Starling temperaments: Difference between revisions

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

* [[Father family #Pater|pater]] ({16/15, 126/125}, father family)

* [[Dicot family #Flat|flat]] ({21/20, 25/24}, dicot family)

* [[Trienstonic clan #Opossum|opossum]] ({28/27, 126/125}, trienstonic clan)

* [[Jubilismic clan #Diminished|diminished]] ({36/35, 50/49}, dimipent family / jubilismic clan)

* [[Kleismic family #Keemun|keemun]] ({49/48, 126/125}, kleismic family / slendro clan)

* [[Augmented family #Augene|augene]] ({64/63, 126/125}, augmented family / archytas clan)

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

* [[Pelogic family #Mavila|mavila]] ({126/125, 135/128}, pelogic family)

* [[Sensipent family #Sensi|sensi]] ({126/125, 245/243}, sensipent family / sensamagic clan)

* [[Shibboleth family #Gilead|gilead]] ({126/125, 343/324}, shibboleth family)

* [[Magic family #Muggles|muggles]] ({126/125, 525/512}, magic family)

* [[Diaschismic family #Diaschismic|diaschismic]] ({126/125, 2048/2025}, diaschismic family)

* [[Tetracot family #Wollemia|wollemia]] ({126/125, 2240/2187}, tetracot family)

* [[Unicorn family|unicorn]] ({126/125, 10976/10935}, unicorn family)

* [[Cloudy clan #Coblack|coblack]] ({126/125, 16807/16384}, cloudy clan)

* [[Schismatic family #Grackle|grackle]] ({126/125, 32805/32768}, schismatic family)

* [[Würschmidt family #Worschmidt|worschmidt]] ({126/125, 33075/32768}, würschmidt 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]] 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.  

{{main| Myna }}

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]] 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]]: 0.249965

 

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

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

[[Badness]]: 0.027044

== 11-limit ==

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

POTE generator: ~6/5 = 310.144

Vals: {{Val list| 27e, 31, 58, 89 }}

Badness: 0.016842

=== 13-limit ===

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

POTE generator: ~6/5 = 310.276

Vals: {{Val list| 27e, 31, 58 }}

Badness: 0.017125

=== Minah ===

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

POTE generator: ~6/5 = 310.381

Vals: {{Val list| 27e, 31f, 58f }}

Badness: 0.027568

=== Maneh ===

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

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

POTE generator: ~6/5 = 309.804

Vals: {{Val list| 27eff, 31 }}

Badness: 0.029868

== Myno ==

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

POTE generator: ~6/5 = 309.737

Vals: {{Val list| 27, 31 }}

Badness: 0.033434

== Coleto ==

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

POTE generator: ~6/5 = 310.853

Vals: {{Val list| 4, 23bc, 27e }}

Badness: 0.048687

{{main| Valentine }}

{{see also| Gamelismic clan #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]], [[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.

Subgroup: 2.3.5

[[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]]: 0.122765

== 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

[[Minimax tuning]]:

: [{{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 }}]

: [{{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]]: 0.031056

== 11-limit ==

Subgroup: 2.3.5.7.11

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

: [{{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: 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.

Vals: {{Val list| 15, 31, 46, 77, 262cdee, 339cdeee }}

Subgroup: 2.3.5.7.11.13

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

Vals: {{Val list| 15, 31f, 46 }}

Badness: 0.023461

=== Lupercalia ===

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

Vals: {{Val list| 15, 31, 77ff, 108eff, 139efff }}

Badness: 0.021328

=== Valentino ===

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

Vals: {{Val list| 15f, 31, 46, 77 }}

Badness: 0.020665

=== Semivalentine ===

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

Vals: {{Val list| 16, 30, 46, 62, 108ef }}

Badness: 0.032749

Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described 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.

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 589824/588245

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

{{Multival|legend=1| 19 14 4 -22 -47 -30 }}

[[POTE generator]]: ~35/24 = 657.818

{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}

[[Badness]]: 0.101191

Subgroup: 2.3.5.7.11

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

Vals: {{Val list| 11b, 20b, 31 }}

Badness: 0.067291

== Marrakesh ==

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

Vals: {{Val list| 31, 73, 104c, 135c }}

Badness: 0.040539

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: [{{val| 1 12 10 5 21 -10 }}, {{val| 0 -19 -14 -4 -32 25 }}]

POTE generator: ~22/15 = 657.756

Vals: {{Val list| 31, 73, 104c, 135c, 239ccf }}

Badness: 0.040774

=== Murakuc ===

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: [{{val| 1 12 10 5 21 7 }}, {{val| 0 -19 -14 -4 -32 -6 }}]

POTE generator: ~22/15 = 657.700

Vals: {{Val list| 31, 104cff, 135cff }}

Badness: 0.041395

= Nusecond =

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70. 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.

