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 &lt;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
 - and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
@@ Line 29: / Line 29: @@
 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.
-==[[Minkowski_blocks|Minkowski blocks]]==
+== [[Minkowski blocks]] ==
 : 25/24, 81/80
@@ Line 62: / Line 62: @@
 : 15625/15552, 2048/2025
-==11-limit==
+== 11-limit ==
 Commas: 126/125, 385/384
@@ Line 73: / Line 73: @@
 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]]
 and 9 limit minimax
@@ Line 83: / Line 83: @@
 |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
@@ Line 93: / Line 93: @@
 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
@@ Line 101: / Line 101: @@
 Projection pairs: 7 125/18 11 3125/288
-Scales: [[thrush12|thrush12]]
+Scales: [[thrush12]]
-==13-limit==
+== 13-limit ==
 Commas: 126/125, 176/175, 196/195
@@ Line 112: / Line 112: @@
 Badness: 0.000677
-==Bluebird==
+== Bluebird ==
 Commas 126/125, 176/175, 144/143
@@ Line 123: / Line 123: @@
 Projection pairs: 7 125/18 11 3125/288 13 41472/3125
-==Nightingale==
+== Nightingale ==
 Commas: 126/125, 176/175, 66/65
@@ Line 132: / Line 132: @@
 Badness: 0.000837
-==Veery==
+== Veery ==
 Commas: 91/90, 126/125, 176/175
@@ Line 141: / Line 141: @@
 Badness: 0.000991
-=Thrasher=
+= Thrasher =
-[[Comma|Commas]]: 56/55, 100/99
+[[Comma]]s: 56/55, 100/99
 -limit minimax
@@ Line 150: / Line 150: @@
 |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
@@ Line 160: / Line 160: @@
 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
@@ Line 166: / Line 166: @@
 Badness: 0.000480
-==13-limit==
+== 13-limit ==
 Commas: 126/125, 100/99, 91/90
@@ Line 175: / Line 175: @@
 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|]
@@ Line 184: / Line 184: @@
 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|]
@@ Line 193: / Line 193: @@
 Badness: 0.000905
-=Aplonis=
+= Aplonis =
 Commas: 126/125, 540/539
@@ Line 202: / Line 202: @@
 Badness: 0.000648
-==13-limit==
+== 13-limit ==
 Commas: 126/125, 144/143, 196/195
@@ Line 211: / Line 211: @@
 Badness: 0.000821
-=Oxpecker=
+= Oxpecker =
 Commas: 121/120, 126/125
@@ Line 220: / Line 220: @@
 Badness: 0.000699
-==Woodpecker==
+== Woodpecker ==
 Commas: 66/65, 121/120, 126/125
@@ Line 229: / Line 229: @@
 Badness: 0.001093
-=Treecreeper=
+= Treecreeper =
 Commas: 126/125, 1232/1215
@@ Line 238: / Line 238: @@
 Badness: 0.001585
-==13-limit==
+== 13-limit ==
 Commas: 91/90, 126/125, 352/351
@@ Line 247: / Line 247: @@
 Badness: 0.001588
-=Cuckoo=
+= Cuckoo =
 Commas: 126/125, 243/242
 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
-==13-limit==
+== 13-limit ==
 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|]
-EDOs: 31 58 65f 89f 92e 154 185 216f
+EDOs: 31, 58, 65f, 89f, 92e, 154, 185, 216f
 [[Category:Theory]]
 [[Category:Temperament family]]
-[[Category:Planar temperament]]
 [[Category:Starling]]
+[[Category:Thrush]]
+[[Category:Rank 3]]
 [[Category:Listen]]
-[[Category:Thrush]]