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
- Erato (+81/80) → Didymus rank-3 family
- Sensigh (+245/243) → Sensamagic family
- Oxpecker (+121/120) → Biyatismic clan
- Cuckoo (+243/242) → Rastmic rank-3 clan
Considered below are starling, thrush, thrasher, aplonis, and treecreeper.
Starling
- Main article: Starling and thrush
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
- 7- and 9-odd-limit: 3 and 7 just, 5 1/3-comma sharp
- [[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
Optimal ET sequence: 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c
Badness: 0.0699 × 10-3
Projection pair: 7 125/18
- 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]]
Optimal ET sequence: 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de
Badness: 0.677 × 10-3
Thrush
- Main article: Starling and 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
- 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⟩]
- eigenmonzo (unchanged-interval) basis: 2.3.7
Optimal ET 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
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]]
Optimal ET sequence: 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]]
Optimal ET sequence: 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]]
Optimal ET sequence: 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]]
Optimal ET sequence: 12, 15, 19e, 27e, 31f, 46
Badness: 0.991 × 10-3
Thrasher
- See also: Ptolemismic clan #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
Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e *
Badness: 0.480 × 10-3
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 ET sequence: 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, 56/55, 100/99
Mapping: [⟨1 0 0 -1 2 3], ⟨0 1 0 -2 -2 -1], ⟨0 0 1 3 2 1]]
Optimal ET sequence: 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]]
Optimal ET sequence: 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 ET 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]]
Optimal ET sequence: 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff
Badness: 0.821 × 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 ET 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]]
Optimal ET sequence: 7d, 12e, 19e, 27e, 46
Badness: 1.588 × 10-3