# Starling comma

This page discusses some of the rank two temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess the starling tetrad, the 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. 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.

# Myna temperament

In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&31 temperament, or in terms of its wedgie <<10 9 7 -9 -17 -9||. It has 6/5 as a generator, and 58edo can be used as a tuning, with 89edo being a better one, and fans of round amounts in cents may like 120edo. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

## 5-limit (Mynic)

Comma: 10077696/9765625

POTE generator: ~6/5 = 310.140

Map: [<1 9 9|, <0 -10 -9|]

EDOs: 27, 31, 58, 89, 325c

## 7-limit

Commas: 126/125, 1728/1715

7 and 9 limit minimax

[|1 0 0 0>, |0 1 0 0 >, |9/10 9/10 0 0>, |17/10 7/10 0 0>]

Eigenmonzos: 2, 3

POTE generator: 310.146

Map: [<1 9 9 8|, <0 -10 -9 -7|]

Generators: 2, 5/3

EDOs: 27, 31, 58, 89

## 11-limit

Commas: 126/125, 176/175, 243/242

POTE generator: ~6/5 = 310.144

Map: [<1 9 9 8 22|, <0 -10 -9 -7 -25|]

EDOs: 31, 58, 89

## 13-limit

Commas: 126/125, 144/143, 176/175, 196/195

POTE generator: ~6/5 = 310.276

Map: [<1 9 9 8 22 0|, <0 -10 -9 -7 -25 5|]

EDOs: 27, 31, 58

## Minah

Commas: 78/77, 91/90, 126/125, 176/175

POTE generator: ~6/5 = 310.381

Map: [<1 9 9 8 22 20|, <0 -10 -9 -7 -25 -22|]

EDOs: 27e, 31f, 58f, 116cef

## Maneh

Commas: 66/65, 105/104, 126/125, 540/539

POTE generator: ~6/5 = 309.804

Map: [<1 9 9 8 22 23|, <0 -10 -9 -7 -25 -26|]

EDOs: 31

## Myno

Commas: 99/98, 126/125, 385/384

POTE generator: ~6/5 = 309.737

Map: [<1 9 9 8 -1|, <0 -10 -9 -7 6|]

EDOs: 27, 31

## Coleto

Commas: 56/55, 100/99, 1728/1715

POTE generator: ~6/5 = 310.853

Map: [<1 9 9 8 2|, <0 -10 -9 -7 2|]

EDOs: 23bc, 27e

# Sensi temperament

Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. 46edo is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.

Commas: 126/125, 245/243

7-limit minimax

[|1 0 0 0>, |1/13 0 0 7/13>, |5/13 0 0 9/13>, |0 0 0 1>]

Eigenmonzos: 2, 7

9-limit minimax

[|1 0 0 0>, |2/5 14/5 -7/5 0>, |4/5 18/5 -9/5 0>, |3/5 26/5 -13/5 0>]

Eigenmonzos: 2, 9/5

POTE generator: ~9/7 = 443.383

Algebraic generator: Calista, the real root of x^7-2x^2-1, at 340.6467 cents.

Map: [<1 6 8 11|, <0 -7 -9 -13|]

Generators: 2, 14/9

EDOs: 19, 27, 46, 249, 295

## Sensor

Commas: 126/125, 245/243, 385/384

POTE generator: ~9/7 = 443.294

Map: [<1 6 8 11 -6|, <0 -7 -9 -13 15|]

EDOs: 8, 19, 27, 46, 111, 157

### 13-limit

Commas: 91/90, 126/125, 169/168, 385/384

POTE generator: ~9/7 = 443.321

Map: [<1 6 8 11 -6 10|, <0 -7 -9 -13 15 -10|]

EDOs: 8, 19, 27, 46, 157

## Sensis

Commas: 56/55, 100/99, 245/243

POTE generator: 443.962

Map: [<1 6 8 11 6|, <0 -7 -9 -13 -4|]

EDOs: 19, 27, 73, 100

### 13-limit

Commas: 56/55, 78/77, 91/90, 100/99

POTE generator: 443.945

Map: [<1 6 8 11 6 10|, <0 -7 -9 -13 -4 -10|]

EDOs: 19, 27, 73, 100

## Sensus

Commas: 126/125, 176/175, 245/243

POTE generator: ~9/7 = 443.626

Map: [<1 6 8 11 23|, <0 -7 -9 -13 -31|]

EDOs: 8, 19, 27, 46, 165

### 13-limit

Commas: 91/90, 126/125, 169/168, 352/351

POTE generator: ~9/7 = 443.559

Map: [<1 6 8 11 23 10|, <0 -7 -9 -13 -31 -10|]

EDOs: 8, 19, 27, 46, 303

# Valentine temperament

Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the lattice of 7-limit tetrads. Valentine can also be described as the 31&46 temperament, and 77edo, 108edo or 185edo make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as <<9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).

