Starling family: Difference between revisions

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The head of the starling [[family|family]] is starling, which tempers out [[126/125|126/125]], the [[starling_comma|starling comma]] or [[septimal_semicomma|septimal semicomma]]. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo|77edo]]. Other possible tunings are [[108edo|108edo]] and [[185edo|185edo]], and the nonpatent [[135edo|135edo]] val <135 214 314 379|.
The head of the '''starling family''' is starling, which tempers out [[126/125]], the starling comma or septimal semicomma. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and the nonpatent [[135edo]] val {{val| 135 214 314 379 }}.


In starling, (6/5)^3 = 126/125 * 12/7, and minor thirds/major sixths are low complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5-6/5-6/5-7/6 versions of the diminished seventh chord which is very characteristic of it. 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.
In starling, (6/5)<sup>3</sup> = 126/125 × 12/7, and minor thirds/major sixths are low complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5-6/5-6/5-7/6 versions of the diminished seventh chord which is very characteristic of it. 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.


Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to [[Starling_temperaments|valentine temperament]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to [[Starling temperaments #Valentine|valentine temperament]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.


=7-limit starling=
= Starling =
[[Comma|Comma]] c = 126/125
[[Comma]] c = 126/125


7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
Line 29: Line 29:
Badness: 0.0000699
Badness: 0.0000699


[[Projection_pair|Projection pair]]: 7 125/18
[[Projection pair]]: 7 125/18


Scales: [[starling7|starling7]], [[starling8|starling8]], [[starling9|starling9]], [[starling11|starling11]], [[starling12|starling12]], [[starling15|starling15]], [[starling16|starling16]], [[starling17|starling17]], [[starling19|starling19]]
Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]


Music: [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by Jake Freivald. The melody depends on tempering out 126/125.
Music: [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by Jake Freivald. The melody depends on tempering out 126/125.


==[[Minkowski_blocks|Minkowski blocks]]==
== [[Minkowski blocks]] ==
7: 25/24, 81/80
7: 25/24, 81/80


Line 62: Line 62:
34: 15625/15552, 2048/2025
34: 15625/15552, 2048/2025


==11-limit==
== 11-limit ==
Commas: 126/125, 385/384
Commas: 126/125, 385/384


Line 73: Line 73:
Badness: .000677
Badness: .000677


=Thrush=
= Thrush =
[[Comma|Commas]]: 126/125, 176/175
[[Commas]]: 126/125, 176/175


Associated linear temperament: [[Starling_temperaments|myna]]
Associated linear temperament: [[Starling temperaments #Myna|myna]]


7 and 9 limit minimax
7 and 9 limit minimax
Line 83: Line 83:
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
[[Eigenmonzo]]s: 2, 7/6, 4/3


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
Lattice basis 5/4 length 0.8576 6/5 length 0.9314
Line 93: Line 93:
Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]


[[generator|Generators]]: 2, 3, 5
[[Generator]]s: 2, 3, 5


EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
Line 101: Line 101:
Projection pairs: 7 125/18 11 3125/288
Projection pairs: 7 125/18 11 3125/288


Scales: [[thrush12|thrush12]]
Scales: [[thrush12]]


==13-limit==
== 13-limit ==
Commas: 126/125, 176/175, 196/195
Commas: 126/125, 176/175, 196/195


Line 112: Line 112:
Badness: 0.000677
Badness: 0.000677


==Bluebird==
== Bluebird ==
Commas 126/125, 176/175, 144/143
Commas 126/125, 176/175, 144/143


Line 123: Line 123:
Projection pairs: 7 125/18 11 3125/288 13 41472/3125
Projection pairs: 7 125/18 11 3125/288 13 41472/3125


==Nightingale==
== Nightingale ==
Commas: 126/125, 176/175, 66/65
Commas: 126/125, 176/175, 66/65


Line 132: Line 132:
Badness: 0.000837
Badness: 0.000837


==Veery==
== Veery ==
Commas: 91/90, 126/125, 176/175
Commas: 91/90, 126/125, 176/175


Line 141: Line 141:
Badness: 0.000991
Badness: 0.000991


=Thrasher=
= Thrasher =
[[Comma|Commas]]: 56/55, 100/99
[[Comma]]s: 56/55, 100/99


11-limit minimax
11-limit minimax
Line 150: Line 150:
|2 -1/2 0 1/2 1/4&gt;]
|2 -1/2 0 1/2 1/4&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9
[[Eigenmonzo]]s: 2, 8/7, 11/9


Lattice basis: 6/5 length 0.9089 5/4 length 1.2007
Lattice basis: 6/5 length 0.9089 5/4 length 1.2007
Line 160: Line 160:
Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]


[[generator|Generators]]: 2, 3, 5
[[Generator]]s: 2, 3, 5


EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
Line 166: Line 166:
Badness: 0.000480
Badness: 0.000480


==13-limit==
== 13-limit ==
Commas: 126/125, 100/99, 91/90
Commas: 126/125, 100/99, 91/90


