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Starling temperaments: Difference between revisions

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{{main| Myna }}
{{main| Myna }}


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 {{multival| 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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
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. 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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.


== 5-limit (mynic) ==
== 5-limit (mynic) ==
Line 31: Line 31:


Mapping generators: ~2, ~5/3
Mapping generators: ~2, ~5/3
{{Multival|legend=1| 10 9 7 -9 -17 -9 }}


[[POTE generator]]: ~6/5 = 310.146
[[POTE generator]]: ~6/5 = 310.146
Line 431: Line 433:


= Alicorn =
= Alicorn =
{{see also|Unicorn family #Alicorn}}
{{see also| Unicorn family #Alicorn }}
 
Subgroup: 2.3.5.7


Commas: 126/125, 10976/10935
[[Comma list]]: 126/125, 10976/10935


POTE generator: ~28/27 = 62.278
[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -8 -13 -23 }}]


Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
{{Multival|legend=1| 8 13 23 2 14 17 }}


Wedgie: &lt;&lt;8 13 23 2 14 17||
[[POTE generator]]: ~28/27 = 62.278


EDOs: {{EDOs|19, 39d, 58, 77, 135c}}
{{Val list|legend=1| 19, 39d, 58, 77, 135c }}


Badness: 0.0409
[[Badness]]: 0.0409


== 11-limit ==
== 11-limit ==
Commas: 126/125, 540/539, 896/891
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 540/539, 896/891
 
Mapping: [{{val| 1 2 3 4 3 }}, {{val| 0 -8 -13 -23 9 }}]


POTE generator: ~28/27 = 62.101
POTE generator: ~28/27 = 62.101


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
{{Val list|legend=1| 19, 39d, 58 }}
 
EDOs: {{EDOs|19, 39d, 58}}


Badness: 0.0392
Badness: 0.0392


=== 13-limit ===
=== 13-limit ===
Commas: 126/125, 144/143, 196/195, 676/675
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 196/195, 676/675
 
Mapping: [{{val| 1 2 3 4 3 5 }}, {{val| 0 -8 -13 -23 9 -25 }}]


POTE generator: ~28/27 = 62.119
POTE generator: ~28/27 = 62.119


Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
{{Val list|legend=1| 19, 39df, 58 }}
 
EDOs: {{EDOs|19, 39df, 58}}


Badness: 0.0237
Badness: 0.0237


== Camahueto ==
== Camahueto ==
Commas: 126/125, 10976/10935, 385/384
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 10976/10935
 
Mapping: [{{val| 1 2 3 4 2 }}, {{val| 0 -8 -13 -23 28 }}]


POTE generator: ~28/27 = 62.431
POTE generator: ~28/27 = 62.431


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
 
EDOs: {{EDOs|19, 58e, 77, 96d, 173d}}


Badness: 0.0659
Badness: 0.0659


=== 13-limit ===
=== 13-limit ===
Commas: 126/125, 196/195, 385/384, 676/675
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 385/384, 676/675
 
Mapping: [{{val| 1 2 3 4 2 5 }}, {{val| 0 -8 -13 -23 28 -25 }}]


POTE generator: ~28/27 = 62.434
POTE generator: ~28/27 = 62.434


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
 
EDOs: {{EDOs|19, 58e, 77, 96d, 173d}}


Badness: 0.0362
Badness: 0.0362
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{{see also|Trisedodge family #Coblack}}
{{see also|Trisedodge family #Coblack}}


In addition to 126/125, the 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.
In addition to 126/125, the 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.
 
Subgroup: 2.3.5.7


Commas: 126/125, 16807/16384
[[Comma list]]: 126/125, 16807/16384


POTE generator: ~21/20 = 73.044
[[POTE generator]]: ~21/20 = 73.044


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


EDOs: {{EDOs|15, 35, 50, 65, 115d}}
{{Val list|legend=1| 15, 35, 50, 65, 115d }}


Badness: 0.1073
[[Badness]]: 0.1073


==11-limit==
== 11-limit ==
Commas: 126/125, 245/242, 385/384
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 245/242, 385/384


POTE generator: ~21/20 = 73.264
POTE generator: ~21/20 = 73.264


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


EDOs: {{EDOs|15, 35, 50, 65, 115d}}
{{Val list|legend=1| 15, 35, 50, 65, 115d }}


= Casablanca =
= Casablanca =
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.
Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described 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.
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.


