Xenharmonic Wiki

Starling temperaments: Difference between revisions

← Older edit
VisualWikitext
m Need to check edo lists
m Recategorize
 
(101 intermediate revisions by 13 users not shown)
Line 1: Line 1:
__FORCETOC__
{{Technical data page}}
= Starling comma =
This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.  
This page discusses some of the rank two 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 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 =
Temperaments discussed in families and clans are:
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mavling]]'' (+135/128) → [[Mavila family #Mavling|Mavila family]]
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
* [[Opossum]] (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
* ''[[Paraguay]]'' (+12005/11664) → [[Parakleismic family #Paraguay|Parakleismic family]]
* ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnu family #Vishnean|Vishnu family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]]


{{main|Myna}}
Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, supersensi, and cobalt, sorted by increasing [[badness]].


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&31 temperament, or in terms of its wedgie <<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.
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.  


==5-limit (Mynic)==
== Myna ==
Comma: 10077696/9765625
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''


POTE generator: ~6/5 = 310.140
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.  


Map: [&lt;1 9 9|, &lt;0 -10 -9|]
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.


EDOs: 27, 31, 58, 89, 325c
[[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.


Badness: 0.2500
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 126/125, 1728/1715
[[Comma]]s: 126/125, 1728/1715


7 and 9 limit minimax
{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5


[|1 0 0 0&gt;, |0 1 0 0 &gt;, |9/10 9/10 0 0&gt;, |17/10 7/10 0 0&gt;]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
: [[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 }}


[[Eigenmonzo]]s: 2, 3
[[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


[[POTE_tuning|POTE generator]]: 310.146
{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}


Map: [&lt;1 9 9 8|, &lt;0 -10 -9 -7|]
[[Badness]] (Sintel): 0.684


[[Generator]]s: 2, 5/3
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 27, 31, 58, 89
Comma list: 126/125, 176/175, 243/242


Badness: 0.0270
Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}


==11-limit==
Optimal tunings:
Commas: 126/125, 176/175, 243/242
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}


[[POTE_tuning|POTE generator]]: ~6/5 = 310.144
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}


Map: [&lt;1 9 9 8 22|, &lt;0 -10 -9 -7 -25|]
Badness (Sintel): 0.557


EDOs: 27e, 31, 58, 89
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0168
Comma list: 126/125, 144/143, 176/175, 196/195


==13-limit==
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
Commas: 126/125, 144/143, 176/175, 196/195


[[POTE_tuning|POTE generator]]: ~6/5 = 310.276
Optimal tunings:  
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}


Map: [&lt;1 9 9 8 22 0|, &lt;0 -10 -9 -7 -25 5|]
{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}


EDOs: 27e, 31, 58
Badness (Sintel): 0.708


Badness: 0.0171
==== Minah ====
Subgroup: 2.3.5.7.11.13


==Minah==
Comma list: 78/77, 91/90, 126/125, 176/175
Commas: 78/77, 91/90, 126/125, 176/175


POTE generator: ~6/5 = 310.381
Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}


Map: [&lt;1 9 9 8 22 20|, &lt;0 -10 -9 -7 -25 -22|]
Optimal tunings:  
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}


EDOs: 27e, 31f, 58f, 116cef
{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}


Badness: 0.0276
Badness (Sintel): 1.14


==Maneh==
==== Maneh ====
Commas: 66/65, 105/104, 126/125, 540/539
Subgroup: 2.3.5.7.11.13


POTE generator: ~6/5 = 309.804
Comma list: 66/65, 105/104, 126/125, 243/242


Map: [&lt;1 9 9 8 22 23|, &lt;0 -10 -9 -7 -25 -26|]
Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}


EDOs: 31
Optimal tunings:  
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}


Badness: 0.0299
{{Optimal ET sequence|legend=0| 27eff, 31 }}


==Myno==
Badness (Sintel): 1.23
Commas: 99/98, 126/125, 385/384


POTE generator: ~6/5 = 309.737
=== Myno ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 9 9 8 -1|, &lt;0 -10 -9 -7 6|]
Comma list: 99/98, 126/125, 385/384


EDOs: 27, 31
Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}


Badness: 0.0334
Optimal tunings:  
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}


==Coleto==
{{Optimal ET sequence|legend=0| 27, 31 }}
Commas: 56/55, 100/99, 1728/1715


