nowrap|(6/5)<sup>3</sup> {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord.
[|1 0 0 0&gt;, |0 1 0 0 &gt;, |9/10 9/10 0 0&gt;, |17/10 7/10 0 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 3


[[POTE tuning|POTE generator]]: 310.146
== Myna ==
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''


Map: [&lt;1 9 9 8|, &lt;0 -10 -9 -7|]
7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). [[11-limit]] myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]). In that sense, it is opposed to [[keemic temperaments]], in particular [[quasitemp]], where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.  
[[Generator|Generators]]: 2, 5/3
EDOs: 27, 31, 58, 89
Badness: 0.0270


===11-limit===
In terms of vanishing commas, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.
Commas: 126/125, 176/175, 243/242


[[POTE tuning|POTE generator]]: 310.144
[[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round cent values may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11- and 13-limit.


Map: [&lt;1 9 9 8 22|, &lt;0 -10 -9 -7 -25|]: 310.146
[[Subgroup]]: 2.3.5.7
EDOs: 31, 58, 89
Badness: 0.0168


==Sensi temperament==
[[Comma list]]: 126/125, 1728/1715
Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&amp;27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.


[[Comma|Commas]]: 126/125, 245/243
{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5


7-limit minimax
[[Optimal tuning]]s:
[|1 0 0 0&gt;, |1/13 0 0 7/13&gt;, |5/13 0 0 9/13&gt;, |0 0 0 1&gt;]
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
[[Eigenmonzo|Eigenmonzos]]: 2, 7
: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}
: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}


9-limit minimax
[[Minimax tuning]]:
[|1 0 0 0&gt;, |2/5 14/5 -7/5 0&gt;,
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
|4/5 18/5 -9/5 0&gt;, |3/5 26/5 -13/5 0&gt;]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
[[Eigenmonzo|Eigenmonzos]]: 2, 9/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[POTE tuning|POTE generator]]: 443.383
{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}
Algebraic generator: Calista, the [[Algebraic number|real root]] of x^7-2x^2-1, at 340.6467 cents.


Map: [&lt;1 6 8 11|, &lt;0 -7 -9 -13|]
[[Badness]] (Sintel): 0.684
[[Generator|Generators]]: 2, 14/9
EDOs: 19, 27, 46, 249, 295
Badness: 0.0256


===Sensis===
=== 11-limit ===
Commas: 56/55, 100/99, 245/243
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: 443.962
Comma list: 126/125, 176/175, 243/242


Map: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}
EDOs: 19, 27, 73, 100
Badness: 0.0287


====13-limit====
Optimal tunings:
Commas: 56/55, 78/77, 91/90, 100/99
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}


[[POTE tuning|POTE generator]]: 443.945
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}


Map: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
Badness (Sintel): 0.557
EDOs: 19, 27, 73, 100
Badness: 0.0200


==Valentine temperament==
==== 13-limit ====
Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the 31&amp;46 temperament, and [[77edo]], [[108edo]] or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as &lt;&lt;9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).
Subgroup: 2.3.5.7.11.13


Valentine is very closely related to [[Carlos Alpha]], the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in //Beauty in the Beast// suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.
Comma list: 126/125, 144/143, 176/175, 196/195


[[Comma|Commas]]: 1029/1024, 126/125
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}


[[Minimax tuning]]:
Optimal tunings:  
7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,  
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
|17/6 5/12 0 -5/12&gt;, [5/2 -1/4 0 1/4&gt;]
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,  
{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}
|47/21 10/21 0 -5/21&gt;, |20/7 -2/7 0 1/7&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 9/7


[[POTE tuning|POTE generator]]: 77.864
Badness (Sintel): 0.708


Algebraic generator: [[Algebraic number|smaller root]] of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.
==== Minah ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]
Comma list: 78/77, 91/90, 126/125, 176/175
[[Generator|Generators]]: 2, 21/20
EDOs: 15, 31, 46, 77, 185, 262
Badness: 0.0311


===11-limit===
Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}
[[Comma|Commas]]: 121/120, 126/125, 176/175


[[Minimax tuning]]:
Optimal tunings:  
[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;]
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}
[[Eigenmonzo|Eigenmonzos]]: 2, 11/7


