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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>2011-11-03 15:30:07 UTC</tt>.<br>
: The original revision id was <tt>271641420</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]]


[[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]]: ~6/5 = 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|]
[[Subgroup]]: 2.3.5.7
EDOs: 31, 58, 89
Badness: 0.0168


==13-limit==
[[Comma list]]: 126/125, 1728/1715
Commas: 126/125, 144/143, 176/175, 196/195


[[POTE tuning|POTE generator]]: ~6/5 = 310.276
{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5


Map: [&lt;1 9 9 8 22 0|, &lt;0 -10 -9 -7 -25 5|]
[[Optimal tuning]]s:
EDOs: 27, 31, 58
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
Badness: 0.0171
: [[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 }}


[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/89versionof23Myna.mp3|Myna Music]] by [[Igliashon Jones]]
[[Minimax tuning]]:
* 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


=Sensi temperament=
{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}
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): 0.684


7-limit minimax
=== 11-limit ===
[|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
[[Eigenmonzo|Eigenmonzos]]: 2, 7


9-limit minimax
Comma list: 126/125, 176/175, 243/242
[|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 | 0 10 9 7 25 }}
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 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
EDOs: 19, 27, 46, 249, 295
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}
Badness: 0.0256


==Sensor==
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}
Commas: 126/125, 245/243, 385/384


[[POTE tuning|POTE generator]]: ~9/7 = 443.294
Badness (Sintel): 0.557


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


===13-limit===
Comma list: 126/125, 144/143, 176/175, 196/195
Commas: 91/90, 126/125, 169/168, 385/384


[[POTE tuning|POTE generator]]: ~9/7 = 443.321
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}


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 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
Badness: 0.0256
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}


==Sensis==
{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}
Commas: 56/55, 100/99, 245/243


[[POTE tuning|POTE generator]]: 443.962
Badness (Sintel): 0.708


Map: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
==== Minah ====
EDOs: 19, 27, 73, 100
Subgroup: 2.3.5.7.11.13
Badness: 0.0287


===13-limit===
Comma list: 78/77, 91/90, 126/125, 176/175
Commas: 56/55, 78/77, 91/90, 100/99


[[POTE tuning|POTE generator]]: 443.945
Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}


Map: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
Optimal tunings:  
EDOs: 19, 27, 73, 100
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
Badness: 0.0200
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}


==Sensus==
{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}
Commas: 126/125, 176/175, 245/243


POTE generator: ~9/7 = 443.626
Badness (Sintel): 1.14


Map: [&lt;1 6 8 11 23|, &lt;0 -7 -9 -13 -31|]
==== Maneh ====
EDOs: 8, 19, 27, 46, 165
Subgroup: 2.3.5.7.11.13
Badness: 0.0295


===13-limit===
Comma list: 66/65, 105/104, 126/125, 243/242
Commas: 91/90, 126/125, 169/168, 352/351


POTE generator: ~9/7 = 443.559
Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}


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 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
Badness: 0.0208
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}


=Valentine temperament=
{{Optimal ET sequence|legend=0| 27eff, 31 }}
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): 1.23


[[Comma|Commas]]: 1029/1024, 126/125
=== Myno ===
Subgroup: 2.3.5.7.11


[[Minimax tuning]]:
Comma list: 99/98, 126/125, 385/384
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;,
Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}
|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
Optimal tunings:  
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}


Algebraic generator: [[Algebraic number|smaller root]] of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.
{{Optimal ET sequence|legend=0| 27, 31 }}


Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]
Badness (Sintel): 1.11
[[Generator|Generators]]: 2, 21/20
EDOs: 15, 31, 46, 77, 185, 262
Badness: 0.0311


==11-limit==
=== Coleto ===
[[Comma|Commas]]: 121/120, 126/125, 176/175
Subgroup: 2.3.5.7.11


[[Minimax tuning]]:
Comma list: 56/55, 100/99, 1728/1715
[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,
|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/7


Minimax generator: (11/7)^(1/10) = 78.249
Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}
[[POTE tuning|POTE generator]]: 77.881


Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.
Optimal tunings:  
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}


Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]
{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}
[[Edo|Edos]]: [[15edo|15]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[108edo|108]], [[185edo|185]]
Badness: 0.0167


==Dwynwen==
Badness (Sintel): 1.61
Commas: 91/90, 121/120, 126/125, 176/175


POTE generator: ~21/20 = 78.219
== Nusecond ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''


Map: [&lt;1 1 2 3 3 2|, &lt;0 9 5 -3 7 26|]
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.  
EDOs: 15, 46
Badness: 0.0235