POTE generator: ~6/5 = 310.853
Badness (Sintel): 1.11


Map: [&lt;1 9 9 8 2|, &lt;0 -10 -9 -7 2|]
=== Coleto ===
Subgroup: 2.3.5.7.11


EDOs: 23bc, 27e
Comma list: 56/55, 100/99, 1728/1715


Badness: 0.0487
Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}


''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/89versionof23Myna.mp3 Myna Music]'' by [[Igliashon Jones]]
Optimal tunings:  
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}


= Sensi =
{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}
{{main|Sensi}}
{{see also|Sensipent family #Sensi}}


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.
Badness (Sintel): 1.61


[[Comma]]s: 126/125, 245/243
== Nusecond ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''


7-limit minimax
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.


[|1 0 0 0&gt;, |1/13 0 0 7/13&gt;, |5/13 0 0 9/13&gt;, |0 0 0 1&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.


[[Eigenmonzo]]s: 2, 7
[[Subgroup]]: 2.3.5.7


9-limit minimax
[[Comma list]]: 126/125, 2430/2401


[|1 0 0 0&gt;, |2/5 14/5 -7/5 0&gt;,
{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
|4/5 18/5 -9/5 0&gt;, |3/5 26/5 -13/5 0&gt;<nowiki>]</nowiki>
: mapping generators: ~2, ~49/27


[[Eigenmonzo]]s: 2, 9/5
[[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 }}


[[POTE_tuning|POTE generator]]: ~9/7 = 443.383
[[Minimax tuning]]:
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: {{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


Algebraic generator: Calista, the [[Algebraic_number|real root]] of x^7-2x^2-1, at 340.6467 cents.
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


Map: [&lt;1 6 8 11|, &lt;0 -7 -9 -13|]
[[Badness]] (Sintel): 1.28


[[Generator]]s: 2, 14/9
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd
Comma list: 99/98, 121/120, 126/125


Badness: 0.0256
Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}


==Sensor==
Optimal tunings:
Commas: 126/125, 245/243, 385/384
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}


[[POTE_tuning|POTE generator]]: ~9/7 = 443.294
Minimax tuning:
* [[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


Map: [&lt;1 6 8 11 -6|, &lt;0 -7 -9 -13 15|]
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.


EDOs: 19, 27, 46, 111d, 157d
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}


Badness: 0.0379
Badness (Sintel): 0.847


===13-limit===
=== 13-limit ===
Commas: 91/90, 126/125, 169/168, 385/384
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~9/7 = 443.321
Comma list: 66/65, 99/98, 121/120, 126/125


Map: [&lt;1 6 8 11 -6 10|, &lt;0 -7 -9 -13 15 -10|]
Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}


EDOs: 19, 27, 46, 111df, 157df
Optimal tunings:  
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}


Badness: 0.0256
{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}


==Sensis==
Badness (Sintel): 0.964
Commas: 56/55, 100/99, 245/243


[[POTE_tuning|POTE generator]]: 443.962
== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''


Map: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
[[Subgroup]]: 2.3.5.7


EDOs: 19, 27e, 73ee
[[Comma list]]: 126/125, 117649/116640


Badness: 0.0287
{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3


===13-limit===
[[Optimal tuning]]s:
Commas: 56/55, 78/77, 91/90, 100/99
* [[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 }}


[[POTE_tuning|POTE generator]]: 443.945
{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}


Map: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
[[Badness]] (Sintel): 1.86


EDOs: 19, 27e, 46e, 73ee
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0200
Comma list: 126/125, 176/175, 26411/26244


==Sensus==
Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}
Commas: 126/125, 176/175, 245/243


POTE generator: ~9/7 = 443.626
Optimal tunings:  
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}


Map: [&lt;1 6 8 11 23|, &lt;0 -7 -9 -13 -31|]
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


EDOs: 19e, 27e, 46, 119c, 165c
Badness (Sintel): 1.88


Badness: 0.0295
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 126/125, 176/175, 196/195, 13013/12960
Commas: 91/90, 126/125, 169/168, 352/351


POTE generator: ~9/7 = 443.559
Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}


Map: [&lt;1 6 8 11 23 10|, &lt;0 -7 -9 -13 -31 -10|]
Optimal tunings:  
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}


EDOs: 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


Badness: 0.0208
Badness (Sintel): 1.47


= Valentine =
== Vines ==
{{main|Valentine}}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''
{{see also|Gamelismic clan #Valentine}}


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).
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.  