Minimax generator: (11/7)^(1/10) = 78.249
{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}
[[POTE tuning|POTE generator]]: 77.881


Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.
Badness (Sintel): 1.14


Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]
==== Maneh ====
[[Edo|Edos]]: [[15edo|15]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[108edo|108]], [[185edo|185]]
Subgroup: 2.3.5.7.11.13
Badness: 0.0167


==Casablanca temperament==
Comma list: 66/65, 105/104, 126/125, 243/242
Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described by its wedgie, &lt;&lt;19 14 4 -22 -47 -30||, or as 31&amp;73. 74/135 or 91/166 supply good tunings for the generator, and 20 and 31 note MOS are available.


It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a [[Hexany|hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.
Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}


==Nusecond temperament==
Optimal tunings:
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;70, or in terms of its wedgie as &lt;&lt;11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[Patent val|patent vals]] for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}


[[Comma|Commas]]: 126/125, 2430/2401
{{Optimal ET sequence|legend=0| 27eff, 31 }}


7-limit minimax
Badness (Sintel): 1.23
[|1 0 0 0&gt;, |-5/13 0 11/13 0&gt;, |0 0 1 0&gt;, |-3/13 0 17/13 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 5


9-limit minimax
=== Myno ===
[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]
Subgroup: 2.3.5.7.11
[[Eigenmonzo|Eigenmonzos]]: 2, 3


[[POTE tuning|POTE generator]]: 154.579
Comma list: 99/98, 126/125, 385/384


Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]
Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}
[[Generator|Generators]]: 2, 49/45
EDOs: 7, 8, 31, 101, 132, 163
Badness: 0.0504


===11-limit===
Optimal tunings:
[[Comma|Commas]]: 99/98, 121/120, 126/125
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}


11-limit minimax
{{Optimal ET sequence|legend=0| 27, 31 }}
[|1 0 0 0 0&gt;, |19/10 11/5 0 0 -11/10&gt;,
|27/10 13/5 0 0 -13/10&gt;, |33/10 17/5 0 0 -17/10&gt;,
|19/5 12/5 0 0 -6/5&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/9


[[POTE tuning|POTE generator]]: 154.645
Badness (Sintel): 1.11
Algebraic generator: [[Algebraic number|positive root]] of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.


Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]
=== Coleto ===
[[Generator|Generators]]: 2, 11/10
Subgroup: 2.3.5.7.11
EDOs: 7, 8, 31, 101, 194
 
Badness: 0.0256</pre></div>
Comma list: 56/55, 100/99, 1728/1715
<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;Starling temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This page discusses some of the temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.&lt;br /&gt;
Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Myna temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Myna temperament&lt;/h2&gt;
Optimal tunings:  
In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&amp;amp;31 temperament, or in terms of its wedgie &amp;lt;&amp;lt;10 9 7 -9 -17 -9||. It has 6/5 as a generator, and &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; can be used as a tuning, with &lt;a class="wiki_link" href="/89edo"&gt;89edo&lt;/a&gt; being a better one, and fans of round amounts in cents may like &lt;a class="wiki_link" href="/120edo"&gt;120edo&lt;/a&gt;. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.&lt;br /&gt;
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 1728/1715&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}
7 and 9 limit minimax&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |0 1 0 0 &amp;gt;, |9/10 9/10 0 0&amp;gt;, |17/10 7/10 0 0&amp;gt;]&lt;br /&gt;
Badness (Sintel): 1.61
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 3&lt;br /&gt;
 
&lt;br /&gt;
== Nusecond ==
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 310.146&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''
&lt;br /&gt;
 
Map: [&amp;lt;1 9 9 8|, &amp;lt;0 -10 -9 -7|]&lt;br /&gt;
Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 70 }}. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 5/3&lt;br /&gt;
 
EDOs: 27, 31, 58, 89&lt;br /&gt;
[[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.
Badness: 0.0270&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Myna temperament-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h3&gt;
 
Commas: 126/125, 176/175, 243/242&lt;br /&gt;
[[Comma list]]: 126/125, 2430/2401
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 310.144&lt;br /&gt;
{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
&lt;br /&gt;
: mapping generators: ~2, ~49/27
Map: [&amp;lt;1 9 9 8 22|, &amp;lt;0 -10 -9 -7 -25|]: 310.146&lt;br /&gt;
 
