Starling temperaments: Difference between revisions

This page discusses miscellaneous rank-2 temperaments tempering out [[126/125]], the starling comma or septimal semicomma.  

* ''[[Pater]]'', {16/15, 126/125} → [[Father family #Pater|Father family]]

* ''[[Flat]]'', {21/20, 25/24} → [[Dicot family #Flat|Dicot family]]

* ''[[Opossum]]'', {28/27, 126/125} → [[Trienstonic clan #Opossum|Trienstonic clan]]

* ''[[Diminished]]'', {36/35, 50/49} → [[Dimipent family #Diminished|Dimipent family]] / [[Jubilismic clan #Diminished|jubilismic clan]]

* [[Keemun]], {49/48, 126/125} → [[Kleismic family #Keemun|Kleismic family]]

* ''[[Augene]]'', {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]

* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]

* [[Mavila]], {126/125, 135/128} → [[Pelogic family #Mavila|Pelogic family]]

* [[Sensi]], {126/125, 245/243}, [[Sensipent family #Sensi|Sensipent family]] / [[Sensamagic clan #Sensi|sensamagic clan]]

* ''[[Gilead]]'', {126/125, 343/324} → [[Shibboleth family #Gilead|Shibboleth family]]

* [[Muggles]], {126/125, 525/512} → [[Magic family #Muggles|Magic family]]

* ''[[Diaschismic]]'', {126/125, 2048/2025} → [[Diaschismic family #Diaschismic|Diaschismic family]]

* ''[[Wollemia]]'', {126/125, 2240/2187} → [[Tetracot family #Wollemia|Tetracot family]]

* ''[[Unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Unicorn|Unicorn family]]

* ''[[Coblack]]'', {126/125, 16807/16384} → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]

* ''[[Grackle]]'', {126/125, 32805/32768} → [[Schismatic family #Grackle|Schismatic family]]

* ''[[Worschmidt]]'', {126/125, 33075/32768} → [[Würschmidt family #Worschmidt|Würschmidt family]]

* ''[[Passionate]]'', {126/125, 131072/127575} → [[Passion family #Passionate|Passion family]]

* ''[[Vishnean]]'', {126/125, 540225/524288} → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]

* ''[[Ditonic]]'', {126/125, 8751645/8388608} → [[Ditonmic family #Ditonic|Ditonmic family]]

* ''[[Muscogee]]'', {126/125, 33756345/33554432} → [[Mabila family #Muscogee|Mabila family]]

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|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.  

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. It has 6/5 as a generator, and [[58edo|58EDO]] can be used as a tuning, with [[89edo|89EDO]] being a better one, and fans of round amounts in cents may like [[120edo|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.

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

: [{{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 (unchanged intervals): 2, 3

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

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

POTE generator: ~6/5 = 310.144

Optimal GPV sequence: {{Val list| 27e, 31, 58, 89 }}

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

POTE generator: ~6/5 = 310.276

Optimal GPV sequence: {{Val list| 27e, 31, 58 }}

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

POTE generator: ~6/5 = 310.381

Optimal GPV sequence: {{Val list| 27e, 31f, 58f }}

Mapping: [{{val| 1 9 9 8 22 23 }}, {{val| 0 -10 -9 -7 -25 -26 }}]

POTE generator: ~6/5 = 309.804

Optimal GPV sequence: {{Val list| 27eff, 31 }}

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

POTE generator: ~6/5 = 309.737

Optimal GPV sequence: {{Val list| 27, 31 }}

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

POTE generator: ~6/5 = 310.853

Optimal GPV sequence: {{Val list| 4, 23bc, 27e }}

== Valentine ==

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|77EDO]], [[108edo|108EDO]] or [[185edo|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.

Subgroup: 2.3.5

[[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]]: 0.031056

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

Minimax tuning:

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]

: Eigenmonzos (unchanged intervals): 2, 11/7

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.

