Starling temperaments: Difference between revisions

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= 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.
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 =
= Myna =
{{main|Myna}}
{{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, 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.
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)==
==5-limit (mynic)==
Comma: 10077696/9765625
Comma: 10077696/9765625


Line 16: Line 13:
Map: [<1 9 9|, <0 -10 -9|]
Map: [<1 9 9|, <0 -10 -9|]


EDOs: 27, 31, 58, 89, 325c
EDOs: {{EDOs|27, 31, 58, 89, 325cc}}


Badness: 0.2500
Badness: 0.2500
Line 35: Line 32:
[[Generator]]s: 2, 5/3
[[Generator]]s: 2, 5/3


EDOs: 27, 31, 58, 89
EDOs: {{EDOs|27, 31, 58, 89}}


Badness: 0.0270
Badness: 0.0270
Line 46: Line 43:
Map: [<1 9 9 8 22|, <0 -10 -9 -7 -25|]
Map: [<1 9 9 8 22|, <0 -10 -9 -7 -25|]


EDOs: 27e, 31, 58, 89
EDOs: {{EDOs|27e, 31, 58, 89}}


Badness: 0.0168
Badness: 0.0168


==13-limit==
=== 13-limit ===
Commas: 126/125, 144/143, 176/175, 196/195
Commas: 126/125, 144/143, 176/175, 196/195


Line 57: Line 54:
Map: [<1 9 9 8 22 0|, <0 -10 -9 -7 -25 5|]
Map: [<1 9 9 8 22 0|, <0 -10 -9 -7 -25 5|]


EDOs: 27e, 31, 58
EDOs: {{EDOs|27e, 31, 58}}


Badness: 0.0171
Badness: 0.0171


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


Line 68: Line 65:
Map: [<1 9 9 8 22 20|, <0 -10 -9 -7 -25 -22|]
Map: [<1 9 9 8 22 20|, <0 -10 -9 -7 -25 -22|]


EDOs: 27e, 31f, 58f, 116cef
EDOs: {{EDOs|27e, 31f, 58f}}


Badness: 0.0276
Badness: 0.0276


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


Line 79: Line 76:
Map: [<1 9 9 8 22 23|, <0 -10 -9 -7 -25 -26|]
Map: [<1 9 9 8 22 23|, <0 -10 -9 -7 -25 -26|]


EDOs: 31
EDOs: {{EDOs|27eff, 31}}


Badness: 0.0299
Badness: 0.0299
Line 90: Line 87:
Map: [<1 9 9 8 -1|, <0 -10 -9 -7 6|]
Map: [<1 9 9 8 -1|, <0 -10 -9 -7 6|]


EDOs: 27, 31
EDOs: {{EDOs|27, 31}}


Badness: 0.0334
Badness: 0.0334
Line 101: Line 98:
Map: [<1 9 9 8 2|, <0 -10 -9 -7 2|]
Map: [<1 9 9 8 2|, <0 -10 -9 -7 2|]


EDOs: 23bc, 27e
EDOs: {{EDOs|23bc, 27e}}


Badness: 0.0487
Badness: 0.0487
Line 136: Line 133:
[[Generator]]s: 2, 14/9
[[Generator]]s: 2, 14/9


EDOs: 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd
EDOs: {{EDOs|19, 27, 46, 157d, 203cd, 249cdd, 295ccdd}}


Badness: 0.0256
Badness: 0.0256
Line 147: Line 144:
Map: [<1 6 8 11 -6|, <0 -7 -9 -13 15|]
Map: [<1 6 8 11 -6|, <0 -7 -9 -13 15|]


EDOs: 19, 27, 46, 111d, 157d
EDOs: {{EDOs|19, 27, 46, 111d, 157d}}


Badness: 0.0379
Badness: 0.0379
Line 158: Line 155:
Map: [<1 6 8 11 -6 10|, <0 -7 -9 -13 15 -10|]
Map: [<1 6 8 11 -6 10|, <0 -7 -9 -13 15 -10|]