Line 175: Line 175:
Badness: 0.000876
Badness: 0.000876


==Mockingbird==
== Mockingbird ==
Commas: 126/125, 100/99, 40/39
Commas: 40/39, 100/99, 126/125


Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
Line 184: Line 184:
Badness: 0.000859
Badness: 0.000859


==Catbird==
== Catbird ==
Commas: 126/125, 100/99, 78/77
Commas: 78/77, 100/99, 126/125


Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
Line 193: Line 193:
Badness: 0.000905
Badness: 0.000905


=Aplonis=
= Aplonis =
Commas: 126/125, 540/539
Commas: 126/125, 540/539


Line 202: Line 202:
Badness: 0.000648
Badness: 0.000648


==13-limit==
== 13-limit ==
Commas: 126/125, 144/143, 196/195
Commas: 126/125, 144/143, 196/195


Line 211: Line 211:
Badness: 0.000821
Badness: 0.000821


=Oxpecker=
= Oxpecker =
Commas: 121/120, 126/125
Commas: 121/120, 126/125


Line 220: Line 220:
Badness: 0.000699
Badness: 0.000699


==Woodpecker==
== Woodpecker ==
Commas: 66/65, 121/120, 126/125
Commas: 66/65, 121/120, 126/125


Line 229: Line 229:
Badness: 0.001093
Badness: 0.001093


=Treecreeper=
= Treecreeper =
Commas: 126/125, 1232/1215
Commas: 126/125, 1232/1215


Line 238: Line 238:
Badness: 0.001585
Badness: 0.001585


==13-limit==
== 13-limit ==
Commas: 91/90, 126/125, 352/351
Commas: 91/90, 126/125, 352/351


Line 247: Line 247:
Badness: 0.001588
Badness: 0.001588


=Cuckoo=
= Cuckoo =
Commas: 126/125, 243/242
Commas: 126/125, 243/242


Map: [&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|]
Map: [&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|]


EDOs: 31 34 58 65 89 154 185 216
EDOs: 31, 34, 58, 65, 89, 154, 185, 216


Badness: 0.000933
Badness: 0.000933


==13-limit==
== 13-limit ==
Commas: 126/125, 196/195, 243/242
Commas: 126/125, 196/195, 243/242


Map: [&lt;1 1 0 -3 2 -5|, &lt;0 2 0 -4 5 -10|, &lt;0 0 1 3 0 5|]
Map: [&lt;1 1 0 -3 2 -5|, &lt;0 2 0 -4 5 -10|, &lt;0 0 1 3 0 5|]


EDOs: 31 58 65f 89f 92e 154 185 216f
EDOs: 31, 58, 65f, 89f, 92e, 154, 185, 216f


[[Category:Theory]]
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Planar temperament]]
[[Category:Starling]]
[[Category:Starling]]
[[Category:Thrush]]
[[Category:Rank 3]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Thrush]]

Revision as of 06:17, 6 January 2021

The head of the starling family is starling, which tempers out 126/125, the starling comma or septimal semicomma. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by 77edo. Other possible tunings are 108edo and 185edo, and the nonpatent 135edo val ⟨135 214 314 379].

In starling, (6/5)3 = 126/125 × 12/7, and minor thirds/major sixths are low complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5-6/5-6/5-7/6 versions of the diminished seventh chord which is very characteristic of it. 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.

Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to valentine temperament, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.

Starling

Comma c = 126/125

7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp

[<1 0 0 0|, <0 1 0 0|, <1/3 2/3 0 1/3|, <0 0 0 1|]

Eigenmonzos: 2, 8/7, 4/3

Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206

Angle(6/5, 5/4) = 100.364 degrees

Map to lattice: [<0 1 0 -2|, <0 1 1 1|]

Map: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]

Generators: 2, 3, 5

EDOs: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, 77, 108, 166c, 185, 243c

Badness: 0.0000699

Projection pair: 7 125/18

Scales: starling7, starling8, starling9, starling11, starling12, starling15, starling16, starling17, starling19

Music: A Seed Planted by Jake Freivald. The melody depends on tempering out 126/125.

Minkowski blocks

7: 25/24, 81/80

8: 16/15, 648/625

9: 27/25, 128/125

11: 16/15, 15625/15552

12: 128/125, 628/625

15: 128/125, 250/243

16: 648/625, 3125/3072

17: 25/24, 20480/19683

19: 81/80, 3125/3072

27: 128/125, 78732/78125

28: 648/625, 16875/16384

31: 81/80, 1990656/1953125

34: 15625/15552, 2048/2025

11-limit

Commas: 126/125, 385/384

Map: [<1 0 0 -1 8|, <0 1 0 -2 3|, <0 0 1 3 -4]]

Generators: 2, 3, 5

EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de

Badness: .000677

Thrush

Commas: 126/125, 176/175

Associated linear temperament: myna

7 and 9 limit minimax

[|1 0 0 0 0>, |0 1 0 0 0>, |1/3 2/3 0 1/3 0>, |0 0 0 1 0>, |-10/3 4/3 0 5/3 0>]