Optimal GPV sequence: {{Val list| 15, 31, 46, 77, 262cdee, 339cdeee }}

 

==== Dwynwen ====

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

Optimal GPV sequence: {{Val list| 15, 31f, 46 }}

Badness: 0.023461

==== 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

Optimal GPV sequence: {{Val list| 15, 31, 77ff, 108eff, 139efff }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Badness]]: 0.101191

Comma list: 126/125, 385/384, 2420/2401

Mapping: [{{val| 1 12 10 5 4 }}, {{val| 0 -19 -14 -4 -1 }}]

POTE generator: ~16/11 = 657.923

Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}

Badness: 0.067291

==== 13-limit ====

Comma list: 126/125, 196/195, 385/384, 2420/2401

Mapping: [{{val| 1 12 10 5 4 7 }}, {{val| 0 -19 -14 -4 -1 -6 }}]

POTE generator: ~16/11 = 657.854

Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: [{{val| 1 12 10 5 21 }}, {{val| 0 -19 -14 -4 -32 }}]

POTE generator: ~22/15 = 657.791

Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c }}

Badness: 0.040539

==== 13-limit ====

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Badness]]: 0.050389

Comma list: 99/98, 121/120, 126/125

Mapping: [{{val| 1 3 4 5 5 }}, {{val| 0 -11 -13 -17 -12 }}]

Mapping generators: ~2, ~11/10

POTE generator: ~11/10 = 154.645

 

 

 

Badness: 0.025621

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: [{{val| 1 3 4 5 5 5 }}, {{val| 0 -11 -13 -17 -12 -10 }}]

POTE generator: ~11/10 = 154.478

Optimal GPV sequence: {{Val list| 8d, 23de, 31, 70f, 101ff }}

Badness: 0.023323

== Thuja ==

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Thuja]].''

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 65536/64827

[[Mapping]]: [{{val| 1 -4 0 7 }}, {{val| 0 12 5 -9 }}]

{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}

[[POTE generator]]: ~175/128 = 558.605

 

[[Badness]]: 0.088441

Comma list: 126/125, 176/175, 1344/1331

Mapping: [{{val| 1 -4 0 7 3 }}, {{val| 0 12 5 -9 1 }}]

POTE generator: ~11/8 = 558.620

Optimal GPV sequence: {{Val list| 15, 43, 58 }}

Badness: 0.033078

Comma list: 126/125, 144/143, 176/175, 364/363

Mapping: [{{val| 1 -4 0 7 3 -7 }}, {{val| 0 12 5 -9 1 23 }}]

POTE generator: ~11/8 = 558.589

Optimal GPV sequence: {{Val list| 15, 43, 58 }}

Badness: 0.022838

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 7 10 15 }}, {{val| 0 -12 -17 -27 }}]

 

{{Multival|legend=1| 12 17 27 -1 9 15 }}

[[POTE generator]]: ~135/98 = 541.828

{{Val list|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}

Mapping: [{{val| 1 7 10 15 17 }}, {{val| 0 -12 -17 -27 -30 }}]

POTE generator: ~15/11 = 541.772

Optimal GPV sequence: {{Val list| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}

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

Mapping: [{{val| 1 7 10 15 17 15 }}, {{val| 0 -12 -17 -27 -30 -25 }}]

POTE generator: ~15/11 = 541.778

Optimal GPV sequence: {{Val list| 11cdeef, 20cdef, 31 }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 2 1 2 2 }}, {{val| 0 9 11 15 }}]

 

{{Multival|legend=1| 18 22 30 -7 -3 8 }}

[[POTE generator]]: ~35/27 = 455.445

{{Val list|legend=1| 50, 58, 108, 166c, 408ccc }}

Mapping: [{{val| 2 1 2 2 5 }}, {{val| 0 9 11 15 8 }}]

POTE generator: ~35/27 = 455.373

Optimal GPV sequence: {{Val list| 50, 58, 108, 166ce, 224cee }}

Mapping: [{{val| 2 1 2 2 5 5 }}, {{val| 0 9 11 15 8 10 }}]

POTE generator: ~35/27 = 455.347

Optimal GPV sequence: {{Val list| 50, 58, 166cef, 224ceeff }}

== Vines ==

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Vines]].''

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 84035/82944

[[Mapping]]: [{{val| 2 7 8 8 }}, {{val| 0 -8 -7 -5 }}]

[[POTE generator]]: ~6/5 = 312.602

{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}

[[Badness]]: 0.078049

Comma list: 126/125, 385/384, 2401/2376

Mapping: [{{val| 2 7 8 8 5 }}, {{val| 0 -8 -7 -5 4 }}]

POTE generator: ~6/5 = 312.601

Optimal GPV sequence: {{Val list| 42, 46, 96d, 142d, 238dd }}

Badness: 0.044499

=== 13-limit ===

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: [{{val| 2 7 8 8 5 5 }}, {{val| 0 -8 -7 -5 4 5 }}]

POTE generator: ~6/5 = 312.564

Optimal GPV sequence: {{Val list| 42, 46, 96d, 238ddf }}

Badness: 0.029693

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kumonga]].''