Valentine is very closely related to Carlos Alpha, the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in Beauty in the Beast suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.

Commas: 1029/1024, 126/125

7-limit: [|1 0 0 0>, |5/2 3/4 0 -3/4>, |17/6 5/12 0 -5/12>, [5/2 -1/4 0 1/4>]

Eigenmonzos: 2, 7/6

9-limit: [|1 0 0 0>, |10/7 6/7 0 -3/7>, |47/21 10/21 0 -5/21>, |20/7 -2/7 0 1/7>]

Eigenmonzos: 2, 9/7

POTE generator: 77.864

Algebraic generator: smaller root of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.

Map: [<1 1 2 3|, <0 9 5 -3|]

Generators: 2, 21/20

EDOs: 15, 31, 46, 77, 185, 262

## 11-limit

Commas: 121/120, 126/125, 176/175

[|1 0 0 0 0>, |1 0 0 -9/10 9/10>, |2 0 0 -1/2 1/2>, |3 0 0 3/10 -3/10>, |3 0 0 -7/10 7/10>]

Eigenmonzos: 2, 11/7

Minimax generator: (11/7)^(1/10) = 78.249

POTE generator: 77.881

Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.

Map: [<1 1 2 3 3|, <0 9 5 -3 7|]

EDOs: 15, 31, 46, 77, 108, 185

## Dwynwen

Commas: 91/90, 121/120, 126/125, 176/175

POTE generator: ~21/20 = 78.219

Map: [<1 1 2 3 3 2|, <0 9 5 -3 7 26|]

EDOs: 15, 46

## Lupercalia

Commas: 66/65, 105/104, 121/120, 126/125

POTE generator: ~22/21 = 77.709

Map: [<1 1 2 3 3 3|, <0 9 5 -3 7 11|]

EDOs: 15, 31, 108, 139

## Valentino

Commas: 121/120, 126/125, 176/175, 196/195

POTE generator: ~22/21 = 77.958

Map: [<1 1 2 3 3 5|, <0 9 5 -3 7 -20|]

EDOs: 15, 31, 46, 77, 431

## Semivalentine

Commas: 121/120, 126/125, 169/168, 176/175

POTE generator: ~22/21 = ~21/20 = 77.839

Map: [<2 2 4 6 6 7|, <0 9 5 -3 7 3|]

EDOs: 16, 30, 46, 62, 108ef

# Alicorn temperament

Commas: 126/125, 10976/10935

POTE generator: ~28/27 = 62.278

Map: [<1 2 3 4|, <0 -8 -13 -23|]

Wedgie: <<8 13 23 2 14 17||

EDOs: 19, 58, 77, 96

## 11-limit

Commas: 126/125, 540/539, 896/891

POTE generator: ~28/27 = 62.101

Map: [<1 2 3 4 3|, <0 -8 -13 -23 9|]

EDOs: 19, 58

## 13-limit

Commas: 126/125, 144/143, 196/195, 676/675

POTE generator: ~28/27 = 62.119

Map: [<1 2 3 4 3 5|, <0 -8 -13 -23 9 -25|]

EDOs: 19, 58

## Camahueto

Commas: 126/125, 10976/10935, 385/384

POTE generator: ~28/27 = 62.431

Map: [<1 2 3 4 2|, <0 -8 -13 -23 28|]

EDOs: 19, 58, 77, 96

### 13-limit

Commas: 126/125, 196/195, 385/384, 676/675

POTE generator: ~28/27 = 62.434

Map: [<1 2 3 4 2 5|, <0 -8 -13 -23 28 -25|]

EDOs: 19, 58, 77

# Coblack temperament

In addition to 126/125, the coblack temperament tempers out the cloudy comma, 16807/16384, which is the amount by which five septimal supermajor seconds (8/7) fall short of an octave.

