# Starling family

(Redirected from Veery)

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|.

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

Projection pair: 7 125/18

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

# 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

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

## 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

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

## 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

# 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

## 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

## 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

## 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

# 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

## 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

# 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

## 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

# 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

## 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

# 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