EDOs: 19, 27, 46, 111df, 157df
EDOs: {{EDOs|19, 27, 46, 111df, 157df}}


Badness: 0.0256
Badness: 0.0256
Line 169: Line 166:
Map: [<1 6 8 11 6|, <0 -7 -9 -13 -4|]
Map: [<1 6 8 11 6|, <0 -7 -9 -13 -4|]


EDOs: 19, 27e, 73ee
EDOs: {{EDOs|19, 27e, 73ee}}


Badness: 0.0287
Badness: 0.0287
Line 180: Line 177:
Map: [<1 6 8 11 6 10|, <0 -7 -9 -13 -4 -10|]
Map: [<1 6 8 11 6 10|, <0 -7 -9 -13 -4 -10|]


EDOs: 19, 27e, 46e, 73ee
EDOs: {{EDOs|19, 27e, 46e, 73ee}}


Badness: 0.0200
Badness: 0.0200
Line 191: Line 188:
Map: [<1 6 8 11 23|, <0 -7 -9 -13 -31|]
Map: [<1 6 8 11 23|, <0 -7 -9 -13 -31|]


EDOs: 19e, 27e, 46, 119c, 165c
EDOs: {{EDOs|19e, 27e, 46, 119c, 165c}}


Badness: 0.0295
Badness: 0.0295
Line 202: Line 199:
Map: [<1 6 8 11 23 10|, <0 -7 -9 -13 -31 -10|]
Map: [<1 6 8 11 23 10|, <0 -7 -9 -13 -31 -10|]


EDOs: 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff
EDOs: {{EDOs|19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff}}


Badness: 0.0208
Badness: 0.0208
Line 236: Line 233:
[[Generator]]s: 2, 21/20
[[Generator]]s: 2, 21/20


EDOs: 15, 31, 46, 77, 185, 262
EDOs: {{EDOs|15, 31, 46, 77, 185, 262cd}}


Badness: 0.0311
Badness: 0.0311
Line 258: Line 255:
Map: [<1 1 2 3 3|, <0 9 5 -3 7|]
Map: [<1 1 2 3 3|, <0 9 5 -3 7|]


[[EDO]]s: [[15edo|15]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[108edo|108]], [[185edo|185]]
[[EDO]]s: {{EDOs|15, 31, 46, 77}}


Badness: 0.0167
Badness: 0.0167


See also: [[Chords of valentine]]
{{see also|Chords of valentine}}


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


Line 271: Line 268:
Map: [<1 1 2 3 3 2|, <0 9 5 -3 7 26|]
Map: [<1 1 2 3 3 2|, <0 9 5 -3 7 26|]


EDOs: 15, 46
EDOs: {{EDOs|15, 31f, 46}}


Badness: 0.0235
Badness: 0.0235


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


POTE generator: ~22/21 = 77.709
POTE generator: ~21/20 = 77.709


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


EDOs: 15, 31, 108, 139
EDOs: {{EDOs|15, 31, 108eff, 139efff}}


Badness: 0.0213
Badness: 0.0213


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


POTE generator: ~22/21 = 77.958
POTE generator: ~21/20 = 77.958


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


EDOs: 15, 31, 46, 77, 431
EDOs: {{EDOs|15f, 31, 46, 77, 431ccdeeeef}}


Badness: 0.0207
Badness: 0.0207


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


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


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


EDOs: 16, 30, 46, 62, 108ef
EDOs: {{EDOs|16, 30, 46, 62, 108ef}}


Badness: 0.0327
Badness: 0.0327
Line 319: Line 316:
Wedgie: <<8 13 23 2 14 17||
Wedgie: <<8 13 23 2 14 17||


EDOs: 19, 58, 77, 96
EDOs: {{EDOs|19, 39d, 58, 77, 135c}}


Badness: 0.0409
Badness: 0.0409


==11-limit==
== 11-limit ==
Commas: 126/125, 540/539, 896/891
Commas: 126/125, 540/539, 896/891