Commas: 126/125, 16807/16384

POTE generator: ~21/20 = 73.044

Map: [<5 1 7 14|, <0 3 2 0|]

EDOs: 15, 35, 50, 65

## 11-limit

Commas: 126/125, 245/242, 385/384

POTE generator: ~21/20 = 73.264

Map: [<5 1 7 14 15|, <0 3 2 0 1|]

EDOs: 15, 35, 50, 65

# Casablanca temperament

Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described by its wedgie, <<19 14 4 -22 -47 -30||, or as 31&73. 74/135 or 91/166 supply good tunings for the generator, and 20 and 31 note MOS are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a hexany and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.

Commas: 126/125, 589824/588245

POTE generator: ~35/24 = 657.818

Map: [<1 12 10 5|, <0 -19 -14 -4|]

EDOs: 9bc, 11b, 31, 135c, 166c

## 11-limit

Commas: 126/125, 385/384, 2420/2401

POTE generator: ~16/11 = 657.923

Map: [<1 12 10 5 4|, |0 -19 -14 -4 -1>]

EDOs: 9bc, 11b, 31, 259bce, 549bce

## Marrakesh

Commas: 126/125, 176/175, 14641/14580

POTE generator: ~22/15 = 657.791

Map: [<1 12 10 5 21|, |0 -19 -14 -4 -32>]

EDOs: 9bce, 11be, 20be, 31, 42e, 73

### 13-limit

126/125, 176/175, 196/195, 17303/17280

POTE generator: ~22/15 = 657.756

Map: [<1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25>]

EDOs: 31, 73, 104c, 135c, 239cf

### Murakuc

Commas: 126/125, 144/143, 176/175, 1540/1521

POTE generator: ~22/15 = 657.700

Map: [<1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6>]

EDOs: 31, 73f, 104cf

# Nusecond temperament

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70, or in terms of its wedgie as <<11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. 31edo can be used as a tuning, or 132edo with a val which is the sum of the patent vals for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.

## 5-limit

Comma: 51018336/48828125

POTE generator: ~3125/2916 = 154.523

Map: [<1 3 4|, <0 -11 -13|]

EDOs: 8, 23, 31, 70, 101, 132c, 233c, 365bc

## 7-limit

Commas: 126/125, 2430/2401

7-limit minimax

[|1 0 0 0>, |-5/13 0 11/13 0>, |0 0 1 0>, |-3/13 0 17/13 0>]

Eigenmonzos: 2, 5

9-limit minimax

[|1 0 0 0>, |0 1 0 0>, |5/11 13/11 0 0>, |4/11 17/11 0 0>]

Eigenmonzos: 2, 3

POTE generator: 154.579

Map: [<1 3 4 5|, <0 -11 -13 -17|]

Generators: 2, 49/45

EDOs: 7, 8, 31, 101, 132, 163

## 11-limit

Commas: 99/98, 121/120, 126/125

11-limit minimax

[|1 0 0 0 0>, |19/10 11/5 0 0 -11/10>, |27/10 13/5 0 0 -13/10>, |33/10 17/5 0 0 -17/10>, |19/5 12/5 0 0 -6/5>]

Eigenmonzos: 2, 11/9

POTE generator: ~11/10 = 154.645

Algebraic generator: positive root of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.

Map: [<1 3 4 5 5|, <0 -11 -13 -17 -12|]

Generators: 2, 11/10

EDOs: 7, 8, 31, 101, 194

## 13-limit

Commas: 66/65 99/98 121/120 126/125

POTE generator: ~11/10 = 154.478

Map: [<1 3 4 5 5 5|, <0 -11 -13 -17 -12 -10|]

EDOs: 31, 70f, 101f

# Thuja

Commas: 126/125, 65536/64827

POTE generator: ~175/128 = 558.605

Map: [<1 8 5 -2|, <0 -12 -5 9|]

Wedgie: <<12 5 -9 -20 -48 -35||

EDOs: 15, 43, 58

## 11-limit

Commas: 126/125, 176/175, 1344/1331

POTE generator: ~11/8 = 558.620

Map: [<1 8 5 -2 4|, <0 -12 -5 9 -1|]

EDOs: 13, 15, 28, 43, 58

## 13-limit

Commas: 126/125, 144/143, 176/175, 364/363

POTE generator: ~11/8 = 558.589

Map: [<1 8 5 -2 4 16|, <0 -12 -5 9 -1 -23|]

EDOs: 15, 43, 58

## 29-limit

POTE generator: ~11/8 = 558.520

Map: [<1 -4 0 7 3 -7 12 1 5 3|, <0 12 5 -9 1 23 -17 7 -1 4|]

EDOs: 43, 58

(Raison d'etre of this entry being the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.)

