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Starling family: Difference between revisions

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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 }}.
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.


Temperaments discussed elsewhere include [[erato]] (→ [[Didymus rank three family #Erato|Didymus rank-3 family]]) and [[sensigh]] (→ [[Sensamagic family #Sensigh|Sensamagic family]]). Considered below are starling, thrush, thrasher, aplonis, oxpecker, treecreeper, and cuckoo.  
Temperaments discussed elsewhere include [[erato]] (→ [[Didymus rank three family #Erato|Didymus rank-3 family]]) and [[sensigh]] (→ [[Sensamagic family #Sensigh|Sensamagic family]]). Considered below are starling, thrush, thrasher, aplonis, oxpecker, treecreeper, and cuckoo.  


== Starling ==
== Starling ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


[[Mapping 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|eigenmonzo (unchanged-interval) basis]]: 2.3.7


{{Optimal ET sequence|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 }}
Line 38: Line 38:
* [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by [[Jake Freivald]]. The melody depends on tempering out 126/125.
* [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;">
{{Databox|[[Minkowski blocks]]|
<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 55: Line 52:
* 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


{{Optimal ET sequence|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 }}
Line 72: Line 66:


== Thrush ==
== Thrush ==
Subgroup: 2.3.5.7.11
[[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


Mapping 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 }}]
Line 89: Line 81:
* 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|eigenmonzo (unchanged-interval) basis]]: 2.3.7


{{Optimal ET sequence|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 }}
Line 97: Line 89:
[[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]]
Scales: [[thrush12]]
Line 106: Line 98:
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 }}


{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
Line 117: Line 109:
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 }}


{{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
{{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
Line 130: Line 122:
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 }}


{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}
{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}
Line 141: Line 133:
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 }}


{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}
Line 148: Line 140:


== Thrasher ==
== Thrasher ==
Subgroup: 2.3.5.7.11
[[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 }}]
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
Line 165: Line 155:
* [[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|eigenmonzo (unchanged-interval) basis]]: 2.7.11/9


{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
Line 176: Line 166:
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 }}


{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
Line 189: Line 179:
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 }}


{{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
{{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
Line 202: Line 192:
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 }}


{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
Line 209: Line 199:


== 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 }}


{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
Line 224: Line 214:
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 }}


{{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
Line 235: Line 225:
[[Comma list]]: 121/120, 126/125
[[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 }}]
{{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}
 
: mapping generators: ~2, ~3, ~35/32


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
Line 246: Line 238:
Comma list: 66/65, 121/120, 126/125
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 }}]
Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 -1 }}


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}
Line 253: Line 245:


== 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 }}


{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
Line 268: Line 260:
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 }}


{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}
Line 275: Line 267:


== Cuckoo ==
== Cuckoo ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 126/125, 243/242
[[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 }}]
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
 
: mapping generators: ~2, ~11/9, ~5


{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
Line 290: Line 284:
Comma list: 126/125, 196/195, 243/242
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 }}]
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}


{{Optimal ET sequence|legend=1| 27e, 31, 58, 96d, 154 }}
{{Optimal ET sequence|legend=1| 27e, 31, 58, 96d, 154 }}
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