Starling temperaments: Difference between revisions

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.  

Temperaments discussed else where are [[Father family #Pater|pater]], [[Dicot family #Flat|flat]], [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Diminished|diminished]], [[Kleismic family #Keemun|keemun]], [[Augmented family #Augene|augene]], [[Meantone family #Septimal meantone|septimal meantone]], [[Pelogic family #Mavila|mavila]], [[Shibboleth family #Gilead|gilead]], [[Magic family #Muggles|muggles]], [[Diaschismic family #Diaschismic|diaschismic]], [[Tetracot family #Wollemia|wollemia]], [[Schismatic family #Grackle|grackle]] and [[Würschmidt family #Worschmidt|worschmidt]].

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 {{multival| 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) ==

Subgroup: 2.3.5

[[Comma list]]: 10077696/9765625

[[Mapping]]: [{{val| 1 9 9 }}, {{val| 0 -10 -9 }}]

[[POTE generator]]: ~6/5 = 310.140

 

 

[[Badness]]: 0.2500

 

 

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 9 9 8 }}, {{val| 0 -10 -9 -7 }}]

 

Mapping generators: ~2, ~5/3

[[POTE generator]]: ~6/5 = 310.146

* 7- and [[9-odd-limit]]

: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 9/10 9/10 0 0 }}, {{monzo| 17/10 7/10 0 0 }}]

: [[Eigenmonzo]]s: 2, 3

 

Mapping: [{{val| 1 9 9 8 22 }}, {{val| 0 -10 -9 -7 -25 }}]

 

{{Val list|legend=1| 27e, 31, 58, 89 }}

Badness: 0.0168

Mapping: [{{val| 1 9 9 8 22 0 }}, {{val| 0 -10 -9 -7 -25 5 }}]

 

{{Val list|legend=1| 27e, 31, 58 }}

Badness: 0.0171

Mapping: [{{val| 1 9 9 8 22 20 }}, {{val| 0 -10 -9 -7 -25 -22 }}]

 

{{Val list|legend=1| 27e, 31f, 58f }}

Badness: 0.0276

Comma list: 66/65, 105/104, 126/125, 540/539

 

POTE generator: ~6/5 = 309.804

{{Val list|legend=1| 27eff, 31 }}

Badness: 0.0299

Mapping: [{{val| 1 9 9 8 -1 }}, {{val| 0 -10 -9 -7 6 }}]

 

{{Val list|legend=1| 27, 31 }}

Badness: 0.0334

Mapping: [{{val| 1 9 9 8 2 }}, {{val| 0 -10 -9 -7 2 }}]

 

 

{{Val list|legend=1| 4, …, 23bc, 27e }}

Badness: 0.0487

{{main|Sensi}}

{{see also|Sensipent family #Sensi}}

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 the size 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 2.3.5.7.13 sensi (sensation) 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. The name "sensi" is a play on the words "semi-" and "sixth."

== 7-limit ==

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 245/243

[[Mapping]]: [{{val| 1 6 8 11 }}, {{val| 0 -7 -9 -13 }}]

Mapping generators: ~2, ~14/9

{{Multival|legend=1| 7 9 13 -2 1 5 }}

[[POTE generator]]: ~9/7 = 443.383

* [[7-odd-limit]]

: [{{monzo| 1 0 0 0 }}, {{monzo| 1/13 0 0 7/13 }}, {{monzo| 5/13 0 0 9/13 }}, {{monzo| 0 0 0 1 }}]

: [[Eigenmonzo]]s: 2, 7

* [[9-odd-limit]]

: [{{monzo| 1 0 0 0 }}, {{monzo| 2/5 14/5 -7/5 0 }}, {{monzo| 4/5 18/5 -9/5 0 }}, {{monzo| 3/5 26/5 -13/5 0 }}]

: [[Eigenmonzo]]s: 2, 9/5

[[Algebraic generator]]: The real root of ''x''⁵ + ''x''⁴ - 4''x''² + ''x'' - 1, at 443.3783 cents.