Subgroup: 2.3.5

[[Comma list]]: 51018336/48828125

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

[[POTE generator]]: ~3125/2916 = 154.523

[[Badness]]: 0.4665

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 2430/2401

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

Mapping generators: ~2, ~49/45

{{Multival|legend=1| 11 13 17 -5 -4 3 }}

[[POTE generator]]: ~49/45 = 154.579

[[Minimax tuning]]:  

: [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]

: [[Eigenmonzo]]s: 2, 5

{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}

[[Badness]]: 0.0504

== 11-limit ==

Subgroup: 2.3.5.7.11

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

Minimax tuning:  

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]

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

{{Val list|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}

Badness: 0.0256

== 13-limit ==

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

{{Val list|legend=1| 8d, 23de, 31, 70f, 101ff }}

Badness: 0.0233

= Thuja =

Subgroup: 2.3.5.7

Comma list: 126/125, 65536/64827

Mapping: [{{val| 1 8 5 -2 }}, {{val| 0 -12 -5 9 }}]

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

POTE generator: ~175/128 = 558.605

{{Val list|legend=1| 15, 43, 58 }}

Badness: 0.0884

== 11-limit ==

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

Mapping: [{{val| 1 8 5 -2 4 }}, {{val| 0 -12 -5 9 -1 }}]

POTE generator: ~11/8 = 558.620

{{Val list|legend=1| 15, 43, 58 }}

Badness: 0.0331

== 13-limit ==

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

Mapping: [{{val| 1 8 5 -2 4 16 }}, {{val| 0 -12 -5 9 -1 -23 }}]

POTE generator: ~11/8 = 558.589

{{Val list|legend=1| 15, 43, 58 }}

Badness: 0.0228

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

Subgroup: 2.3.5.7.11.13.17.19.23.29

Mapping: [{{val| 1 -4 0 7 3 -7 12 1 5 3 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 4 }}]

POTE generator: ~11/8 = 558.520

{{Val list|legend=1| 43, 58hi }}

= Cypress =

[[Comma list]]: 258280326/244140625

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

[[POTE generator]]: ~4374/3125 = 541.726

{{Val list|legend=1| 11c, 20c, 31, 113c, 144c, 175c, 381bcc }}

[[Badness]]: 0.8166

== 7-limit ==

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 19683/19208

[[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 }}

[[Badness]]: 0.0998

== 11-limit ==

Comma list: 99/98, 126/125, 243/242

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

POTE generator: ~15/11 = 541.772

{{Val list|legend=1| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}

Badness: 0.0427

== 13-limit ==

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

{{Val list|legend=1| 11cdeef, 20cdef, 31 }}

Badness: 0.0378

= Bisemidim =

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 118098/117649

[[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 }}

[[Badness]]: 0.0978

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

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

POTE generator: ~35/27 = 455.373

{{Val list|legend=1| 50, 58, 108, 166ce, 224cee }}

Badness: 0.0412

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

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

POTE generator: ~35/27 = 455.347

{{Val list|legend=1| 50, 58, 166cef, 224ceeff }}

Badness: 0.0239

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: 0.0780

== 11-limit ==

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

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

Badness: 0.0445

== 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

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

Badness: 0.0297

Subgroup: 2.3.5

[[Comma list]]: 1289945088/1220703125

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

[[POTE generator]]: ~144/125 = 222.912

[[Badness]]: 0.7296

Subgroup: 2.3.5.7

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

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

{{Multival|legend=1| 13 9 1 -16 -35 -23 }}

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

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

[[Badness]]: 0.0875

== 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

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

Badness: 0.0433

== 13-limit ==

Subgroup: 2.3.5.7.11.13

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

{{Val list|legend=1| 16, 27e, 43, 70e, 113cdee }}

Badness: 0.0289

Subgroup: 2.3.5.7

[[Comma list]]: 126/125, 2097152/2083725

[[Mapping]]: [{{val| 1 9 3 -10 }}, {{val| 0 -11 -1 19 }}]

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

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

[[Badness]]: 0.1109

== 11-limit ==

Comma list: 126/125, 176/175, 16384/16335