EDOs: 31, 58, 89&lt;br /&gt;
[[Optimal tuning]]s:
Badness: 0.0168&lt;br /&gt;
* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Sensi temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Sensi temperament&lt;/h2&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}
Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&amp;amp;27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt; is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.&lt;br /&gt;
: error map: {{val| 0.000 -2.655 +3.768 +2.819 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 245/243&lt;br /&gt;
[[Minimax tuning]]:
&lt;br /&gt;
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
7-limit minimax&lt;br /&gt;
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
[|1 0 0 0&amp;gt;, |1/13 0 0 7/13&amp;gt;, |5/13 0 0 9/13&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7&lt;br /&gt;
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
&lt;br /&gt;
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
9-limit minimax&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
[|1 0 0 0&amp;gt;, |2/5 14/5 -7/5 0&amp;gt;, &lt;br /&gt;
 
|4/5 18/5 -9/5 0&amp;gt;, |3/5 26/5 -13/5 0&amp;gt;]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 9/5&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.28
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 443.383&lt;br /&gt;
 
Algebraic generator: Calista, the &lt;a class="wiki_link" href="/Algebraic%20number"&gt;real root&lt;/a&gt; of x^7-2x^2-1, at 340.6467 cents. &lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 6 8 11|, &amp;lt;0 -7 -9 -13|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 14/9&lt;br /&gt;
Comma list: 99/98, 121/120, 126/125
EDOs: 19, 27, 46, 249, 295&lt;br /&gt;
 
Badness: 0.0256&lt;br /&gt;
Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Sensi temperament-Sensis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Sensis&lt;/h3&gt;
Optimal tunings:
Commas: 56/55, 100/99, 245/243&lt;br /&gt;
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 443.962&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:  
Map: [&amp;lt;1 6 8 11 6|, &amp;lt;0 -7 -9 -13 -4|]&lt;br /&gt;
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
EDOs: 19, 27, 73, 100&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
Badness: 0.0287&lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.11/9
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc4"&gt;&lt;a name="x-Sensi temperament-Sensis-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h4&gt;
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
Commas: 56/55, 78/77, 91/90, 100/99&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 443.945&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.847
Map: [&amp;lt;1 6 8 11 6 10|, &amp;lt;0 -7 -9 -13 -4 -10|]&lt;br /&gt;
 
EDOs: 19, 27, 73, 100&lt;br /&gt;
=== 13-limit ===
Badness: 0.0200&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Valentine temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Valentine temperament&lt;/h2&gt;
Comma list: 66/65, 99/98, 121/120, 126/125
Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice of 7-limit tetrads&lt;/a&gt;. Valentine can also be described as the 31&amp;amp;46 temperament, and &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;, &lt;a class="wiki_link" href="/108edo"&gt;108edo&lt;/a&gt; or &lt;a class="wiki_link" href="/185edo"&gt;185edo&lt;/a&gt; make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as &amp;lt;&amp;lt;9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}
Valentine is very closely related to &lt;a class="wiki_link" href="/Carlos%20Alpha"&gt;Carlos Alpha&lt;/a&gt;, the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in &lt;em&gt;Beauty in the Beast&lt;/em&gt; suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that &amp;quot;The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before&amp;quot;, and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 1029/1024, 126/125&lt;br /&gt;
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;:&lt;br /&gt;
 
7-limit: [|1 0 0 0&amp;gt;, |5/2 3/4 0 -3/4&amp;gt;, &lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}
|17/6 5/12 0 -5/12&amp;gt;, [5/2 -1/4 0 1/4&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6&lt;br /&gt;
Badness (Sintel): 0.964
&lt;br /&gt;
 