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.
[[Subgroup]]: 2.3.5.7


[[Comma]]s: 1029/1024, 126/125
[[Comma list]]: 126/125, 84035/82944


[[Minimax tuning]]:
{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5


7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
[[Optimal tuning]]s:
|17/6 5/12 0 -5/12&gt;, [5/2 -1/4 0 1/4&gt;<nowiki>]</nowiki>
* [[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 }}


[[Eigenmonzo]]s: 2, 7/6
{{Optimal ET sequence|legend=1| 46, 96d, 142d }}


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
[[Badness]] (Sintel): 1.98
|47/21 10/21 0 -5/21&gt;, |20/7 -2/7 0 1/7&gt;<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 9/7
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: 77.864
Comma list: 126/125, 385/384, 2401/2376


Algebraic generator: [[Algebraic_number|smaller root]] of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.
Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}


Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]
Optimal tunings:  
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}


[[Generator]]s: 2, 21/20
{{Optimal ET sequence|legend=0| 46, 96d, 142d }}


EDOs: 15, 31, 46, 77, 185, 262
Badness (Sintel): 1.47


Badness: 0.0311
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 126/125, 196/195, 364/363, 385/384
[[Comma]]s: 121/120, 126/125, 176/175


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


[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,
Optimal tunings:
|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;<nowiki>]</nowiki>
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}


[[Eigenmonzo]]s: 2, 11/7
{{Optimal ET sequence|legend=0| 46, 96d }}


Minimax generator: (11/7)^(1/10) = 78.249
Badness (Sintel): 1.23


[[POTE_tuning|POTE generator]]: 77.881
== Xenial ==
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''


Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.
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 [[octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.


Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]
[[Subgroup]]: 2.3.5.7


[[EDO]]s: [[15edo|15]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[108edo|108]], [[185edo|185]]
[[Comma list]]: 126/125, 177147/175616


Badness: 0.0167
{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5


See also: [[Chords of valentine]]
[[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 }}


==Dwynwen==
{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}
Commas: 91/90, 121/120, 126/125, 176/175


POTE generator: ~21/20 = 78.219
[[Badness]] (Sintel): 2.13


Map: [&lt;1 1 2 3 3 2|, &lt;0 9 5 -3 7 26|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 15, 46
Comma list: 126/125, 540/539, 16384/16335


Badness: 0.0235
Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}


==Lupercalia==
Optimal tunings:
Commas: 66/65, 105/104, 121/120, 126/125
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}


POTE generator: ~22/21 = 77.709
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


Map: [&lt;1 1 2 3 3 3|, &lt;0 9 5 -3 7 11|]
Badness (Sintel): 2.31


EDOs: 15, 31, 108, 139
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0213
Comma list: 126/125, 169/168, 540/539, 729/728


==Valentino==
Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
Commas: 121/120, 126/125, 176/175, 196/195


POTE generator: ~22/21 = 77.958
Optimal tunings:  
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}


Map: [&lt;1 1 2 3 3 5|, &lt;0 9 5 -3 7 -20|]
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


EDOs: 15, 31, 46, 77, 431
Badness (Sintel): 1.98


Badness: 0.0207
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


==Semivalentine==
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539
Commas: 121/120, 126/125, 169/168, 176/175


POTE generator: ~22/21 = ~21/20 = 77.839
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}


Map: [&lt;2 2 4 6 6 7|, &lt;0 9 5 -3 7 3|]
Optimal tunings:  
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}


EDOs: 16, 30, 46, 62, 108ef
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


Badness: 0.0327
Badness (Sintel): 2.06


= Alicorn =
=== 19-limit ===
{{see also|Unicorn family #Alicorn}}
Subgroup: 2.3.5.7.11.13.17.19


Commas: 126/125, 10976/10935
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539


POTE generator: ~28/27 = 62.278
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}


Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
Optimal tunings:  
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}


Wedgie: &lt;&lt;8 13 23 2 14 17||
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}


EDOs: 19, 58, 77, 96
Badness (Sintel): 2.03


Badness: 0.0409
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


==11-limit==
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230
Commas: 126/125, 540/539, 896/891


