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].

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.

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

Starling

Subgroup: 2.3.5.7

Comma list: 126/125

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]]

Minkowski lattice basis:

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

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

Minimax tuning:

• 7- and 9-odd-limit: 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 (unchanged intervals): 2, 8/7, 4/3

Optimal GPV sequence: 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c

Badness: 0.0699 × 10^-3

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

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]]

Mapping generators: ~2, ~3, ~5

Optimal GPV sequence: 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de

Badness: 0.677 × 10^-3

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 generators: ~2, ~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

Minimax tuning:

• 7- and 9-odd-limit

[[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 (unchanged intervals): 2, 7/6, 4/3

Optimal GPV sequence: 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c

Badness: 0.353 × 10^-3

Projection pairs: 7 125/18 11 3125/288

Associated temperament: myna

Scales: thrush12

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]]

Vals: 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf

Badness: 0.677 × 10^-3

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]]

Vals: 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff

Badness: 0.915 × 10^-3

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]]

Vals: 12f, 15, 19e, 27eff, 31

Badness: 0.837 × 10^-3

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]]

Vals: 12, 15, 19e, 27e, 31f, 46

Badness: 0.991 × 10^-3

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 generators: ~2, ~3, ~5

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

Minimax tuning:

• 11-odd-limit

[[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 (unchanged intervals): 2, 8/7, 11/9

Optimal GPV sequence: 7d, 8d, 12, 15, 19, 27e *

* optimal patent val: 34

Badness: 0.480 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 126/125

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

Vals: 7d, 8d, 12, 15, 19, 27e, 69bceef *

* optimal patent val: 34

Badness: 0.876 × 10^-3

Mockingbird

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 100/99, 126/125

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

Vals: 7d, 8d, 12f, 15, 27eff

Badness: 0.859 × 10^-3

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]]

Vals: 7df, 8d, 12f, 19, 27e, 66cdeeef

Badness: 0.905 × 10^-3

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]]

Optimal GPV sequence: 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: [⟨1 0 0 -1 4 0], ⟨0 1 0 -2 7 -5], ⟨0 0 1 3 -5 5]]

Vals: 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff

Badness: 0.821 × 10^-3

Oxpecker

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125

Mapping: [⟨1 0 1 2 2], ⟨0 1 1 1 1], ⟨0 0 -2 -6 -1]]

Optimal GPV sequence: 7d, 8d, 15, 23de, 24d, 31, 46, 77

Badness: 0.699 × 10^-3

Woodpecker

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 126/125

Mapping: [⟨1 0 1 2 2 2], ⟨0 1 1 1 1 1], ⟨0 0 -2 -6 -1 -1]]

Vals: 7d, 8d, 15, 23de, 24d, 31

Badness: 1.093 × 10^-3

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]]

Optimal GPV sequence: 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: [⟨1 0 0 -1 -3 2], ⟨0 1 0 -2 7 4], ⟨0 0 1 3 -2 -2]]

Vals: 7d, 12e, 19e, 27e, 46

Badness: 1.588 × 10^-3

Cuckoo

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

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

Optimal GPV sequence: 24d, 27e, 31, 58, 89, 154, 185

Badness: 0.933 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]

Vals: 27e, 31, 58, 96d, 154