Starling family: Difference between revisions

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

* ''[[Erato]]'' (+81/80) → [[Didymus rank three family #Erato|Didymus rank-3 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.

{{Main| Starling and thrush }}

[[Subgroup]]: 2.3.5.7

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

 

{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}

 

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

 

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

 

[[Minkowski lattice basis]]:

: 6/5 length = 1.068, 5/4 length = 1.206

 

[[Minimax tuning]]:

* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp

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

 

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

 

[[Badness]]: 0.0699 × 10^-3

{{Databox|[[Minkowski blocks]]|

* 12: 128/125, 628/625

 

[[Subgroup]]: 2.3.5.7.11

 

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

 

 

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

 

[[Badness]]: 0.677 × 10^-3

 

 

[[Subgroup]]: 2.3.5.7.11

 

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

 

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

: 5/4 length = 0.8576, 6/5 length = 0.9314

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

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

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

[[Badness]]: 0.353 × 10^-3

[[Projection pair]]s: 7 125/18 11 3125/288

[[Associated temperament]]: [[myna]]

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195

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 }}

Badness: 0.677 × 10^-3

 

 

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 }}

 

Badness: 0.915 × 10^-3

=== Nightingale ===

 

 

 

 

 

 

Comma list: 91/90, 126/125, 176/175

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

Badness: 0.859 × 10^-3

 

Comma list: 78/77, 100/99, 126/125

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 }}

Badness: 0.905 × 10^-3

[[Subgroup]]: 2.3.5.7.11

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

{{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 }}

[[Badness]]: 0.648 × 10^-3

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195

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 }}

Badness: 0.821 × 10^-3

[[Subgroup]]: 2.3.5.7.11

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

{{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 }}

[[Badness]]: 1.585 × 10^-3

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 352/351

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 }}

Badness: 1.588 × 10^-3

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Starling family| ]] <!-- main article -->

[[Category:Starling| ]] <!-- key article -->

[[Category:Rank 3]]