Commas: 126/125, 589824/588245
Subgroup: 2.3.5.7


POTE generator: ~35/24 = 657.818
[[Comma list]]: 126/125, 589824/588245


Map: [&lt;1 12 10 5|, &lt;0 -19 -14 -4|]
[[Mapping]]: [{{val| 1 12 10 5 }}, {{val| 0 -19 -14 -4 }}]


EDOs: {{EDOs|11b, 20b, 31, 104c, 135c, 166c}}
{{Multival|legend=1| 19 14 4 -22 -47 -30 }}


Badness: 0.1012
[[POTE generator]]: ~35/24 = 657.818


==11-limit==
{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
Commas: 126/125, 385/384, 2420/2401
 
[[Badness]]: 0.1012
 
== 11-limit ==
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 2420/2401
 
Mapping: [{{val| 1 12 10 5 4 }}, {{val| 0 -19 -14 -4 -1 }}]


POTE generator: ~16/11 = 657.923
POTE generator: ~16/11 = 657.923


Map: [&lt;1 12 10 5 4|, |0 -19 -14 -4 -1&gt;]
{{Val list|legend=1| 11b, 20b, 31 }}
 
EDOs: {{EDOs|11b, 20b, 31}}


Badness: 0.0623
Badness: 0.0623


== Marrakesh ==
== Marrakesh ==
Commas: 126/125, 176/175, 14641/14580
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 14641/14580
 
Mapping: [{{val| 1 12 10 5 21 }}, {{val| 0 -19 -14 -4 -32 }}]


POTE generator: ~22/15 = 657.791
POTE generator: ~22/15 = 657.791


Map: [&lt;1 12 10 5 21|, |0 -19 -14 -4 -32&gt;]
{{Val list|legend=1| 31, 73, 104c, 135c }}
 
EDOs: {{EDOs|31, 73, 104c, 135c}}


Badness: 0.0405
Badness: 0.0405


=== 13-limit ===
=== 13-limit ===
Commas: 126/125, 176/175, 196/195, 14641/14580
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195, 14641/14580
 
Mapping: [{{val| 1 12 10 5 21 -10 }}, {{val| 0 -19 -14 -4 -32 25 }}]


POTE generator: ~22/15 = 657.756
POTE generator: ~22/15 = 657.756


Map: [&lt;1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25&gt;]
{{Val list|legend=1| 31, 73, 104c, 135c, 239ccf }}
 
EDOs: {{EDOs|31, 73, 104c, 135c, 239ccf}}


Badness: 0.0408
Badness: 0.0408


=== Murakuc ===
=== Murakuc ===
Commas: 126/125, 144/143, 176/175, 1540/1521
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 176/175, 1540/1521
 
Mapping: [{{val| 1 12 10 5 21 7 }}, {{val| 0 -19 -14 -4 -32 -6 }}]


POTE generator: ~22/15 = 657.700
POTE generator: ~22/15 = 657.700


Map: [&lt;1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6&gt;]
{{Val list|legend=1| 31, 104cf, 135cf, 166c }}
 
EDOs: {{EDOs|31, 104cf, 135cf, 166c}}


Badness: 0.0414
Badness: 0.0414


= Nusecond =
= Nusecond =
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.
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;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. [[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 ==
== 5-limit ==
Comma: 51018336/48828125


POTE generator: ~3125/2916 = 154.523
Subgroup: 2.3.5


Map: [&lt;1 3 4|, &lt;0 -11 -13|]
[[Comma list]]: 51018336/48828125


EDOs: {{EDOs|8, 23, 31, 70, 101, 132c, 233c, 365bcc}}
[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -11 -13 }}]


Badness: 0.4665
[[POTE generator]]: ~3125/2916 = 154.523


==7-limit==
{{Val list|legend=1| 8, 23, 31, 70, 101, 132c, 233c, 365bcc }}
[[Comma]]s: 126/125, 2430/2401


7-limit minimax
[[Badness]]: 0.4665


[|1 0 0 0&gt;, |-5/13 0 11/13 0&gt;, |0 0 1 0&gt;, |-3/13 0 17/13 0&gt;]
== 7-limit ==


[[Eigenmonzo]]s: 2, 5
Subgroup: 2.3.5.7


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


[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]
[[Mapping]]: [{{val| 1 3 4 5 }}, {{val| 0 -11 -13 -17 }}]