9-limit: [|1 0 0 0&amp;gt;, |10/7 6/7 0 -3/7&amp;gt;, &lt;br /&gt;
== Oolong ==
|47/21 10/21 0 -5/21&amp;gt;, |20/7 -2/7 0 1/7&amp;gt;]&lt;br /&gt;
{{Main| Oolong }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 9/7&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 77.864&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Algebraic generator: &lt;a class="wiki_link" href="/Algebraic%20number"&gt;smaller root&lt;/a&gt; of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.&lt;br /&gt;
[[Comma list]]: 126/125, 117649/116640
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 3|, &amp;lt;0 9 5 -3|]&lt;br /&gt;
{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 21/20&lt;br /&gt;
: mapping generators: ~2, ~5/3
EDOs: 15, 31, 46, 77, 185, 262&lt;br /&gt;
 
Badness: 0.0311&lt;br /&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Valentine temperament-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h3&gt;
: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 121/120, 126/125, 176/175&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;:&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |1 0 0 -9/10 9/10&amp;gt;, &lt;br /&gt;
{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}
|2 0 0 -1/2 1/2&amp;gt;, |3 0 0 3/10 -3/10&amp;gt;, |3 0 0 -7/10 7/10&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/7&lt;br /&gt;
[[Badness]] (Sintel): 1.86
&lt;br /&gt;
 
Minimax generator: (11/7)^(1/10) = 78.249&lt;br /&gt;
=== 11-limit ===
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 77.881&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.&lt;br /&gt;
Comma list: 126/125, 176/175, 26411/26244
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 3 3|, &amp;lt;0 9 5 -3 7|]&lt;br /&gt;
Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}
&lt;a class="wiki_link" href="/Edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46&lt;/a&gt;, &lt;a class="wiki_link" href="/77edo"&gt;77&lt;/a&gt;, &lt;a class="wiki_link" href="/108edo"&gt;108&lt;/a&gt;, &lt;a class="wiki_link" href="/185edo"&gt;185&lt;/a&gt;&lt;br /&gt;
 
Badness: 0.0167&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-Casablanca temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Casablanca temperament&lt;/h2&gt;
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}
Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described by its wedgie, &amp;lt;&amp;lt;19 14 4 -22 -47 -30||, or as 31&amp;amp;73. 74/135 or 91/166 supply good tunings for the generator, and 20 and 31 note MOS are available.&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}
It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a &lt;a class="wiki_link" href="/Hexany"&gt;hexany&lt;/a&gt; and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.88
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x-Nusecond temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Nusecond temperament&lt;/h2&gt;
 
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;amp;70, or in terms of its wedgie as &amp;lt;&amp;lt;11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; can be used as a tuning, or &lt;a class="wiki_link" href="/132edo"&gt;132edo&lt;/a&gt; with a val which is the sum of the &lt;a class="wiki_link" href="/Patent%20val"&gt;patent vals&lt;/a&gt; for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 2430/2401&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 176/175, 196/195, 13013/12960
7-limit minimax&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |-5/13 0 11/13 0&amp;gt;, |0 0 1 0&amp;gt;, |-3/13 0 17/13 0&amp;gt;]&lt;br /&gt;
Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
9-limit minimax&lt;br /&gt;
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
[|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |5/11 13/11 0 0&amp;gt;, |4/11 17/11 0 0&amp;gt;]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 3&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 154.579&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.47
Map: [&amp;lt;1 3 4 5|, &amp;lt;0 -11 -13 -17|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 49/45&lt;br /&gt;
== Vines ==
EDOs: 7, 8, 31, 101, 132, 163&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''
Badness: 0.0504&lt;br /&gt;
 
&lt;br /&gt;
Vines may be described as the {{nowrap| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Nusecond temperament-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h3&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 99/98, 121/120, 126/125&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
11-limit minimax&lt;br /&gt;
[[Comma list]]: 126/125, 84035/82944
[|1 0 0 0 0&amp;gt;, |19/10 11/5 0 0 -11/10&amp;gt;, &lt;br /&gt;
 
|27/10 13/5 0 0 -13/10&amp;gt;, |33/10 17/5 0 0 -17/10&amp;gt;, &lt;br /&gt;
{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
|19/5 12/5 0 0 -6/5&amp;gt;]&lt;br /&gt;
: mapping generators: ~343/240, ~6/5
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 154.645&lt;br /&gt;
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
Algebraic generator: &lt;a class="wiki_link" href="/Algebraic%20number"&gt;positive root&lt;/a&gt; of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.&lt;br /&gt;
: [[error map]]: {{val| +0.487 -0.363 +3.036 -4.448 }}
&lt;br /&gt;
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}
Map: [&amp;lt;1 3 4 5 5|, &amp;lt;0 -11 -13 -17 -12|]&lt;br /&gt;
: error map: {{val| 0.000 -0.717 +2.269 -5.552 }}
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 11/10&lt;br /&gt;
 