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 12288/12005

[[Mapping]]: [{{val| 1 4 4 3 }}, {{val| 0 -13 -9 -1 }}]

[[POTE generator]]: ~8/7 = 222.797

{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}

[[Badness]]: 0.087500

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: [{{val| 1 4 4 3 7 }}, {{val| 0 -13 -9 -1 -19 }}]

POTE generator: ~8/7 = 222.898

 

 

 

=== 13-limit ===

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: [{{val| 1 4 4 3 7 5 }}, {{val| 0 -13 -9 -1 -19 -7 }}]

POTE generator: ~8/7 = 222.961

Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e, 113cdee }}

Badness: 0.028920

{{see also| High badness temperaments #Magus }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 -2 2 9 }}, {{val| 0 11 1 -19 }}]

 

{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}

[[POTE generator]]: ~5/4 = 391.094

{{Val list|legend=1| 43, 46, 89, 135c, 359cc }}

Mapping: [{{val| 1 -2 2 9 9 }}, {{val| 0 11 1 -19 -17 }}]

POTE generator: ~5/4 = 391.075

Optimal GPV sequence: {{Val list| 43, 46, 89, 135c, 224c }}

Mapping: [{{val| 1 -2 2 9 9 5 }}, {{val| 0 11 1 -19 -17 -4 }}]

POTE generator: ~5/4 = 391.073

Optimal GPV sequence: {{Val list| 43, 46, 89, 135cf, 224cf }}

== Oolong ==

{{Main| Oolong }}

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Oolong]].''

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 117649/116640

[[Mapping]]: [{{val| 1 6 7 8 }}, {{val| 0 -17 -18 -20 }}]

{{Multival|legend=1| 17 18 20 -11 -16 -4 }}

[[POTE generator]]: ~6/5 = 311.679

{{Val list|legend=1| 27, 50, 77 }}

[[Badness]]: 0.073509

Comma list: 126/125, 176/175, 26411/26244

Mapping: [{{val| 1 6 7 8 18 }}, {{val| 0 -17 -18 -20 -56 }}]

POTE generator: ~6/5 = 311.587

Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}

Badness: 0.056915

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: [{{val| 1 6 7 8 18 5 }}, {{val| 0 -17 -18 -20 -56 -5 }}]

POTE generator: ~6/5 = 311.591

Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}

Badness: 0.035582

The name of ''cobalt temperament'' comes from Cobalt, the 27th element.

Cobalt (27 & 81) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]].  

Subgroup: 2.3.5.7

[[Mapping]]: [{{val|27 43 63 76}}, {{val|0 -1 -1 -1}}]

[[POTE generator]]: ~3/2 = 701.244

{{Val list|legend=1| 27, 81, 108, 135c, 243c }}

Mapping: [{{val|27 43 63 76 94}}, {{val|0 -1 -1 -1 -2}}]

POTE generator: ~3/2 = 700.001

Optimal GPV sequence: {{Val list| 27e, 81, 108 }}

Mapping: [{{val|27 43 63 76 94 100}}, {{val|0 -1 -1 -1 -2 0}}]

POTE generator: ~3/2 = 700.867

Optimal GPV sequence: {{Val list| 27e, 81, 108, 243ceef }}

Mapping: [{{val|27 43 63 76 94 100 111}}, {{val|0 -1 -1 -1 -2 0 -2}}]

POTE generator: ~3/2 = 700.397

Optimal GPV sequence: {{Val list| 27eg, 81, 108g }}

Mapping: [{{val|27 43 63 76 94 100 111 115}}, {{val|0 -1 -1 -1 -2 0 -2 -1}}]

POTE generator: ~3/2 = 700.429

Optimal GPV sequence: {{Val list| 27eg, 81, 108g }}

Mapping: [{{val|27 43 63 76 94 100 111}}, {{val|0 -1 -1 -1 -2 0 -3}}]

POTE generator: ~3/2 = 701.595

Optimal GPV sequence: {{Val list| 27eg, 81gg, 108, 135ce }}

Mapping: [{{val|27 43 63 76 94 100 111 115}}, {{val|0 -1 -1 -1 -2 0 -3 -1}}]

POTE generator: ~3/2 = 701.673

Optimal GPV sequence: {{Val list| 27eg, 81gg, 108, 135ceh }}

Mapping: [{{val|27 43 63 76 94 100}}, {{val|0 -1 -1 -1 -2 -1}}]

POTE generator: ~3/2 = 699.179

Optimal GPV sequence: {{Val list| 27e, 54bdef, 81f, 108f }}