{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}

[[Badness]]: 0.0256

Subgroup: 2.3.5.7.13

Comma list: 91/90, 126/125, 169/168

Sval mapping: [{{val| 1 6 8 11 10 }}, {{val| 0 -7 -9 -13 -10 }}]

Gencom mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}]

POTE generator: ~9/7 = 443.322

{{Val list|legend=1| 19, 27, 46, 111de, 157de }}

 

== Sensor ==

 

Subgroup: 2.3.5.7.11

 

Mapping: [{{val| 1 6 8 11 -6 }}, {{val| 0 -7 -9 -13 15 }}]

POTE generator: ~9/7 = 443.294

{{Val list|legend=1| 19, 27, 46, 111d, 157d, 268cdd }}

Badness: 0.0379

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 385/384

Mapping: [{{val| 1 6 8 11 -6 10 }}, {{val| 0 -7 -9 -13 15 -10 }}]

POTE generator: ~9/7 = 443.321

{{Val list|legend=1| 19, 27, 46, 111df, 157df }}

Comma list: 56/55, 100/99, 245/243

Mapping: [{{val| 1 6 8 11 6 }}, {{val| 0 -7 -9 -13 -4 }}]

POTE generator: ~9/7 = 443.962

{{Val list|legend=1| 19, 27e, 73ee }}

Badness: 0.0287

Comma list: 56/55, 78/77, 91/90, 100/99

Mapping: [{{val| 1 6 8 11 6 10 }}, {{val| 0 -7 -9 -13 -4 -10 }}]

POTE generator: ~9/7 = 443.945

{{Val list|legend=1| 19, 27e, 46e, 73ee }}

Badness: 0.0200

== Sensus ==

Subgroup: 2.3.5.7.11

 

Comma list: 126/125, 176/175, 245/243

Mapping: [{{val| 1 6 8 11 23 }}, {{val| 0 -7 -9 -13 -31 }}]

POTE generator: ~9/7 = 443.626

{{Val list|legend=1| 19e, 27e, 46, 119c, 165c }}

Badness: 0.0295

=== 13-limit ===

 

 

Mapping: [{{val| 1 6 8 11 23 10 }}, {{val| 0 -7 -9 -13 -31 -10 }}]

 

POTE generator: ~9/7 = 443.559

 

{{Val list|legend=1| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}

 

Badness: 0.0208

 

Comma list: 55/54, 77/75, 99/98

Mapping: [{{val| 1 6 8 11 11 }}, {{val| 0 -7 -9 -13 -12 }}]

POTE generator: ~9/7 = 443.518

{{Val list|legend=1| 19e, 27, 46ee }}

Badness: 0.0368

Comma list: 55/54, 66/65, 77/75, 143/140

 

 

 

 

 

 

 

 

 

 

 

 

 

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 [[The Seven Limit Symmetrical Lattices|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 {{multival| 9 5 -3 7 … }}, tempering out 121/120 and 441/440; 46edo 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 "[t]he 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.

 

 

 

[[Comma list]]: 1990656/1953125

[[Mapping]]: [{{val| 1 1 2 }}, {{val| 0 9 5 }}]

[[POTE generator]]: ~25/24 = 78.039

{{Val list|legend=1| 15, 31, 46, 77, 123 }}

[[Badness]]: 0.1228

== 7-limit ==

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 1029/1024

[[Mapping]]: [{{val| 1 1 2 3 }}, {{val| 0 9 5 -3 }}]

Mapping generators: ~2, ~21/20

[[POTE generator]]: ~21/20 = 77.864

: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]

: [[Eigenmonzo]]s: 2, 7/6

: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]

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

 

 

[[Badness]]: 0.0311

 

Comma list: 121/120, 126/125, 176/175

Mapping: [{{val| 1 1 2 3 3 }}, {{val| 0 9 5 -3 7 }}]

Mapping generators: ~2, ~21/20

POTE generator: ~21/20 = 77.881

 

Algebraic generator: positive root of 4''x''³ + 15''x''² - 21, or else Gontrand2, the smallest positive root of 4''x''⁷ - 8''x''⁶ + 5.

 

{{Val list|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}

 

 

Comma list: 91/90, 121/120, 126/125, 176/175

Mapping: [{{val| 1 1 2 3 3 2 }}, {{val| 0 9 5 -3 7 26 }}]

POTE generator: ~21/20 = 78.219

{{Val list|legend=1| 15, 31f, 46 }}

Badness: 0.0235

=== Lupercalia ===

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 126/125

Mapping: [{{val| 1 1 2 3 3 3 }}, {{val| 0 9 5 -3 7 11 }}]

POTE generator: ~21/20 = 77.709

Badness: 0.0213

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 196/195

Mapping: [{{val| 1 1 2 3 3 5 }}, {{val| 0 9 5 -3 7 -20 }}]

