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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 }}.
{{Technical data page}}
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 }} (135c).


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 [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.


== Starling ==
Temperaments discussed elsewhere include
Subgroup: 2.3.5.7
* ''[[Erato]]'' (+81/80) → [[Didymus rank three family #Erato|Didymus rank-3 family]]
* ''[[Sensigh]]'' (+245/243) → [[Sensamagic family #Sensigh|Sensamagic family]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank three clan #Cuckoo|Rastmic rank-3 clan]]
 
Considered below are starling, thrush, thrasher, aplonis, and treecreeper.
 
== Starling ==
{{Main| Starling and thrush }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[126/125]]
[[Comma list]]: [[126/125]]


[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}


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


Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]


Minkowski lattice basis:  
[[Minkowski lattice basis]]:  
: 6/5 length = 1.068, 5/4 length = 1.206
: 6/5 length = 1.068, 5/4 length = 1.206
: Angle (6/5, 5/4) = 100.364 degrees
: Angle (6/5, 5/4) = 100.364 degrees


[[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 1/3-comma sharp
: [{{val| 1 0 0 0 }}, {{val| 0 1 0 0 }}, {{val| 1/3 2/3 0 1/3 }}, {{val| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
: Eigenmonzos (unchanged intervals): 2, 8/7, 4/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


{{Val list|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}


[[Badness]]: 0.0699 × 10<sup>-3</sup>
[[Badness]]: 0.0699 × 10<sup>-3</sup>
Line 31: Line 42:
[[Projection pair]]: 7 125/18
[[Projection pair]]: 7 125/18


Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
{{Databox|[[Minkowski blocks]]|
 
; Music
* [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by [[Jake Freivald]]. The melody depends on tempering out 126/125.
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
<div class="mw-collapsible-content">
 
* 7: 25/24, 81/80
* 7: 25/24, 81/80
* 8: 16/15, 648/625
* 8: 16/15, 648/625
Line 53: Line 56:
* 31: 81/80, 1990656/1953125
* 31: 81/80, 1990656/1953125
* 34: 15625/15552, 2048/2025
* 34: 15625/15552, 2048/2025
 
}}
</div></div>


== Undecimal starling ==
== Undecimal starling ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 0 -1 8 }}, {{val| 0 1 0 -2 3 }}, {{val| 0 0 1 3 -4]]
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
 
Mapping generators: ~2, ~3, ~5


{{Val list|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}


[[Badness]]: 0.677 × 10<sup>-3</sup>
[[Badness]]: 0.677 × 10<sup>-3</sup>


== Thrush ==
== Thrush ==
Subgroup: 2.3.5.7.11
{{Main| Starling and thrush }}
 
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 126/125, 176/175
[[Comma list]]: 126/125, 176/175


[[Mapping]]: [{{val| 1 0 0 -1 -5 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 3 5 }}]
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
 
Mapping generators: ~2, ~3, ~5


Map to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]


Lattice basis:  
Lattice basis:  
Line 87: Line 87:
* 7- and [[9-odd-limit]]
* 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 }}]
: [{{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 (unchanged intervals): 2, 7/6, 4/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


{{Val list|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}


[[Badness]]: 0.353 × 10<sup>-3</sup>
[[Badness]]: 0.353 × 10<sup>-3</sup>
Line 95: Line 95:
[[Projection pair]]s: 7 125/18 11 3125/288
[[Projection pair]]s: 7 125/18 11 3125/288


[[Associated temperament]]: [[Starling temperaments #Myna|myna]]
[[Associated temperament]]: [[myna]]
 
Scales: [[thrush12]]


=== 13-limit ===
=== 13-limit ===
Line 104: Line 102:
Comma list: 126/125, 176/175, 196/195
Comma list: 126/125, 176/175, 196/195


Mapping: [{{val| 1 0 0 -1 -5 0 }}, {{val| 0 1 0 -2 -2 -5 }}, {{val| 0 0 1 3 5 5 }}]
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}


Vals: {{Val list| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}


Badness: 0.677 × 10<sup>-3</sup>
Badness: 0.677 × 10<sup>-3</sup>
Line 115: Line 113:
Comma list: 126/125, 144/143, 176/175
Comma list: 126/125, 144/143, 176/175


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


Vals: {{Val list| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
{{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}


Badness: 0.915 × 10<sup>-3</sup>
Badness: 0.915 × 10<sup>-3</sup>
Line 128: Line 126:
Comma list: 66/65, 126/125, 176/175
Comma list: 66/65, 126/125, 176/175


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


Vals: {{Val list| 12f, 15, 19e, 27eff, 31 }}
{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}


Badness: 0.837 × 10<sup>-3</sup>
Badness: 0.837 × 10<sup>-3</sup>
Line 139: Line 137:
Comma list: 91/90, 126/125, 176/175
Comma list: 91/90, 126/125, 176/175