Eigenmonzos: 2, 7/6, 4/3

Lattice basis 5/4 length 0.8576 6/5 length 0.9314

Angle(5/4, 6/5) = 74.6239 degrees

Map to lattice: [<0 1 1 1 3|, <0 1 0 -2 -2|]

Map: [<1 0 0 -1 -5|, <0 1 0 -2 -2|, <0 0 1 3 5|]

Generators: 2, 3, 5

EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c

Badness: 0.000353

Projection pairs: 7 125/18 11 3125/288

Scales: thrush12

13-limit

Commas: 126/125, 176/175, 196/195

Map: [<1 0 0 -1 -5 0|, <0 1 0 -2 -2 -5|, <0 0 1 3 5 5|]

EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf

Badness: 0.000677

Bluebird

Commas 126/125, 176/175, 144/143

Map: [<1 0 0 -1 -5 9|, <0 1 0 -2 -2 4|, <0 0 1 3 5 -5|]

EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef

Badness: 0.000915

Projection pairs: 7 125/18 11 3125/288 13 41472/3125

Nightingale

Commas: 126/125, 176/175, 66/65

Map: [<1 0 0 -1 -5 -4|, <0 1 0 -2 -2 -1|, <0 0 1 3 5 4|]

EDOs: 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef

Badness: 0.000837

Veery

Commas: 91/90, 126/125, 176/175

Map: [<1 0 0 -1 -5 2|, <0 1 0 -2 -2 4|, <0 0 1 3 5 -2|]

EDOs: 12, 15, 46

Badness: 0.000991

Thrasher

Commas: 56/55, 100/99

11-limit minimax

[|1 0 0 0 0>, |1 3/4 0 1/4 -3/8>, |1 1/2 0 1/2 -1/4>, |0 0 0 1 0>, |2 -1/2 0 1/2 1/4>]

Eigenmonzos: 2, 8/7, 11/9

Lattice basis: 6/5 length 0.9089 5/4 length 1.2007

Angle(6/5, 5/4) = 98.8447

Map to lattice: [<0 1 0 -2 -2|, <0 1 1 1 0|]

Map: [<1 0 0 -1 2|, <0 1 0 -2 -2|, <0 0 1 3 2|]

Generators: 2, 3, 5

EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e

Badness: 0.000480

13-limit

Commas: 126/125, 100/99, 91/90

Map: [<1 0 0 -1 2 2|, <0 1 0 -2 -2 4|, <0 0 1 3 2 -2|]

EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e

Badness: 0.000876

Mockingbird

Commas: 40/39, 100/99, 126/125

Map: [<1 0 0 -1 2 3|, <0 1 0 -2 -2 -1|, <0 0 1 3 2 1|]

EDOs: 12f, 15, 27ef

Badness: 0.000859

Catbird

Commas: 78/77, 100/99, 126/125

Map: [<1 0 0 -1 2 0|, <0 1 0 -2 -2 -5|, <0 0 1 3 2 5|]

EDOs: 12, 15, 17c, 19, 22ef, 27e, 34, 58e

Badness: 0.000905

Aplonis

Commas: 126/125, 540/539

Map: [<1 0 0 -1 4|, <0 1 0 -2 7|, <0 0 1 3 -5|]

EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce

Badness: 0.000648

13-limit

Commas: 126/125, 144/143, 196/195

Map: [<1 0 0 -1 4 0|, <0 1 0 -2 7 -5|, <0 0 1 3 -5 5|]

EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef

Badness: 0.000821

Oxpecker

Commas: 121/120, 126/125

Map: [<1 0 1 2 2|, <0 1 1 1 1|, <0 0 -2 -6 -1|]

EDOs: 8d, 15, 31, 46, 77, 185e, 262cde

Badness: 0.000699

Woodpecker

Commas: 66/65, 121/120, 126/125

Map: [<1 0 1 2 2 2|, <0 1 1 1 1 1|, <0 0 -2 -6 -1 -1|]

EDOs: 15, 31, 46f, 70f, 101f

Badness: 0.001093

Treecreeper

Commas: 126/125, 1232/1215

Map: [<1 0 0 -1 -3|, <0 1 0 -2 7|, <0 0 1 3 -2|]

EDOs: 34, 46, 119c, 165c

Badness: 0.001585

13-limit

Commas: 91/90, 126/125, 352/351

Map: [<1 0 0 -1 -3 2|, <0 1 0 -2 7 4|, <0 0 1 3 -2 -2|]

EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf

Badness: 0.001588

Cuckoo

Commas: 126/125, 243/242

Map: [<1 1 0 -3 2|, <0 2 0 -4 5|, <0 0 1 3 0|]

EDOs: 31, 34, 58, 65, 89, 154, 185, 216

Badness: 0.000933

13-limit

Commas: 126/125, 196/195, 243/242

Map: [<1 1 0 -3 2 -5|, <0 2 0 -4 5 -10|, <0 0 1 3 0 5|]

EDOs: 31, 58, 65f, 89f, 92e, 154, 185, 216f

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