[[Eigenmonzo]]s: 2, 3
Mapping generators: ~2, ~49/45


[[POTE_tuning|POTE generator]]: 154.579
{{Multival|legend=1| 11 13 17 -5 -4 3 }}


Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]
[[POTE generator]]: ~49/45 = 154.579


[[Generator]]s: 2, 49/45
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]
: [[Eigenmonzo]]s: 2, 5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]
: [[Eigenmonzo]]s: 2, 3


EDOs: {{EDOs|8d, 23d, 31, 101, 132c, 163c}}
{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


Badness: 0.0504
[[Badness]]: 0.0504


==11-limit==
== 11-limit ==
[[Comma]]s: 99/98, 121/120, 126/125


11-limit minimax
Subgroup: 2.3.5.7.11


[|1 0 0 0 0&gt;, |19/10 11/5 0 0 -11/10&gt;,
Comma list: 99/98, 121/120, 126/125
|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
Mapping: [{{val| 1 3 4 5 5 }}, {{val| 0 -11 -13 -17 -12 }}]


[[POTE_tuning|POTE generator]]: ~11/10 = 154.645
Mapping generators: ~2, ~11/10


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.
POTE generator: ~11/10 = 154.645


Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]
Minimax tuning:
* 11-odd-limit
: [{{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 }}]
: Eigenmonzos: 2, 11/9


[[Generator]]s: 2, 11/10
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: {{EDOs|8d, 23de, 31, 101, 132ce, 163ce, 194cee}}
{{Val list|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}


Badness: 0.0256
Badness: 0.0256


==13-limit==
== 13-limit ==
Commas: 66/65, 99/98, 121/120, 126/125
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 99/98, 121/120, 126/125
 
Mapping: [{{val| 1 3 4 5 5 5 }}, {{val| 0 -11 -13 -17 -12 -10 }}]


POTE generator: ~11/10 = 154.478
POTE generator: ~11/10 = 154.478


Map: [&lt;1 3 4 5 5 5|, &lt;0 -11 -13 -17 -12 -10|]
{{Val list|legend=1| 8d, 23de, 31, 70f, 101ff }}
 
Badness: 0.0233


EDOs: {{EDOs|8d, 23de, 31, 70f, 101ff}}
= Thuja =
Comma list: 126/125, 65536/64827


Badness: 0.0233
Mapping: [{{val| 1 8 5 -2 }}, {{val| 0 -12 -5 9 }}]


=Thuja=
{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}
Commas: 126/125, 65536/64827


POTE generator: ~175/128 = 558.605
POTE generator: ~175/128 = 558.605


Map: [&lt;1 8 5 -2|, &lt;0 -12 -5 9|]
{{Val list|legend=1| 15, 43, 58 }}


Wedgie: &lt;&lt;12 5 -9 -20 -48 -35||
Badness: 0.0884


EDOs: {{EDOs|15, 43, 58}}
== 11-limit ==


Badness: 0.0884
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 1344/1331


==11-limit==
Mapping: [{{val| 1 8 5 -2 4 }}, {{val| 0 -12 -5 9 -1 }}]
Commas: 126/125, 176/175, 1344/1331


POTE generator: ~11/8 = 558.620
POTE generator: ~11/8 = 558.620


Map: [&lt;1 8 5 -2 4|, &lt;0 -12 -5 9 -1|]
{{Val list|legend=1| 15, 43, 58 }}


EDOs: {{EDOs|15, 43, 58}}
Badness: 0.0331


Badness: 0.0331
== 13-limit ==
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 176/175, 364/363


==13-limit==
Mapping: [{{val| 1 8 5 -2 4 16 }}, {{val| 0 -12 -5 9 -1 -23 }}]
Commas: 126/125, 144/143, 176/175, 364/363


POTE generator: ~11/8 = 558.589
POTE generator: ~11/8 = 558.589


Map: [&lt;1 8 5 -2 4 16|, &lt;0 -12 -5 9 -1 -23|]
{{Val list|legend=1| 15, 43, 58 }}
 
Badness: 0.0228


EDOs: {{EDOs|15, 43, 58}}
== 29-limit ==


Badness: 0.0228
The ''raison d'etre'' of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.


==29-limit==
Subgroup: 2.3.5.7.11.13.17.19.23.29
POTE generator: ~11/8 = 558.520


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


EDOs: {{EDOs|43, 58hi}}
POTE generator: ~11/8 = 558.520


(''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.)
{{Val list|legend=1| 43, 58hi }}


= Cypress =
= Cypress =
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"