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


POTE generator: ~22/21 = 77.709
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 2 3 3 3|, &lt;0 9 5 -3 7 11|]
[[Comma list]]: 126/125, 2430/2401
EDOs: 15, 31, 108, 139
Badness: 0.0213


==Valentino==
{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
Commas: 121/120, 126/125, 176/175, 196/195
: mapping generators: ~2, ~49/27


POTE generator: ~22/21 = 77.958
[[Optimal tuning]]s:  
* [[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 }}


Map: [&lt;1 1 2 3 3 5|, &lt;0 9 5 -3 7 -20|]
[[Minimax tuning]]:  
EDOs: 15, 31, 46, 77, 431
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
Badness: 0.0207
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: [[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


=Alicorn temperament=
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
Commas: 126/125, 10976/10935


POTE generator: ~28/27 = 62.278
[[Badness]] (Sintel): 1.28


Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
=== 11-limit ===
Wedgie: &lt;&lt;8 13 23 2 14 17||
Subgroup: 2.3.5.7.11
EDOs: 19, 58, 77, 96
Badness: 0.0409


==11-limit==
Comma list: 99/98, 121/120, 126/125
Commas: 126/125, 10976/10935, 385/384


POTE generator: ~28/27 = 62.431
Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
Optimal tunings:  
EDOs: 19, 58, 77, 96
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
Badness: 0.0659
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}


==13-limit==
Minimax tuning:
Commas: 126/125, 196/195, 385/384, 676/675
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
: [{{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


POTE generator: ~28/27 = 62.434
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.


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}
EDOs: 19, 58, 77
Badness: 0.0362


Badness (Sintel): 0.847


=Casablanca temperament=
=== 13-limit ===
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.
Subgroup: 2.3.5.7.11.13


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.
Comma list: 66/65, 99/98, 121/120, 126/125


Commas: 126/125, 589824/588245
Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}


POTE generator: ~35/24 = 657.818
Optimal tunings:  
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}


Map: [&lt;1 12 10 5|, &lt;0 -19 -14 -4|]
{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}
EDOs: 9, 11, 31, 135, 166
Badness: 0.1012


==11-limit==
Badness (Sintel): 0.964
Commas: 126/125, 385/384, 2420/2401


POTE generator: ~16/11 = 657.923
== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''


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, 117649/116640
Commas: 126/125, 176/175, 14641/14580


POTE generator: ~22/15 = 657.791
{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3


Map: [&lt;1 12 10 5 21|, |0 -19 -14 -4 -32&gt;]
[[Optimal tuning]]s:  
EDOs: 9, 11, 20, 31, 42, 73
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
Badness: 0.0405
: [[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 }}


=Nusecond temperament=
{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}
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): 1.86


7-limit minimax
=== 11-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
[[Eigenmonzo|Eigenmonzos]]: 2, 5


9-limit minimax
Comma list: 126/125, 176/175, 26411/26244
[|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| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}


Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]
Optimal tunings:  
[[Generator|Generators]]: 2, 49/45
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
EDOs: 7, 8, 31, 101, 132, 163
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}
Badness: 0.0504


==11-limit==
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}
[[Comma|Commas]]: 99/98, 121/120, 126/125


11-limit minimax
Badness (Sintel): 1.88
[|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
=== 13-limit ===
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.
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]
Comma list: 126/125, 176/175, 196/195, 13013/12960
[[Generator|Generators]]: 2, 11/10
EDOs: 7, 8, 31, 101, 194
Badness: 0.0256


=Thuja=
Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}
Commas: 126/125, 65536/64827


POTE generator: ~175/128 = 558.605
Optimal tunings:  
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}


Map: [&lt;1 8 5 -2|, &lt;0 -12 -5 9|]
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}
Wedgie: &lt;&lt;12 5 -9 -20 -48 -35||
EDOs: 15, 43, 58
Badness: 0.0884


==11-limit==
Badness (Sintel): 1.47
Commas: 126/125, 176/175, 1344/1331


POTE generator: ~11/8 = 558.620
== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''


Map: [&lt;1 8 5 -2 4|, &lt;0 -12 -5 9 -1|]
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.  
EDOs: 15, 43, 58
Badness: 0.0331


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 126/125, 144/143, 176/175, 364/363


POTE generator: ~11/8 = 558.589
[[Comma list]]: 126/125, 84035/82944


Map: [&lt;1 8 5 -2 4 16|, &lt;0 -12 -5 9 -1 -23|]
{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
EDOs: 15, 43, 58
: mapping generators: ~343/240, ~6/5
Badness: 0.0228</pre></div>
 
<h4>Original HTML content:</h4>
[[Optimal tuning]]s:
<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:54:&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
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
: [[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 }}
 
{{Optimal ET sequence|legend=1| 46, 96d, 142d }}
 
[[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
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"