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Starling family: Difference between revisions

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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.
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|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.


== Starling  ==
== Starling  ==
Line 10: Line 10:
[[Comma list]]: [[126/125]]
[[Comma list]]: [[126/125]]


[[Mapping]]: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]
[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]


Mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]


Minkowski lattice basis:  
Minkowski lattice basis:  
Line 22: Line 22:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3c sharp
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3c sharp
: [&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]
: [{{val| 1 0 0 0 }}, {{val| 0 1 0 0 }}, {{val| 1/3 2/3 0 1/3 }}, {{val| 0 0 0 1 }}]
: Eigenmonzos: 2, 8/7, 4/3
: Eigenmonzos: 2, 8/7, 4/3


Line 61: Line 61:
[[Comma list]]: 126/125, 385/384
[[Comma list]]: 126/125, 385/384


[[Mapping]]: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
[[Mapping]]: [{{val| 1 0 0 -1 8 }}, {{val| 0 1 0 -2 3 }}, {{val| 0 0 1 3 -4]]


Mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5
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[[Comma list]]: 126/125, 176/175
[[Comma list]]: 126/125, 176/175


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


Mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


[[Minimax tuning]]:
Map to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
* 7- and [[9-odd-limit]]
: [|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;, |0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
: [[Eigenmonzo]]s: 2, 7/6, 4/3
 
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]


Lattice basis:  
Lattice basis:  
: 5/4 length = 0.8576, 6/5 length = 0.9314
: 5/4 length = 0.8576, 6/5 length = 0.9314
: Angle(5/4, 6/5) = 74.6239 degrees
: Angle(5/4, 6/5) = 74.6239 degrees
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1/3 2/3 0 1/3 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -10/3 4/3 0 5/3 0 }}]
: [[Eigenmonzo]]s: 2, 7/6, 4/3


{{Val list|legend=1| 12, 15, 31, 46, 58, 89, 135c, 224c }}
{{Val list|legend=1| 12, 15, 31, 46, 58, 89, 135c, 224c }}
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Comma list: 126/125, 176/175, 196/195
Comma list: 126/125, 176/175, 196/195


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


Vals: {{Val list| 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf }}
Vals: {{Val list| 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf }}
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Comma list: 126/125, 176/175, 144/143
Comma list: 126/125, 176/175, 144/143


Mapping: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -5|]
Mapping: [{{val| 1 0 0 -1 -5 9 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 0 0 1 3 5 -5 }}]


Vals: {{Val list| 12, 15, 31, 43, 48c, 58, 147cf, 205cef }}
Vals: {{Val list| 12, 15, 31, 43, 48c, 58, 147cf, 205cef }}
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Comma list: 126/125, 176/175, 66/65
Comma list: 126/125, 176/175, 66/65


Mapping: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]
Mapping: [{{val| 1 0 0 -1 -5 -4 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 0 0 1 3 5 4 }}]


Vals: {{Val list| 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef }}
Vals: {{Val list| 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef }}
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Comma list: 91/90, 126/125, 176/175
Comma list: 91/90, 126/125, 176/175


Mapping: [&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]
Mapping: [{{val| 1 0 0 -1 -5 2 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 0 0 1 3 5 -2 }}]


Vals: {{Val list| 12, 15, 46 }}
Vals: {{Val list| 12, 15, 46 }}
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[[Comma list]]: 56/55, 100/99
[[Comma list]]: 56/55, 100/99


[[Mapping]]: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
[[Mapping]]: [{{val| 1 0 0 -1 2 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 3 2 }}]


Mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]
Map to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]


Lattice basis:  
Lattice basis:  
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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;, |1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;, |2 -1/2 0 1/2 1/4&gt;]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
: [[Eigenmonzo]]s: 2, 8/7, 11/9
: [[Eigenmonzo]]s: 2, 8/7, 11/9


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Comma list: 126/125, 100/99, 91/90
Comma list: 126/125, 100/99, 91/90


Mapping: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]
Mapping: [{{val| 1 0 0 -1 2 2 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 0 0 1 3 2 -2 }}]


Vals: {{Val list| 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e }}
Vals: {{Val list| 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e }}
Line 185: Line 185:
Comma list: 40/39, 100/99, 126/125
Comma list: 40/39, 100/99, 126/125


