# 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] (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.

Temperaments discussed elsewhere include

Considered below are starling, thrush, thrasher, aplonis, and treecreeper.

## Starling

Subgroup: 2.3.5.7

Mapping[1 0 0 -1], 0 1 0 -2], 0 0 1 3]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [0 1 0 -2], 0 1 1 1]]

6/5 length = 1.068, 5/4 length = 1.206
Angle (6/5, 5/4) = 100.364 degrees
[[1 0 0 0, [0 1 0 0, [1/3 2/3 0 1/3, [0 0 0 1]
eigenmonzo (unchanged-interval) basis: 2.3.7

Projection pair: 7 125/18

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

## Undecimal starling

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384

Mapping[1 0 0 -1 8], 0 1 0 -2 3], 0 0 1 3 -4]]

## Thrush

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175

Mapping[1 0 0 -1 -5], 0 1 0 -2 -2], 0 0 1 3 5]]

Mapping to lattice: [0 1 1 1 3], 0 1 0 -2 -2]]

Lattice basis:

5/4 length = 0.8576, 6/5 length = 0.9314
Angle(5/4, 6/5) = 74.6239 degrees
[[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]
eigenmonzo (unchanged-interval) basis: 2.3.7

Projection pairs: 7 125/18 11 3125/288

### 13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 0 0 -1 -5 0], 0 1 0 -2 -2 -5], 0 0 1 3 5 5]]

### Bluebird

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175

Mapping: [1 0 0 -1 -5 9], 0 1 0 -2 -2 4], 0 0 1 3 5 -5]]

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

### Nightingale

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 126/125, 176/175

Mapping: [1 0 0 -1 -5 -4], 0 1 0 -2 -2 -1], 0 0 1 3 5 4]]

### Veery

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 0 0 -1 -5 2], 0 1 0 -2 -2 4], 0 0 1 3 5 -2]]

## Thrasher

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99

Mapping[1 0 0 -1 2], 0 1 0 -2 -2], 0 0 1 3 2]]

Mapping to lattice: [0 1 0 -2 -2], 0 1 1 1 0]]

Lattice basis:

6/5 length = 0.9089, 5/4 length = 1.2007
Angle (6/5, 5/4) = 98.8447
[[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]
eigenmonzo (unchanged-interval) basis: 2.7.11/9

### 13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 0 0 -1 2 2], 0 1 0 -2 -2 4], 0 0 1 3 2 -2]]

* optimal patent val: 34

### Mockingbird

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 0 0 -1 2 3], 0 1 0 -2 -2 -1], 0 0 1 3 2 1]]

### Catbird

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 0 0 -1 2 0], 0 1 0 -2 -2 -5], 0 0 1 3 2 5]]

## Aplonis

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539

Mapping[1 0 0 -1 4], 0 1 0 -2 7], 0 0 1 3 -5]]

### 13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 0 0 -1 4 0], 0 1 0 -2 7 -5], 0 0 1 3 -5 5]]

## Treecreeper

Subgroup: 2.3.5.7.11

Comma list: 126/125, 1232/1215

Mapping[1 0 0 -1 -3], 0 1 0 -2 7], 0 0 1 3 -2]]