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Starling 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]]s tempering out [[126/125]], the starling comma or septimal semicomma.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-28 11:52:58 UTC</tt>.<br>
: The original revision id was <tt>326391418</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]]
=Starling comma=
This page discusses some of the temperaments tempering out [[126_125|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 the [[starling tetrad]], the 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]]


=5-limit=
Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].
Comma: 10077696/9765625


POTE generator: ~6/5 = 310.140
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.  


Map: [&lt;1 9 9|, &lt;0 -10 -9|]
== Myna ==
EDOs: 27, 31, 58, 89, 325c
{{Main| Myna }}
Badness: 0.2500
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''


==7-limit==
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.
[[Comma|Commas]]: 126/125, 1728/1715


7 and 9 limit minimax
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.
[|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
[[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|, &lt;0 -10 -9 -7|]
[[Subgroup]]: 2.3.5.7
[[Generator|Generators]]: 2, 5/3
EDOs: 27, 31, 58, 89
Badness: 0.0270


==11-limit==
[[Comma list]]: 126/125, 1728/1715
Commas: 126/125, 176/175, 243/242


[[POTE tuning|POTE generator]]: ~6/5 = 310.144
{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5


Map: [&lt;1 9 9 8 22|, &lt;0 -10 -9 -7 -25|]
[[Optimal tuning]]s:
EDOs: 31, 58, 89
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
Badness: 0.0168
: [[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 }}


==13-limit==
[[Minimax tuning]]:
Commas: 126/125, 144/143, 176/175, 196/195
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[POTE tuning|POTE generator]]: ~6/5 = 310.276
{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}


Map: [&lt;1 9 9 8 22 0|, &lt;0 -10 -9 -7 -25 5|]
[[Badness]] (Sintel): 0.684
EDOs: 27, 31, 58
Badness: 0.0171


==Minah==
=== 11-limit ===
Commas: 78/77, 91/90, 126/125, 176/175
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 310.381
Comma list: 126/125, 176/175, 243/242


Map: [&lt;1 9 9 8 22 20|, &lt;0 -10 -9 -7 -25 -22|]
Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}
EDOs: 27e, 31f, 58f, 116cef
Badness: 0.0276


==Myno==
Optimal tunings:
Commas: 99/98, 126/125, 385/384
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}


POTE generator: ~6/5 = 309.737
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}


Map: [&lt;1 9 9 8 -1|, &lt;0 -10 -9 -7 6|]
Badness (Sintel): 0.557
EDOs: 27, 31
Badness: 0.0334


==Coleto==
==== 13-limit ====
Commas: 56/55, 100/99, 1728/1715
Subgroup: 2.3.5.7.11.13


POTE generator: ~6/5 = 310.853
Comma list: 126/125, 144/143, 176/175, 196/195


Map: [&lt;1 9 9 8 2|, &lt;0 -10 -9 -7 2|]
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
EDOs: 23bc, 27e
Badness: 0.0487


[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/89versionof23Myna.mp3|Myna Music]] by [[Igliashon Jones]]
Optimal tunings:  
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}


=Sensi temperament=
{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}
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
Badness (Sintel): 0.708


7-limit minimax
==== Minah ====
[|1 0 0 0&gt;, |1/13 0 0 7/13&gt;, |5/13 0 0 9/13&gt;, |0 0 0 1&gt;]
Subgroup: 2.3.5.7.11.13
[[Eigenmonzo|Eigenmonzos]]: 2, 7


9-limit minimax
Comma list: 78/77, 91/90, 126/125, 176/175
[|1 0 0 0&gt;, |2/5 14/5 -7/5 0&gt;,
|4/5 18/5 -9/5 0&gt;, |3/5 26/5 -13/5 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 9/5


[[POTE tuning|POTE generator]]: ~9/7 = 443.383
Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}
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|]
Optimal tunings:  
[[Generator|Generators]]: 2, 14/9
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
EDOs: 19, 27, 46, 249, 295
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}
Badness: 0.0256


==Sensor==
{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}
Commas: 126/125, 245/243, 385/384


[[POTE tuning|POTE generator]]: ~9/7 = 443.294
Badness (Sintel): 1.14


Map: [&lt;1 6 8 11 -6|, &lt;0 -7 -9 -13 15|]
==== Maneh ====
EDOs: 8, 19, 27, 46, 111, 157
Subgroup: 2.3.5.7.11.13
Badness: 0.0379


===13-limit===
Comma list: 66/65, 105/104, 126/125, 243/242
Commas: 91/90, 126/125, 169/168, 385/384


[[POTE tuning|POTE generator]]: ~9/7 = 443.321
Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}


