Starling and thrush
| Starling; thrush |
126/125, 176/175 (11-limit)
11-limit 21-odd-limit: 5.27 ¢
11-limit 21-odd-limit: ? notes
Starling is a rank-3 temperament with the same lattice structure as 5-limit JI, while identifying the harmonic seventh (7/4) as a stack of a perfect fifth (3/2) and three classical major sixths (5/3) octave-reduced, tempering out 126/125. It is the head of the starling family.
In starling, classical minor thirds and 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 version of the diminished seventh chord, which is very characteristic of it. Since this is a chord of meantone 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.
Starling's best 11-limit extension, thrush, adding 176/175 and 441/440 to the comma list, makes it a member of both valinorsmic clan and werckismic temperaments. It also has an obvious 13-limit extension tempering out 196/195, 351/350, and 352/351.
Starling was named by Herman Miller in 1999[1][2], and thrush was named by Gene Ward Smith in 2004[3].
See Starling family #Thrush for technical details.
Interval lattice
-
11-limit thrush -
13-limit thrush
Notation
Starling/thrush can be notated the same as 5-limit just intonation since they share the same lattice structure. One way to do this is to take the conventional circle-of-fifths notation with an additional module of accidentals for the 81/80 comma. In this system, 5/4 is a major third, 7/4 an augmented sixth, and 11/8 a double diminished 5th.
| Ratio | Nominal | Example |
|---|---|---|
| 3/2 | Perfect fifth | C-G |
| 5/4 | Down major third | C-vE |
| 7/4 | Triple-down augmented sixth | C-v3A# |
| 11/8 | Quintuple-down double-augmented third | C-v5Ex |
| 13/8 | Quintuple-down double-augmented fifth | C-v5Gx |
Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental pair for each prime. That makes some intervals more intuitive, at the cost of hiding the structure of starling/thrush tempering. For example, it is customary of the 5/4 to be a major third, and 7/4 to be a minor seventh. As a result, the fact that the 12/7 is a stack of three 6/5's is not revealed, and the related chords can be less convenient.
Chords and harmony
Starling/thrush enables essentially tempered chords of starling, valinorsmic, werckismic, and thrush. Extending the temperament to the 13-limit enables chords of mynucumic, ratwolfsmic and minthmic.
Scales
Starling hobbit scales
Thrush hobbit scales
Other scales
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 700.9120 ¢, ~5/4 = 389.6713 ¢ | CWE: ~3/2 = 701.0811 ¢, ~5/4 = 389.8833 ¢ | POTE: ~3/2 = 701.2515 ¢, ~5/4 = 390.0970 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 701.1852 ¢, ~5/4 = 390.3842 ¢ | CWE: ~3/2 = 701.6298 ¢, ~5/4 = 390.8558 ¢ | POTE: ~3/2 = 701.9454 ¢, ~5/4 = 391.1906 ¢ |
Music
- A Seed Planted (2013) – the melody depends on tempering out 126/125.