Mapping: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
Mapping: [{{val| 1 0 0 -1 2 3 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 0 0 1 3 2 1 }}]


Vals: {{Val list| 12f, 15, 27ef }}
Vals: {{Val list| 12f, 15, 27ef }}
Line 196: Line 196:
Comma list: 78/77, 100/99, 126/125
Comma list: 78/77, 100/99, 126/125


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


Vals: {{Val list| 12, 15, 17c, 19, 22ef], 27e, 34, 58e }}
Vals: {{Val list| 12, 15, 17c, 19, 22ef], 27e, 34, 58e }}
Line 207: Line 207:
[[Comma list]]: 126/125, 540/539
[[Comma list]]: 126/125, 540/539


[[Mapping]]: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
[[Mapping]]: [{{val| 1 0 0 -1 4 }}, {{val| 0 1 0 -2 7 }}, {{val| 0 0 1 3 -5 }}]


{{Val list|legend=1| 19, 31, 58, 67, 89, 197c, 286ce, 375ce }}
{{Val list|legend=1| 19, 31, 58, 67, 89, 197c, 286ce, 375ce }}
Line 218: Line 218:
Comma list: 126/125, 144/143, 196/195
Comma list: 126/125, 144/143, 196/195


Mapping: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]
Mapping: [{{val| 1 0 0 -1 4 0 }}, {{val| 0 1 0 -2 7 -5 }}, {{val| 0 0 1 3 -5 5 }}]


Vals: {{Val list| 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef }}
Vals: {{Val list| 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef }}
Line 229: Line 229:
[[Comma list]]: 121/120, 126/125
[[Comma list]]: 121/120, 126/125


[[Mapping]]: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
[[Mapping]]: [{{val| 1 0 1 2 2 }}, {{val| 0 1 1 1 1 }}, {{val| 0 0 -2 -6 -1 }}]


{{Val list|legend=1| 8d, 15, 31, 46, 77, 185e, 262cde }}
{{Val list|legend=1| 8d, 15, 31, 46, 77, 185e, 262cde }}
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Comma list: 66/65, 121/120, 126/125
Comma list: 66/65, 121/120, 126/125


Mapping: [&lt;1 0 1 2 2 2|, &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]
Mapping: [{{val| 1 0 1 2 2 2 }}, {{val| 0 1 1 1 1 1 }}, {{val| 0 0 -2 -6 -1 -1 }}]


Vals: {{Val list| 15, 31, 46f, 70f, 101f }}
Vals: {{Val list| 15, 31, 46f, 70f, 101f }}
Line 251: Line 251:
[[Comma list]]: 126/125, 1232/1215
[[Comma list]]: 126/125, 1232/1215


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


{{Val list|legend=1| 34, 46, 119c, 165c }}
{{Val list|legend=1| 34, 46, 119c, 165c }}
Line 262: Line 262:
Comma list: 91/90, 126/125, 352/351
Comma list: 91/90, 126/125, 352/351


Mapping: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
Mapping: [{{val| 1 0 0 -1 -3 2 }}, {{val| 0 1 0 -2 7 4 }}, {{val| 0 0 1 3 -2 -2 }}]


Vals: {{Val list| 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf }}
Vals: {{Val list| 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf }}
Line 273: Line 273:
[[Comma list]]: 126/125, 243/242
[[Comma list]]: 126/125, 243/242


[[Mapping]]: [&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|]
[[Mapping]]: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 0 -4 5 }}, {{val| 0 0 1 3 0 }}]


{{Val list|legend=1| 31, 34, 58, 65, 89, 154, 185, 216 }}
{{Val list|legend=1| 31, 34, 58, 65, 89, 154, 185, 216 }}
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Comma list: 126/125, 196/195, 243/242
Comma list: 126/125, 196/195, 243/242


Mapping: [&lt;1 1 0 -3 2 -5|, &lt;0 2 0 -4 5 -10|, &lt;0 0 1 3 0 5|]
Mapping: [{{val| 1 1 0 -3 2 -5 }}, {{val| 0 2 0 -4 5 -10 }}, {{val| 0 0 1 3 0 5 }}]


Vals: {{Val list| 31, 58, 65f, 89f, 92e, 154, 185, 216f }}
Vals: {{Val list| 31, 58, 65f, 89f, 92e, 154, 185, 216f }}
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