POTE generator: ~21/20 = 77.958

{{Val list|legend=1| 15f, 31, 46, 77, 431ccdeeeef }}

Badness: 0.0207

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 169/168, 176/175

Mapping: [{{val| 2 2 4 6 6 7 }}, {{val| 0 9 5 -3 7 3 }}]

POTE generator: ~21/20 = 77.839

Badness: 0.0327

{{see also|Unicorn family #Alicorn}}

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||

 

 

Badness: 0.0409

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: {{EDOs|19, 39d, 58}}

Badness: 0.0392

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: {{EDOs|19, 39df, 58}}

Badness: 0.0237

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: {{EDOs|19, 58e, 77, 96d, 173d}}

Badness: 0.0659

=== 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: {{EDOs|19, 58e, 77, 96d, 173d}}

Badness: 0.0362

{{see also|Trisedodge family #Coblack}}

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: {{EDOs|15, 35, 50, 65, 115d}}

Badness: 0.1073

==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: {{EDOs|15, 35, 50, 65, 115d}}

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.

POTE generator: ~35/24 = 657.818

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

EDOs: {{EDOs|11b, 20b, 31, 104c, 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>]

 

 

Badness: 0.0623

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: {{EDOs|31, 73, 104c, 135c}}

Badness: 0.0405

Commas: 126/125, 176/175, 196/195, 14641/14580

POTE generator: ~22/15 = 657.756

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

EDOs: {{EDOs|31, 73, 104c, 135c, 239ccf}}

Badness: 0.0408

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: {{EDOs|31, 104cf, 135cf, 166c}}

Badness: 0.0414

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 val]]s 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 ==

POTE generator: ~3125/2916 = 154.523

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

EDOs: {{EDOs|8, 23, 31, 70, 101, 132c, 233c, 365bcc}}

[[Comma]]s: 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>]

[[Eigenmonzo]]s: 2, 5

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

[[Eigenmonzo]]s: 2, 3

[[POTE_tuning|POTE generator]]: 154.579

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

[[Generator]]s: 2, 49/45

EDOs: {{EDOs|8d, 23d, 31, 101, 132c, 163c}}

Badness: 0.0504

[[Comma]]s: 99/98, 121/120, 126/125

11-limit minimax

|27/10 13/5 0 0 -13/10>, |33/10 17/5 0 0 -17/10>,

|19/5 12/5 0 0 -6/5&gt;<nowiki>]</nowiki>

[[Eigenmonzo]]s: 2, 11/9

[[POTE_tuning|POTE generator]]: ~11/10 = 154.645

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

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

[[Generator]]s: 2, 11/10

EDOs: {{EDOs|8d, 23de, 31, 101, 132ce, 163ce, 194cee}}

Badness: 0.0256

==13-limit==

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

POTE generator: ~11/10 = 154.478

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

EDOs: {{EDOs|8d, 23de, 31, 70f, 101ff}}

Badness: 0.0233

Commas: 126/125, 65536/64827

POTE generator: ~175/128 = 558.605

Map: [&lt;1 8 5 -2|, &lt;0 -12 -5 9|]

Wedgie: &lt;&lt;12 5 -9 -20 -48 -35||

EDOs: {{EDOs|15, 43, 58}}

Badness: 0.0884

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

POTE generator: ~11/8 = 558.620

Map: [&lt;1 8 5 -2 4|, &lt;0 -12 -5 9 -1|]

EDOs: {{EDOs|15, 43, 58}}

Badness: 0.0331

==13-limit==

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

POTE generator: ~11/8 = 558.589

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

EDOs: {{EDOs|15, 43, 58}}

Badness: 0.0228

POTE generator: ~11/8 = 558.520

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

EDOs: {{EDOs|43, 58hi}}

(''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.)

== 5-limit ==

Comma: 258280326/244140625

POTE generator: ~4374/3125 = 541.726

Map: [&lt;1 7 10|, &lt;0 -12 -17|]

EDOs: {{EDOs|11c, 20c, 31, 113c, 144c, 175c, 381bcc}}

Badness: 0.8166

Commas: 126/125, 19683/19208

POTE generator: ~135/98 = 541.828

Map: [&lt;1 7 10 15|, &lt;0 -12 -17 -27|]

Wedgie: &lt;&lt;12 17 27 -1 9 15||

EDOs: {{EDOs|11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd}}

Badness: 0.0998

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

POTE generator: ~15/11 = 541.772

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

EDOs: {{EDOs|11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde}}

Badness: 0.0427

== 13-limit ==

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