EDOs: 7, 8, 31, 101, 194&lt;br /&gt;
{{Optimal ET sequence|legend=1| 46, 96d, 142d }}
Badness: 0.0256&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[Badness]] (Sintel): 1.98
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 2401/2376
 
Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}
 
Optimal tunings:
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}
 
{{Optimal ET sequence|legend=0| 46, 96d, 142d }}
 
Badness (Sintel): 1.47
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 364/363, 385/384
 
Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}
 
Optimal tunings:
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}
 
{{Optimal ET sequence|legend=0| 46, 96d }}
 
Badness (Sintel): 1.23
 
== Xenial ==
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''
 
Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 177147/175616
 
{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}
 
{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}
 
[[Badness]] (Sintel): 2.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 540/539, 16384/16335
 
Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}
 
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}
 
Badness (Sintel): 2.31
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 169/168, 540/539, 729/728
 
Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
 
Optimal tunings:  
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}
 
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}
 
Badness (Sintel): 1.98
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539
 
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}
 
Optimal tunings:  
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}
 
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}
 
Badness (Sintel): 2.06
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539
 
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}
 
Optimal tunings:
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}
 
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}
 
Badness (Sintel): 2.03
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230
 
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}
 
Optimal tunings:
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}
 
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}
 
Badness (Sintel): 1.93
 
== Kumonga ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 12288/12005
 
{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0653{{c}}, ~7/4 = 975.6277{{c}}
: [[error map]]: {{val| -1.935 -1.382 +4.009 +2.932 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 977.1096{{c}}
: error map: {{val| 0.000 +0.470 +7.673 +8.284 }}
 
{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}
 
[[Badness]] (Sintel): 2.21
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 864/847
 
Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}
 
Optimal tunings:
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}
 
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}
 
Badness (Sintel): 1.43
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 126/125, 144/143, 176/175
 
Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}
 
Optimal tunings:
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}
 
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}
 
Badness (Sintel): 1.19
 
== Paraguay ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''
 
Named by [[User:Xenllium|Xenllium]] in 2026, paraguay tempers out [[12005/11664]] and may be described as the {{nowrap| 19 & 61 }} temperament. It is a variant of [[parakleismic]], mapping 7th harmonic to 16 generators.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 12005/11664
 
{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}
 
{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}
 
[[Badness]] (Sintel): 2.47
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 100/99, 12005/11664
 
Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}
 
Optimal tunings:
* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}
 
{{Optimal ET sequence|legend=0| 19, 42e, 61e }}
 
Badness (Sintel): 2.49
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 91/90, 100/99, 343/338
 
Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}
 
Optimal tunings:
* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}
 
{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}
 
Badness (Sintel): 1.86
 
==== Uruguay ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 100/99, 1183/1152
 
Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}
 
{{Optimal ET sequence|legend=0| 19, 42e }}
 
Badness (Sintel): 2.51
 
== Bisemidim ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 118098/117649
 
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45
 
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}
 
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
 
[[Badness]] (Sintel): 2.47
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 540/539, 1344/1331
 
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
 
Optimal tunings:  
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
 
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
 
Badness (Sintel): 1.36
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 196/195, 364/363
 
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
 
Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
 
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
 
Badness (Sintel): 0.987
 
== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 19683/19208
 
{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}
 
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}
 
[[Badness]] (Sintel): 2.53
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 126/125, 243/242
 
Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
 
Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}
 
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}
 
Badness (Sintel): 1.41
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 99/98, 126/125, 243/242
 
Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
 
Optimal tunings:
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}
 
{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}
 
Badness (Sintel): 1.56
 
== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''
 
Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.
 
It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.
 
If we add 385/384 to the list of commas, 48/35 is identified with [[11/8]], and casablanca is revealed as an [[11-limit]] temperament with a very low complexity for [[11/1|11]] and not too high a one for [[7/1|7]]; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit [[meantone]].
 