POTE generator: ~28/27 = 62.101
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
Optimal tunings:  
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}


EDOs: 19, 58
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}


Badness: 0.0392
Badness (Sintel): 1.93


==13-limit==
== Kumonga ==
Commas: 126/125, 144/143, 196/195, 676/675
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''


POTE generator: ~28/27 = 62.119
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
[[Comma list]]: 126/125, 12288/12005


EDOs: 19, 58
{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4


Badness: 0.0237
[[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 }}


==Camahueto==
{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}
Commas: 126/125, 10976/10935, 385/384


POTE generator: ~28/27 = 62.431
[[Badness]] (Sintel): 2.21


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 19, 58, 77, 96
Comma list: 126/125, 176/175, 864/847


Badness: 0.0659
Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}


===13-limit===
Optimal tunings:
Commas: 126/125, 196/195, 385/384, 676/675
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}


POTE generator: ~28/27 = 62.434
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
Badness (Sintel): 1.43


EDOs: 19, 58, 77
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0362
Comma list: 78/77, 126/125, 144/143, 176/175


= Coblack =
Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}
In addition to 126/125, the [[Trisedodge family|coblack temperament]] tempers out the cloudy comma, 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave.


Commas: 126/125, 16807/16384
Optimal tunings:  
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}


POTE generator: ~21/20 = 73.044
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}


Map: [&lt;5 1 7 14|, &lt;0 3 2 0|]
Badness (Sintel): 1.19


EDOs: 15, 35, 50, 65
== 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.


Badness: 0.1073
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 126/125, 118098/117649
Commas: 126/125, 245/242, 385/384


POTE generator: ~21/20 = 73.264
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45


Map: [&lt;5 1 7 14 15|, &lt;0 3 2 0 1|]
[[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 }}


EDOs: 15, 35, 50, 65
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}


= Casablanca =
[[Badness]] (Sintel): 2.47
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]] 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.
=== 11-limit ===
Subgroup: 2.3.5.7.11


Commas: 126/125, 589824/588245
Comma list: 126/125, 540/539, 1344/1331


POTE generator: ~35/24 = 657.818
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}


Map: [&lt;1 12 10 5|, &lt;0 -19 -14 -4|]
Optimal tunings:  
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}


EDOs: 9bc, 11b, 31, 135c, 166c
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}


Badness: 0.1012
Badness (Sintel): 1.36


==11-limit==
=== 13-limit ===
Commas: 126/125, 385/384, 2420/2401
Subgroup: 2.3.5.7.11.13


POTE generator: ~16/11 = 657.923
Comma list: 126/125, 144/143, 196/195, 364/363


Map: [&lt;1 12 10 5 4|, |0 -19 -14 -4 -1&gt;]
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}


EDOs: 9bc, 11b, 31, 259bce, 549bce
Optimal tunings:  
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}


Badness: 0.0623
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}


==Marrakesh==
Badness (Sintel): 0.987
Commas: 126/125, 176/175, 14641/14580


POTE generator: ~22/15 = 657.791
== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''


Map: [&lt;1 12 10 5 21|, |0 -19 -14 -4 -32&gt;]
[[Subgroup]]: 2.3.5.7


EDOs: 9bce, 11be, 20be, 31, 42e, 73
[[Comma list]]: 126/125, 19683/19208


Badness: 0.0405
{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}


===13-limit===
[[Optimal tuning]]s:
126/125, 176/175, 196/195, 17303/17280
* [[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 }}


POTE generator: ~22/15 = 657.756
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}


Map: [&lt;1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25&gt;]
[[Badness]] (Sintel): 2.53


EDOs: 31, 73, 104c, 135c, 239cf
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0408
Comma list: 99/98, 126/125, 243/242


===Murakuc===
Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
Commas: 126/125, 144/143, 176/175, 1540/1521


POTE generator: ~22/15 = 657.700
Optimal tunings:  
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}


Map: [&lt;1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6&gt;]
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}


EDOs: 31, 73f, 104cf
Badness (Sintel): 1.41


Badness: 0.0414
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


= Nusecond =
Comma list: 66/65, 99/98, 126/125, 243/242
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]]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. 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.