# Cypress

Comma: 258280326/244140625

POTE generator: ~4374/3125 = 541.726

Map: [<1 7 10|, <0 -12 -17|]

EDOs: 20c, 31, 113c, 144c, 175c, 381bc

## 7-limit

Commas: 126/125, 19683/19208

POTE generator: ~135/98 = 541.828

Map: [<1 7 10 15|, <0 -12 -17 -27|]

Wedgie: <<12 17 27 -1 9 15||

EDOs: 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bcd

## 11-limit

Commas: 99/98, 126/125, 243/242

POTE generator: ~15/11 = 541.772

Map: [<1 7 10 15 17|, <0 -12 -17 -27 -30|]

EDOs: 31, 144cd, 175cd, 206bcde, 237bcde

## 13-limit

Commas: 66/65, 99/98. 126/125, 243/242

POTE generator: ~15/11 = 541.778

Map: [<1 7 10 15 17 15|, <0 -12 -17 -27 -30 -25|]

EDOs: 31

# Bisemidim

Commas: 126/125, 118098/117649

POTE generator: ~35/27 = 455.445

Map: [<2 1 2 2|, <0 9 11 15|]

Wedgie: <<18 22 30 -7 -3 8||

EDOs: 50, 58, 108, 166c, 408c

## 11-limit

Commas: 126/125, 540/539, 1344/1331

POTE generator: ~35/27 = 455.373

Map: [<2 1 2 2 5|, <0 9 11 15 8|]

EDOs: 50, 58, 108, 166ce, 224ce

## 13-limit

Commas: 126/125, 144/143, 196/195, 364/363

POTE generator: ~35/27 = 455.347

Map: [<2 1 2 2 5 5|, <0 9 11 15 8 10|]

EDOs: 50, 58, 166cef, 224cef

# Vines

Commas: 126/125, 84035/82944

POTE generator: ~6/5 = 312.602

Map: [<2 7 8 8|, <0 -8 -7 -5|]

EDOs: 4, 42, 46, 96d, 142d, 238d

## 11-limit

Commas: 126/125, 385/384, 2401/2376

POTE generator: ~6/5 = 312.601

Map: [<2 7 8 8 5|, <0 -8 -7 -5 4|]

EDOs: 4, 42, 46, 96d, 142d, 238d

## 13-limit

Commas: 126/125, 196/195, 364/363, 385/384

POTE generator: ~6/5 = 312.564

Map: [<2 7 8 8 5 5|, <0 -8 -7 -5 4 5|]

EDOs: 4, 42, 46, 96d, 238df

# Kumonga

Comma: 1289945088/1220703125

POTE generator: ~144/125 = 222.912

Map: [<1 4 4|, <0 -13 -9|]

EDOs: 16, 27, 43, 70, 183c

## 7-limit

Commas: 126/125, 12288/12005

POTE generator: ~8/7 = 222.797

Map: [<1 4 4 3|, <0 -13 -9 -1|]

Wedgie: <<13 9 1 -16 -35 -23||

EDOs: 16, 27, 43, 70, 167cd

## 11-limit

Commas: 126/125, 176/175, 864/847

POTE generator: ~8/7 = 222.898

Map: [<1 4 4 3 7|, <0 -13 -9 -1 -19|]

EDOs: 16, 27e, 43, 70e

## 13-limit

Commas: 78/77, 126/125, 144/143, 176/175

POTE generator: ~8/7 = 222.961

Map: [<1 4 4 3 7 5|, <0 -13 -9 -1 -19 -7|]

EDOs: 16, 27e, 43, 70e, 113cde

# Amigo

Commas: 126/125, 2097152/2083725

POTE generator: ~5/4 = 391.094

Map: [<1 9 3 -10|, <0 -11 -1 19|]

EDOs: 43, 46, 89, 135c, 359c

## 11-limit

Commas: 126/125, 176/175, 16384/16335

POTE generator: ~5/4 = 391.075

Map: [<1 9 3 -10 -8|, <0 -11 -1 19 17|]

EDOs: 43, 46, 89, 135c, 224c