Starling family

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 <135 214 314 379|. It is the most peaceful temperament.

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

7-limit starling

Comma c = 126/125

7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp

[<1 0 0 0|, <0 1 0 0|, <1/3 2/3 0 1/3|, <0 0 0 1|]

Eigenmonzos: 2, 8/7, 4/3

Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206

Angle(6/5, 5/4) = 100.364 degrees

Map to lattice: [<0 1 0 -2|, <0 1 1 1|]

Map: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]

Generators: 2, 3, 5

EDOs: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, 77, 108, 166c, 185, 243c

Badness: 0.0000699

Projection pair: 7 125/18

Scales: starling7, starling8, starling9, starling11, starling12, starling15, starling16, starling17, starling19

Music: A Seed Planted by Jake Freivald. The melody depends on tempering out 126/125.

Minkowski blocks

7: 25/24, 81/80

8: 16/15, 648/625

9: 27/25, 128/125

11: 16/15, 15625/15552

12: 128/125, 628/625

15: 128/125, 250/243

16: 648/625, 3125/3072

17: 25/24, 20480/19683

19: 81/80, 3125/3072

27: 128/125, 78732/78125

28: 648/625, 16875/16384

31: 81/80, 1990656/1953125

34: 15625/15552, 2048/2025

11-limit

Commas: 126/125, 385/384

Map: [<1 0 0 -1 8|, <0 1 0 -2 3|, <0 0 1 3 -4]]

Generators: 2, 3, 5

EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de

Badness: .000677

Thrush

Commas: 126/125, 176/175

Associated linear temperament: myna

7 and 9 limit minimax

[|1 0 0 0 0>, |0 1 0 0 0>, |1/3 2/3 0 1/3 0>, |0 0 0 1 0>, |-10/3 4/3 0 5/3 0>]

Eigenmonzos: 2, 7/6, 4/3

Lattice basis 5/4 length 0.8576 6/5 length 0.9314

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

Map to lattice: [<0 1 1 1 3|, <0 1 0 -2 -2|]

Map: [<1 0 0 -1 -5|, <0 1 0 -2 -2|, <0 0 1 3 5|]

Generators: 2, 3, 5

EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c

Badness: 0.000353

Projection pairs: 7 125/18 11 3125/288

Scales: thrush12

13-limit

Commas: 126/125, 176/175, 196/195

Map: [<1 0 0 -1 -5 0|, <0 1 0 -2 -2 -5|, <0 0 1 3 5 5|]

EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf

Badness: 0.000677

Bluebird

Commas 126/125, 176/175, 144/143

Map: [<1 0 0 -1 -5 9|, <0 1 0 -2 -2 4|, <0 0 1 3 5 -5|]

EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef

Badness: 0.000915

Projection pairs: 7 125/18 11 3125/288 13 41472/3125

Nightingale

Commas: 126/125, 176/175, 66/65

Map: [<1 0 0 -1 -5 -4|, <0 1 0 -2 -2 -1|, <0 0 1 3 5 4|]

EDOs: 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef

Badness: 0.000837

Veery

Commas: 91/90, 126/125, 176/175

Map: [<1 0 0 -1 -5 2|, <0 1 0 -2 -2 4|, <0 0 1 3 5 -2|]

EDOs: 12, 15, 46

Badness: 0.000991

Thrasher

Commas: 56/55, 100/99

11-limit minimax

[|1 0 0 0 0>, |1 3/4 0 1/4 -3/8>, |1 1/2 0 1/2 -1/4>, |0 0 0 1 0>, |2 -1/2 0 1/2 1/4>]

Eigenmonzos: 2, 8/7, 11/9

Lattice basis: 6/5 length 0.9089 5/4 length 1.2007

Angle(6/5, 5/4) = 98.8447

Map to lattice: [<0 1 0 -2 -2|, <0 1 1 1 0|]

Map: [<1 0 0 -1 2|, <0 1 0 -2 -2|, <0 0 1 3 2|]

Generators: 2, 3, 5

EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e

Badness: 0.000480

13-limit

Commas: 126/125, 100/99, 91/90

Map: [<1 0 0 -1 2 2|, <0 1 0 -2 -2 4|, <0 0 1 3 2 -2|]

EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e

Badness: 0.000876

Mockingbird

Commas: 126/125, 100/99, 40/39

Map: [<1 0 0 -1 2 3|, <0 1 0 -2 -2 -1|, <0 0 1 3 2 1|]

EDOs: 12f, 15, 27ef

Badness: 0.000859

Catbird

Commas: 126/125, 100/99, 78/77

Map: [<1 0 0 -1 2 0|, <0 1 0 -2 -2 -5|, <0 0 1 3 2 5|]

EDOs: 12, 15, 17c, 19, 22ef, 27e, 34, 58e

Badness: 0.000905

Aplonis

Commas: 126/125, 540/539

Map: [<1 0 0 -1 4|, <0 1 0 -2 7|, <0 0 1 3 -5|]

EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce

Badness: 0.000648

13-limit

Commas: 126/125, 144/143, 196/195

Map: [<1 0 0 -1 4 0|, <0 1 0 -2 7 -5|, <0 0 1 3 -5 5|]

EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef

Badness: 0.000821

Oxpecker

Commas: 121/120, 126/125

Map: [<1 0 1 2 2|, <0 1 1 1 1|, <0 0 -2 -6 -1|]

EDOs: 8d, 15, 31, 46, 77, 185e, 262cde

Badness: 0.000699

Woodpecker

Commas: 66/65, 121/120, 126/125

Map: [<1 0 1 2 2 2|, <0 1 1 1 1 1|, <0 0 -2 -6 -1 -1|]

EDOs: 15, 31, 46f, 70f, 101f

Badness: 0.001093

Treecreeper

Commas: 126/125, 1232/1215

Map: [<1 0 0 -1 -3|, <0 1 0 -2 7|, <0 0 1 3 -2|]

EDOs: 34, 46, 119c, 165c

Badness: 0.001585

13-limit

Commas: 91/90, 126/125, 352/351

Map: [<1 0 0 -1 -3 2|, <0 1 0 -2 7 4|, <0 0 1 3 -2 -2|]

EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf

Badness: 0.001588

Cuckoo

Commas: 126/125, 243/242

Map: [<1 1 0 -3 2|, <0 2 0 -4 5|, <0 0 1 3 0|]

EDOs: 31 34 58 65 89 154 185 216

Badness: 0.000933

13-limit

Commas: 126/125, 196/195, 243/242

Map: [<1 1 0 -3 2 -5|, <0 2 0 -4 5 -10|, <0 0 1 3 0 5|]

EDOs: 31 58 65f 89f 92e 154 185 216f