Map: [&lt;1 6 8 11 -6 10|, &lt;0 -7 -9 -13 15 -10|]
Optimal tunings:  
EDOs: 8, 19, 27, 46, 157
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
Badness: 0.0256
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}


==Sensis==
{{Optimal ET sequence|legend=0| 27eff, 31 }}
Commas: 56/55, 100/99, 245/243


[[POTE tuning|POTE generator]]: 443.962
Badness (Sintel): 1.23


Map: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
=== Myno ===
EDOs: 19, 27, 73, 100
Subgroup: 2.3.5.7.11
Badness: 0.0287


===13-limit===
Comma list: 99/98, 126/125, 385/384
Commas: 56/55, 78/77, 91/90, 100/99


[[POTE tuning|POTE generator]]: 443.945
Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}


Map: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
Optimal tunings:  
EDOs: 19, 27, 73, 100
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
Badness: 0.0200
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}


==Sensus==
{{Optimal ET sequence|legend=0| 27, 31 }}
Commas: 126/125, 176/175, 245/243


POTE generator: ~9/7 = 443.626
Badness (Sintel): 1.11


Map: [&lt;1 6 8 11 23|, &lt;0 -7 -9 -13 -31|]
=== Coleto ===
EDOs: 8, 19, 27, 46, 165
Subgroup: 2.3.5.7.11
Badness: 0.0295


===13-limit===
Comma list: 56/55, 100/99, 1728/1715
Commas: 91/90, 126/125, 169/168, 352/351


POTE generator: ~9/7 = 443.559
Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}


Map: [&lt;1 6 8 11 23 10|, &lt;0 -7 -9 -13 -31 -10|]
Optimal tunings:  
EDOs: 8, 19, 27, 46, 303
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
Badness: 0.0208
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}


=Valentine temperament=
{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}
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).


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.
Badness (Sintel): 1.61


[[Comma|Commas]]: 1029/1024, 126/125
== Nusecond ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''


[[Minimax tuning]]:
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.
7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
|17/6 5/12 0 -5/12&gt;, [5/2 -1/4 0 1/4&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&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.
|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
[[Subgroup]]: 2.3.5.7


Algebraic generator: [[Algebraic number|smaller root]] of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.
[[Comma list]]: 126/125, 2430/2401


Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]
{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
[[Generator|Generators]]: 2, 21/20
: mapping generators: ~2, ~49/27
EDOs: 15, 31, 46, 77, 185, 262
Badness: 0.0311


==11-limit==
[[Optimal tuning]]s:
[[Comma|Commas]]: 121/120, 126/125, 176/175
* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}
: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}
: error map: {{val| 0.000 -2.655 +3.768 +2.819 }}


[[Minimax tuning]]:
[[Minimax tuning]]:  
[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;]
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
[[Eigenmonzo|Eigenmonzos]]: 2, 11/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


Minimax generator: (11/7)^(1/10) = 78.249
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
[[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): 1.28


Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]
=== 11-limit ===
[[Edo|Edos]]: [[15edo|15]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[108edo|108]], [[185edo|185]]
Subgroup: 2.3.5.7.11
Badness: 0.0167


See also: [[Chords of valentine]]
Comma list: 99/98, 121/120, 126/125


==Dwynwen==
Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}
Commas: 91/90, 121/120, 126/125, 176/175


POTE generator: ~21/20 = 78.219
Optimal tunings:  
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}


Map: [&lt;1 1 2 3 3 2|, &lt;0 9 5 -3 7 26|]
Minimax tuning:  
EDOs: 15, 46
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
Badness: 0.0235
: [{{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 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9


==Lupercalia==
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: 66/65, 105/104, 121/120, 126/125


POTE generator: ~22/21 = 77.709
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}


Map: [&lt;1 1 2 3 3 3|, &lt;0 9 5 -3 7 11|]
Badness (Sintel): 0.847
EDOs: 15, 31, 108, 139
Badness: 0.0213


==Valentino==  
=== 13-limit ===
Commas: 121/120, 126/125, 176/175, 196/195
Subgroup: 2.3.5.7.11.13


POTE generator: ~22/21 = 77.958
Comma list: 66/65, 99/98, 121/120, 126/125


Map: [&lt;1 1 2 3 3 5|, &lt;0 9 5 -3 7 -20|]
Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}
EDOs: 15, 31, 46, 77, 431
Badness: 0.0207


=Alicorn temperament=  
Optimal tunings:
Commas: 126/125, 10976/10935
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}


POTE generator: ~28/27 = 62.278
{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}


Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
Badness (Sintel): 0.964
Wedgie: &lt;&lt;8 13 23 2 14 17||
EDOs: 19, 58, 77, 96
Badness: 0.0409


==11-limit==  
== Oolong ==
Commas: 126/125, 540/539, 896/891
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''


POTE generator: ~28/27 = 62.101
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
[[Comma list]]: 126/125, 117649/116640
EDOs: 19, 58
Badness: 0.0392


==13-limit==
{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
Commas: 126/125, 144/143, 196/195, 676/675
: mapping generators: ~2, ~5/3


POTE generator: ~28/27 = 62.119
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}
: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}


Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}
EDOs: 19, 58


Badness: 0.0237
[[Badness]] (Sintel): 1.86


==Camahueto==  
=== 11-limit ===
Commas: 126/125, 10976/10935, 385/384
Subgroup: 2.3.5.7.11


POTE generator: ~28/27 = 62.431
Comma list: 126/125, 176/175, 26411/26244


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}
EDOs: 19, 58, 77, 96
Badness: 0.0659


===13-limit===
Optimal tunings:
Commas: 126/125, 196/195, 385/384, 676/675
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}


POTE generator: ~28/27 = 62.434
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
Badness (Sintel): 1.88
EDOs: 19, 58, 77
Badness: 0.0362


=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=Casablanca temperament=
Comma list: 126/125, 176/175, 196/195, 13013/12960
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 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}


Commas: 126/125, 589824/588245
Optimal tunings:  
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}


POTE generator: ~35/24 = 657.818
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


Map: [&lt;1 12 10 5|, &lt;0 -19 -14 -4|]
Badness (Sintel): 1.47
EDOs: 9, 11, 31, 135, 166
Badness: 0.1012


==11-limit==  
== Vines ==
Commas: 126/125, 385/384, 2420/2401
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''


POTE generator: ~16/11 = 657.923
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.  


Map: [&lt;1 12 10 5 4|, |0 -19 -14 -4 -1&gt;]
[[Subgroup]]: 2.3.5.7
EDOs: 9, 11, 31, 259, 549
Badness: 0.0623


==Marrakesh==
[[Comma list]]: 126/125, 84035/82944
Commas: 126/125, 176/175, 14641/14580


POTE generator: ~22/15 = 657.791
{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5


Map: [&lt;1 12 10 5 21|, |0 -19 -14 -4 -32&gt;]
[[Optimal tuning]]s:  
EDOs: 9, 11, 20, 31, 42, 73
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
Badness: 0.0405
: [[error map]]: {{val| +0.487 -0.363 +3.036 -4.448 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}
: error map: {{val| 0.000 -0.717 +2.269 -5.552 }}


===13-limit===
{{Optimal ET sequence|legend=1| 46, 96d, 142d }}
126/125, 176/175, 196/195, 17303/17280


POTE generator: ~22/15 = 657.756
[[Badness]] (Sintel): 1.98


Map: [&lt;1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25&gt;]
=== 11-limit ===
EDOs: 31, 73, 104c, 135c, 239cf
Subgroup: 2.3.5.7.11
Badness: 0.0408


===Murakuc===
Comma list: 126/125, 385/384, 2401/2376
Commas: 126/125, 144/143, 176/175, 1540/1521


POTE generator: ~22/15 = 657.700
Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}


Map: [&lt;1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6&gt;]
Optimal tunings:  
EDOs: 31, 73f, 104cf
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
Badness: 0.0414
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}


=Nusecond temperament=
{{Optimal ET sequence|legend=0| 46, 96d, 142d }}
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.


[[Comma|Commas]]: 126/125, 2430/2401
Badness (Sintel): 1.47


7-limit minimax
=== 13-limit ===
[|1 0 0 0&gt;, |-5/13 0 11/13 0&gt;, |0 0 1 0&gt;, |-3/13 0 17/13 0&gt;]
Subgroup: 2.3.5.7.11.13
[[Eigenmonzo|Eigenmonzos]]: 2, 5


9-limit minimax
Comma list: 126/125, 196/195, 364/363, 385/384
[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 3


[[POTE tuning|POTE generator]]: 154.579
Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}


Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]
Optimal tunings:  
[[Generator|Generators]]: 2, 49/45
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
EDOs: 7, 8, 31, 101, 132, 163
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}
Badness: 0.0504


==11-limit==
{{Optimal ET sequence|legend=0| 46, 96d }}
[[Comma|Commas]]: 99/98, 121/120, 126/125


11-limit minimax
Badness (Sintel): 1.23
[|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]]: ~11/10 = 154.645
== Xenial ==
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.
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''


Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]
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.
[[Generator|Generators]]: 2, 11/10
EDOs: 7, 8, 31, 101, 194
Badness: 0.0256


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 66/65 99/98 121/120 126/125


POTE generator: ~11/10 = 154.478
[[Comma list]]: 126/125, 177147/175616


Map: [&lt;1 3 4 5 5 5|, &lt;0 -11 -13 -17 -12 -10|]
{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
EDOs: 31, 70f, 101f
: mapping generators: ~2, ~9/5
Badness: 0.0233


=Thuja=  
[[Optimal tuning]]s:
Commas: 126/125, 65536/64827
* [[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 }}


POTE generator: ~175/128 = 558.605
{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}


Map: [&lt;1 8 5 -2|, &lt;0 -12 -5 9|]
[[Badness]] (Sintel): 2.13
Wedgie: &lt;&lt;12 5 -9 -20 -48 -35||
EDOs: 15, 43, 58
Badness: 0.0884


==11-limit==  
=== 11-limit ===
Commas: 126/125, 176/175, 1344/1331
Subgroup: 2.3.5.7.11


POTE generator: ~11/8 = 558.620
Comma list: 126/125, 540/539, 16384/16335


Map: [&lt;1 8 5 -2 4|, &lt;0 -12 -5 9 -1|]
Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
EDOs: 15, 43, 58
Badness: 0.0331


==13-limit==
Optimal tunings:
Commas: 126/125, 144/143, 176/175, 364/363
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}


POTE generator: ~11/8 = 558.589
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


Map: [&lt;1 8 5 -2 4 16|, &lt;0 -12 -5 9 -1 -23|]
Badness (Sintel): 2.31
EDOs: 15, 43, 58
Badness: 0.0228


=Bisemidim=
=== 13-limit ===
Commas: 126/125, 118098/117649
Subgroup: 2.3.5.7.11.13


POTE generator: ~35/27 = 455.445
Comma list: 126/125, 169/168, 540/539, 729/728


Map: [&lt;2 1 2 2|, &lt;0 9 11 15|]
Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
Wedgie: &lt;&lt;18 22 30 -7 -3 8||
EDOs: 50, 58, 108, 166c, 408c
Badness: 0.0978


==11-limit==
Optimal tunings:
Commas: 126/125, 540/539, 1344/1331
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}


POTE generator: ~35/27 = 455.373
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


Map: [&lt;2 1 2 2 5|, &lt;0 9 11 15 8|]
Badness (Sintel): 1.98
EDOs: 50, 58, 108, 166ce, 224ce
Badness: 0.0412


==13-limit==
=== 17-limit ===
Commas: 126/125, 144/143, 196/195, 364/363
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~35/27 = 455.347
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539


Map: [&lt;2 1 2 2 5 5|, &lt;0 9 11 15 8 10|]
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}
EDOs: 50, 58, 166cef, 224cef
Badness: 0.0239


=Kumonga=
Optimal tunings:
Comma: 1289945088/1220703125
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}


POTE generator: ~144/125 = 222.912
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


Map: [&lt;1 4 4|, &lt;0 -13 -9|]
Badness (Sintel): 2.06
EDOs: 16, 27, 43, 70, 183c
Badness: 0.7296


==7-limit==
=== 19-limit ===
Commas: 126/125, 12288/12005
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~8/7 = 222.797
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539


Map: [&lt;1 4 4 3|, &lt;0 -13 -9 -1|]
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}
Wedgie: &lt;&lt;13 9 1 -16 -35 -23||
EDOs: 16, 27, 43, 70, 167cd
Badness: 0.0875


==11-limit==
Optimal tunings:
Commas: 126/125, 176/175, 864/847
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}


POTE generator: ~8/7 = 222.898
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}


Map: [&lt;1 4 4 3 7|, &lt;0 -13 -9 -1 -19|]
Badness (Sintel): 2.03
EDOs: 16, 27e, 43, 70e
Badness: 0.0433


==13-limit==
=== 23-limit ===
Commas: 78/77, 126/125, 144/143, 176/175
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~8/7 = 222.961
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230


Map: [&lt;1 4 4 3 7 5|, &lt;0 -13 -9 -1 -19 -7|]
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}
EDOs: 16, 27e, 43, 70e, 113cde
 
Badness: 0.0289</pre></div>
Optimal tunings:
<h4>Original HTML content:</h4>
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
<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;&lt;!-- ws:start:WikiTextTocRule:88:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Starling comma"&gt;Starling comma&lt;/a&gt;&lt;/div&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}
&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Myna temperament"&gt;Myna temperament
 
{{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/
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