==5-limit==
Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
Comma: 51018336/48828125


POTE generator: ~3125/2916 = 154.523
Optimal tunings:  
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}


Map: [&lt;1 3 4|, &lt;0 -11 -13|]
{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}


EDOs: 8, 23, 31, 70, 101, 132c, 233c, 365bc
Badness (Sintel): 1.56


Badness: 0.4665
== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''


==7-limit==
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.
[[Comma]]s: 126/125, 2430/2401


7-limit minimax
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.


[|1 0 0 0&gt;, |-5/13 0 11/13 0&gt;, |0 0 1 0&gt;, |-3/13 0 17/13 0&gt;]
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]].


[[Eigenmonzo]]s: 2, 5
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.


9-limit minimax
[[Subgroup]]: 2.3.5.7


[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]
[[Comma list]]: 126/125, 589824/588245


[[Eigenmonzo]]s: 2, 3
{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35


[[POTE_tuning|POTE generator]]: 154.579
[[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 }}


Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]
{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}


[[Generator]]s: 2, 49/45
[[Badness]] (Sintel): 2.56


EDOs: 7, 8, 31, 101, 132, 163
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0504
Comma list: 126/125, 385/384, 2420/2401


==11-limit==
Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}
[[Comma]]s: 99/98, 121/120, 126/125


11-limit minimax
Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}


[|1 0 0 0 0&gt;, |19/10 11/5 0 0 -11/10&gt;,
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
|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;<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 11/9
Badness (Sintel): 2.22


[[POTE_tuning|POTE generator]]: ~11/10 = 154.645
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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.
Comma list: 126/125, 196/195, 385/384, 2420/2401


Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]
Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}


[[Generator]]s: 2, 11/10
Optimal tunings:  
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}


EDOs: 7, 8, 31, 101, 194
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


Badness: 0.0256
Badness (Sintel): 2.31


==13-limit==
=== Marrakesh ===
Commas: 66/65 99/98 121/120 126/125
Subgroup: 2.3.5.7.11


POTE generator: ~11/10 = 154.478
Comma list: 126/125, 176/175, 14641/14580


Map: [&lt;1 3 4 5 5 5|, &lt;0 -11 -13 -17 -12 -10|]
Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}


EDOs: 31, 70f, 101f
Optimal tunings:  
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}


Badness: 0.0233
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}


=Thuja=
Badness (Sintel): 1.34
Commas: 126/125, 65536/64827


POTE generator: ~175/128 = 558.605
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 8 5 -2|, &lt;0 -12 -5 9|]
Comma list: 126/125, 176/175, 196/195, 14641/14580


Wedgie: &lt;&lt;12 5 -9 -20 -48 -35||
Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}


EDOs: 15, 43, 58
Optimal tunings:  
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}


Badness: 0.0884
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}


==11-limit==
Badness (Sintel): 1.68
Commas: 126/125, 176/175, 1344/1331


POTE generator: ~11/8 = 558.620
==== Murakuc ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 8 5 -2 4|, &lt;0 -12 -5 9 -1|]
Comma list: 126/125, 144/143, 176/175, 1540/1521


EDOs: 13, 15, 28, 43, 58
Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}


Badness: 0.0331
Optimal tunings:  
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}
Commas: 126/125, 144/143, 176/175, 364/363


POTE generator: ~11/8 = 558.589
Badness (Sintel): 1.71


Map: [&lt;1 8 5 -2 4 16|, &lt;0 -12 -5 9 -1 -23|]
== Amigo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''


EDOs: 15, 43, 58
[[Subgroup]]: 2.3.5.7


Badness: 0.0228
[[Comma list]]: 126/125, 2097152/2083725


==29-limit==
{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
POTE generator: ~11/8 = 558.520
: mapping generators: ~2, ~5/4


Map: [&lt;1 -4 0 7 3 -7 12 1 5 3|, &lt;0 12 5 -9 1 23 -17 7 -1 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 }}


EDOs: 43, 58
{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}


(''Raison d'etre'' of this entry being the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.)
[[Badness]] (Sintel): 2.81


=Cypress=
=== 11-limit ===
Comma: 258280326/244140625
Subgroup: 2.3.5.7.11


POTE generator: ~4374/3125 = 541.726
Comma list: 126/125, 176/175, 16384/16335


Map: [&lt;1 7 10|, &lt;0 -12 -17|]
Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}


