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.
Starling
Subgroup: 2.3.5.7
Mapping: [⟨1 0 0 -5], ⟨0 1 0 2], ⟨0 0 1 2]]
Mapping generators: ~2, ~3, ~5
Map 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
- 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: 2, 8/7, 4/3
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.
- 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], {{val| 0 0 1 3 -4]]
Mapping generators: ~2, ~3, ~5
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
Map 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
- 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: 2, 7/6, 4/3
Badness: 0.353 × 10-3
Projection pairs: 7 125/18 11 3125/288
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: Template:Val list
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: Template:Val list
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: Template:Val list
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: Template:Val list
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
Map 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⟩]
- Eigenmonzos: 2, 8/7, 11/9
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: Template:Val list *
* 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: Template:Val list
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: Template:Val list
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]]
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: Template:Val list
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]]
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: Template:Val list
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]]
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: Template:Val list
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]]
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: Template:Val list