Marrakesh, named by [[Herman Miller]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, is a more accurate 11-limit extension where the generator is identified with [[15/11]] as opposed to 11/8 in casablanca.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 589824/588245
 
{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}
: [[error map]]: {{val| -0.371 -1.087 +3.370 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 542.1684{{c}}
: error map: {{val| 0.000 -0.756 +4.044 -0.152 }}
 
{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
 
[[Badness]] (Sintel): 2.56
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 2420/2401
 
Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}
 
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
 
Badness (Sintel): 2.22
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 385/384, 2420/2401
 
Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}
 
Optimal tunings:
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}
 
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
 
Badness (Sintel): 2.31
 
=== Marrakesh ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 14641/14580
 
Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}
 
Optimal tunings:
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}
 
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}
 
Badness (Sintel): 1.34
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195, 14641/14580
 
Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}
 
Optimal tunings:
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}
 
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}
 
Badness (Sintel): 1.68
 
==== Murakuc ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 176/175, 1540/1521
 
Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}
 
Optimal tunings:
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}
 
{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}
 
Badness (Sintel): 1.71
 
== Amigo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 2097152/2083725
 
{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}
: [[error map]]: {{val| -0.565 -0.811 +3.467 -1.206 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.0937{{c}}
: error map: {{val| 0.000 +0.076 +4.780 +0.393 }}
 
{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}
 
[[Badness]] (Sintel): 2.81
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 16384/16335
 
Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}
 
{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}
 
Badness (Sintel): 1.44
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 169/168, 176/175, 364/363
 
Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}
 
{{Optimal ET sequence|legend=0| 43, 46, 89 }}
 
Badness (Sintel): 1.27
 
== Gilead ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 343/324
 
{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4516{{c}}, ~5/3 = 879.6394{{c}}
: [[error map]]: {{val| +1.452 +7.542 +2.823 -21.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.7223{{c}}
: error map: {{val| 0.000 +6.545 +0.909 -24.159 }}
 
{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}
 
[[Badness]] (Sintel): 2.92
 
== Supersensi ==
Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 17496/16807
 
{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}
: [[error map]]: {{val| -0.859 -4.800 +3.337 +6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/270 = 446.5163{{c}}
: error map: {{val| 0.000 -4.210 +4.464 +8.017 }}
 
{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}
 
[[Badness]] (Sintel): 3.76
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 126/125, 864/847
 
Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
 
Optimal tunings:
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}
 
Badness (Sintel): 1.97
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 99/98, 126/125, 144/143
 
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
 
Optimal tunings:
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}
 
Badness (Sintel): 1.46
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 78/77, 99/98, 120/119, 126/125, 144/143
 
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
 
Optimal tunings:
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}
 
Badness (Sintel): 1.32
 
== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''
 
Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.
 
Cobalt was named by [[Xenllium]] in 2022 after the 27th element.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 40353607/40310784
 
{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 44.4363{{c}}, ~3/2 = 701.1154{{c}}
: [[error map]]: {{val| -0.221 -1.060 +3.307 -1.534 }}
* [[CWE]]: ~36/35 = 44.4444{{c}}, ~3/2 = 701.0414{{c}}
: error map: {{val| 0.000 -0.914 +3.617 -1.118 }}
 
{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}
 
[[Badness]] (Sintel): 4.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 540/539, 21609/21296
 
Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}
 
Optimal tunings:
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 81, 108 }}
 
Badness (Sintel): 2.58
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 196/195, 21609/21296
 
Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}
 
Optimal tunings:
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}
 
Badness (Sintel): 2.36
 
===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445
 
Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}
 
Optimal tunings:
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}
 
Badness (Sintel): 2.14
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968
 
Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}
 
Optimal tunings:
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}
 
Badness (Sintel): 1.85
 
===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968
 
Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}
 
Optimal tunings:
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}
 
Badness (Sintel): 2.40
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083
 
Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}
 
Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}
 
Badness (Sintel): 2.08
 
==== Cobaltite ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 169/168, 540/539, 975/968
 
Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}
 
Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}
 
Badness (Sintel): 2.18
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Starling temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"