EDOs: 20c, 31, 113c, 144c, 175c, 381bc
Optimal tunings:  
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}


Badness: 0.8166
{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}


==7-limit==
Badness (Sintel): 1.44
Commas: 126/125, 19683/19208


POTE generator: ~135/98 = 541.828
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 7 10 15|, &lt;0 -12 -17 -27|]
Comma list: 126/125, 169/168, 176/175, 364/363


Wedgie: &lt;&lt;12 17 27 -1 9 15||
Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}


EDOs: 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bcd
Optimal tunings:  
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}


Badness: 0.0998
{{Optimal ET sequence|legend=0| 43, 46, 89 }}


==11-limit==
Badness (Sintel): 1.27
Commas: 99/98, 126/125, 243/242


POTE generator: ~15/11 = 541.772
== 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.  


Map: [&lt;1 7 10 15 17|, &lt;0 -12 -17 -27 -30|]
[[Subgroup]]: 2.3.5.7


EDOs: 31, 144cd, 175cd, 206bcde, 237bcde
[[Comma list]]: 126/125, 17496/16807


Badness: 0.0427
{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270


==13-limit==
[[Optimal tuning]]s:
Commas: 66/65, 99/98. 126/125, 243/242
* [[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 }}


POTE generator: ~15/11 = 541.778
{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}


Map: [&lt;1 7 10 15 17 15|, &lt;0 -12 -17 -27 -30 -25|]
[[Badness]] (Sintel): 3.76


EDOs: 31
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0378
Comma list: 99/98, 126/125, 864/847


=Bisemidim=
Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
Commas: 126/125, 118098/117649


POTE generator: ~35/27 = 455.445
Optimal tunings:  
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}


Map: [&lt;2 1 2 2|, &lt;0 9 11 15|]
{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}


Wedgie: &lt;&lt;18 22 30 -7 -3 8||
Badness (Sintel): 1.97


EDOs: 50, 58, 108, 166c, 408c
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0978
Comma list: 78/77, 99/98, 126/125, 144/143


==11-limit==
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
Commas: 126/125, 540/539, 1344/1331


POTE generator: ~35/27 = 455.373
Optimal tunings:  
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}


Map: [&lt;2 1 2 2 5|, &lt;0 9 11 15 8|]
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


EDOs: 50, 58, 108, 166ce, 224ce
Badness (Sintel): 1.46


Badness: 0.0412
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 78/77, 99/98, 120/119, 126/125, 144/143
Commas: 126/125, 144/143, 196/195, 364/363


POTE generator: ~35/27 = 455.347
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}


Map: [&lt;2 1 2 2 5 5|, &lt;0 9 11 15 8 10|]
Optimal tunings:  
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}


EDOs: 50, 58, 166cef, 224cef
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


Badness: 0.0239
Badness (Sintel): 1.32


=Vines=
== Cobalt ==
Commas: 126/125, 84035/82944
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''


POTE generator: ~6/5 = 312.602
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 }}.  


Map: [&lt;2 7 8 8|, &lt;0 -8 -7 -5|]
Cobalt was named by [[Xenllium]] in 2022 after the 27th element.


EDOs: 4, 42, 46, 96d, 142d, 238d
[[Subgroup]]: 2.3.5.7


Badness: 0.0780
[[Comma list]]: 126/125, 40353607/40310784


==11-limit==
{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
Commas: 126/125, 385/384, 2401/2376
: mapping generators: ~36/35, ~3


POTE generator: ~6/5 = 312.601
[[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 }}


Map: [&lt;2 7 8 8 5|, &lt;0 -8 -7 -5 4|]
{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}


EDOs: 4, 42, 46, 96d, 142d, 238d
[[Badness]] (Sintel): 4.39


Badness: 0.0445
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 126/125, 540/539, 21609/21296
Commas: 126/125, 196/195, 364/363, 385/384


POTE generator: ~6/5 = 312.564
Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}


Map: [&lt;2 7 8 8 5 5|, &lt;0 -8 -7 -5 4 5|]
Optimal tunings:  
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}


EDOs: 4, 42, 46, 96d, 238df
{{Optimal ET sequence|legend=0| 27e, 81, 108 }}


Badness: 0.0297
Badness (Sintel): 2.58


=Kumonga=
==== 13-limit ====
Comma: 1289945088/1220703125
Subgroup: 2.3.5.7.11.13