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


Vals: {{Val list| 12, 15, 19e, 27e, 31f, 46 }}
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}


Badness: 0.991 × 10<sup>-3</sup>
Badness: 0.991 × 10<sup>-3</sup>


== Thrasher ==
== Thrasher ==
Subgroup: 2.3.5.7.11
{{See also| Ptolemismic clan #Thrasher }}
 
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 56/55, 100/99
[[Comma list]]: 56/55, 100/99


[[Mapping]]: [{{val| 1 0 0 -1 2 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 3 2 }}]
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}


Mapping generators: ~2, ~3, ~5
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
 
Map to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]


Lattice basis:  
Lattice basis:  
Line 163: Line 161:
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{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 }}]
: [{{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 (unchanged intervals): 2, 8/7, 11/9
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9


{{Val list|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>


<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]


[[Badness]]: 0.480 × 10<sup>-3</sup>
[[Badness]]: 0.480 × 10<sup>-3</sup>
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 100/99, 126/125
Comma list: 56/55, 91/90, 100/99


Mapping: [{{val| 1 0 0 -1 2 2 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 0 0 1 3 2 -2 }}]
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}


Vals: {{Val list| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>


<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
Line 187: Line 187:
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 40/39, 100/99, 126/125
Comma list: 40/39, 56/55, 100/99


Mapping: [{{val| 1 0 0 -1 2 3 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 0 0 1 3 2 1 }}]
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}


Vals: {{Val list| 7d, 8d, 12f, 15, 27eff }}
{{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}


Badness: 0.859 × 10<sup>-3</sup>
Badness: 0.859 × 10<sup>-3</sup>
Line 200: Line 200:
Comma list: 78/77, 100/99, 126/125
Comma list: 78/77, 100/99, 126/125


Mapping: [{{val| 1 0 0 -1 2 0 }}, {{val| 0 1 0 -2 -2 -5 }}, {{val| 0 0 1 3 2 5 }}]
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}


Vals: {{Val list| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}


Badness: 0.905 × 10<sup>-3</sup>
Badness: 0.905 × 10<sup>-3</sup>


== Aplonis ==
== Aplonis ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 126/125, 540/539
[[Comma list]]: 126/125, 540/539


[[Mapping]]: [{{val| 1 0 0 -1 4 }}, {{val| 0 1 0 -2 7 }}, {{val| 0 0 1 3 -5 }}]
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}


{{Val list|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}


[[Badness]]: 0.648 × 10<sup>-3</sup>
[[Badness]]: 0.648 × 10<sup>-3</sup>
Line 222: Line 222:
Comma list: 126/125, 144/143, 196/195
Comma list: 126/125, 144/143, 196/195


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


Vals: {{Val list| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}


Badness: 0.821 × 10<sup>-3</sup>
Badness: 0.821 × 10<sup>-3</sup>
== Oxpecker ==
Subgroup: 2.3.5.7.11
[[Comma list]]: 121/120, 126/125
[[Mapping]]: [{{val| 1 0 1 2 2 }}, {{val| 0 1 1 1 1 }}, {{val| 0 0 -2 -6 -1 }}]
{{Val list|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
[[Badness]]: 0.699 × 10<sup>-3</sup>
=== Woodpecker ===
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 121/120, 126/125
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| 7d, 8d, 15, 23de, 24d, 31 }}
Badness: 1.093 × 10<sup>-3</sup>


== Treecreeper ==
== Treecreeper ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 126/125, 1232/1215
[[Comma list]]: 126/125, 1232/1215


[[Mapping]]: [{{val| 1 0 0 -1 -3 }}, {{val| 0 1 0 -2 7 }}, {{val| 0 0 1 3 -2 }}]
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}


{{Val list|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}


[[Badness]]: 1.585 × 10<sup>-3</sup>
[[Badness]]: 1.585 × 10<sup>-3</sup>
Line 266: Line 244:
Comma list: 91/90, 126/125, 352/351
Comma list: 91/90, 126/125, 352/351


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


Vals: {{Val list| 7d, 12e, 19e, 27e, 46 }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}


Badness: 1.588 × 10<sup>-3</sup>
Badness: 1.588 × 10<sup>-3</sup>


== Cuckoo ==
[[Category:Temperament families]]
Subgroup: 2.3.5.7.11
[[Category:Pages with mostly numerical content]]
 
[[Comma list]]: 126/125, 243/242
 
[[Mapping]]: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 0 -4 5 }}, {{val| 0 0 1 3 0 }}]
 
{{Val list|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
 
[[Badness]]: 0.933 × 10<sup>-3</sup>
 
=== 13-limit  ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 243/242
 
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| 27e, 31, 58, 96d, 154 }}
 
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Starling family| ]] <!-- main article -->
[[Category:Thrush]]
[[Category:Starling| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Starling_family"