POTE generator: ~144/125 = 222.912
Comma list: 126/125, 144/143, 196/195, 21609/21296


Map: [&lt;1 4 4|, &lt;0 -13 -9|]
Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}


EDOs: 16, 27, 43, 70, 183c
Optimal tunings:  
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}


Badness: 0.7296
{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}


==7-limit==
Badness (Sintel): 2.36
Commas: 126/125, 12288/12005


POTE generator: ~8/7 = 222.797
===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 4 4 3|, &lt;0 -13 -9 -1|]
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445


Wedgie: &lt;&lt;13 9 1 -16 -35 -23||
Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}


EDOs: 16, 27, 43, 70, 167cd
Optimal tunings:  
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}


Badness: 0.0875
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


==11-limit==
Badness (Sintel): 2.14
Commas: 126/125, 176/175, 864/847


POTE generator: ~8/7 = 222.898
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 4 4 3 7|, &lt;0 -13 -9 -1 -19|]
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968


EDOs: 16, 27e, 43, 70e
Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}


Badness: 0.0433
Optimal tunings:  
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}
Commas: 78/77, 126/125, 144/143, 176/175


POTE generator: ~8/7 = 222.961
Badness (Sintel): 1.85


Map: [&lt;1 4 4 3 7 5|, &lt;0 -13 -9 -1 -19 -7|]
===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17


EDOs: 16, 27e, 43, 70e, 113cde
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968


Badness: 0.0289
Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}


=Amigo=
Optimal tunings:
Commas: 126/125, 2097152/2083725
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}


POTE generator: ~5/4 = 391.094
{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}


Map: [&lt;1 9 3 -10|, &lt;0 -11 -1 19|]
Badness (Sintel): 2.40


EDOs: 43, 46, 89, 135c, 359c
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.1109
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083


==11-limit==
Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}
Commas: 126/125, 176/175, 16384/16335


POTE generator: ~5/4 = 391.075
Optimal tunings:  
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}


Map: [&lt;1 9 3 -10 -8|, &lt;0 -11 -1 19 17|]
{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}


EDOs: 43, 46, 89, 135c, 224c
Badness (Sintel): 2.08


Badness: 0.0434
==== Cobaltite ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 126/125, 169/168, 540/539, 975/968
Commas: 126/125, 169/168, 176/175, 364/363


POTE generator: ~5/4 = 391.072
Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}


Map: [&lt;1 9 3 -10 -8 1|, &lt;0 -11 -1 19 17 4|]
Optimal tunings:  
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}


EDOs: 43, 46, 89, 135cf, 224cf
{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}


Badness: 0.0307
Badness (Sintel): 2.18


=Oolong=
== References ==
{{main|Oolong}}
== 5-limit ==
Comma: [11 18 -17>


POTE generator: ~6/5 = 311.6942
[[Category:Starling temperaments| ]] <!-- main article -->
 
[[Category:Temperament collections]]
Map: [<1 6 7|, <0 -17 -18|]
[[Category:Catalogs of rank-2 temperaments]]
 
EDOs: 23, 27, 50, 77
 
Badness: 0.9428
 
==7-limit==
Commas: 126/125, 117649/116640
 
POTE generator: ~6/5 = 311.6793
 
Map: [&lt;1 6 7 8|, &lt;0 -17 -18 -20|]
 
EDOs: 27, 50, 77
 
Badness: 0.0735
 
== 11-limit ==
Commas: 126/125, 176/175, 26411/26244
 
POTE generator: ~6/5 = 311.5873
 
Map: [<1 6 7 8 18|, <0 -17 -18 -20 -56|]
 
EDOs: 27e, 77, 104c, 181c
 
Badness: 0.0569
 
== 13-limit ==
Commas: 126/125, 176/175, 196/195, 13013/12960
 
POTE generator: ~6/5 = 311.5908
 
Map: [<1 6 7 8 18 5|, <0 -17 -18 -20 -56 -5|]
 
EDOs: 27e, 77, 104c, 181c
 
Badness: 0.0356
 
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Starling]]
[[Category:Myna]]
[[Category:Listen]]
 
[[Category:Todo:improve layout]]
[[Category:Todo:review]]
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"