Line 330: Line 327:
Map: [<1 2 3 4 3|, <0 -8 -13 -23 9|]
Map: [<1 2 3 4 3|, <0 -8 -13 -23 9|]


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


Badness: 0.0392
Badness: 0.0392


==13-limit==
=== 13-limit ===
Commas: 126/125, 144/143, 196/195, 676/675
Commas: 126/125, 144/143, 196/195, 676/675


Line 341: Line 338:
Map: [<1 2 3 4 3 5|, <0 -8 -13 -23 9 -25|]
Map: [<1 2 3 4 3 5|, <0 -8 -13 -23 9 -25|]


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


Badness: 0.0237
Badness: 0.0237


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


Line 352: Line 349:
Map: [<1 2 3 4 2|, <0 -8 -13 -23 28|]
Map: [<1 2 3 4 2|, <0 -8 -13 -23 28|]


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


Badness: 0.0659
Badness: 0.0659


===13-limit===
=== 13-limit ===
Commas: 126/125, 196/195, 385/384, 676/675
Commas: 126/125, 196/195, 385/384, 676/675


Line 363: Line 360:
Map: [<1 2 3 4 2 5|, <0 -8 -13 -23 28 -25|]
Map: [<1 2 3 4 2 5|, <0 -8 -13 -23 28 -25|]


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


Badness: 0.0362
Badness: 0.0362


= Coblack =
= Coblack =
In addition to 126/125, the [[Trisedodge family|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.
{{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
Commas: 126/125, 16807/16384
Line 376: Line 375:
Map: [<5 1 7 14|, <0 3 2 0|]
Map: [<5 1 7 14|, <0 3 2 0|]


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


Badness: 0.1073
Badness: 0.1073
Line 387: Line 386:
Map: [<5 1 7 14 15|, <0 3 2 0 1|]
Map: [<5 1 7 14 15|, <0 3 2 0 1|]


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


= Casablanca =
= Casablanca =
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.
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.
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.
Line 400: Line 399:
Map: [<1 12 10 5|, <0 -19 -14 -4|]
Map: [<1 12 10 5|, <0 -19 -14 -4|]


EDOs: 9bc, 11b, 31, 135c, 166c
EDOs: {{EDOs|11b, 20b, 31, 104c, 135c, 166c}}


Badness: 0.1012
Badness: 0.1012
Line 411: Line 410:
Map: [<1 12 10 5 4|, |0 -19 -14 -4 -1>]
Map: [<1 12 10 5 4|, |0 -19 -14 -4 -1>]


EDOs: 9bc, 11b, 31, 259bce, 549bce
EDOs: {{EDOs|11b, 20b, 31}}


Badness: 0.0623
Badness: 0.0623


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


Line 422: Line 421:
Map: [<1 12 10 5 21|, |0 -19 -14 -4 -32>]
Map: [<1 12 10 5 21|, |0 -19 -14 -4 -32>]


EDOs: 9bce, 11be, 20be, 31, 42e, 73
EDOs: {{EDOs|31, 73, 104c, 135c}}


Badness: 0.0405
Badness: 0.0405


===13-limit===
=== 13-limit ===
126/125, 176/175, 196/195, 17303/17280
Commas: 126/125, 176/175, 196/195, 14641/14580


POTE generator: ~22/15 = 657.756
POTE generator: ~22/15 = 657.756
Line 433: Line 432:
Map: [<1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25>]
Map: [<1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25>]


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


Badness: 0.0408
Badness: 0.0408


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


Line 444: Line 443:
Map: [<1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6>]
Map: [<1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6>]


EDOs: 31, 73f, 104cf
EDOs: {{EDOs|31, 104cf, 135cf, 166c}}


Badness: 0.0414
Badness: 0.0414
Line 451: Line 450:
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.
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==
== 5-limit ==
Comma: 51018336/48828125
Comma: 51018336/48828125


Line 458: Line 457:
Map: [<1 3 4|, <0 -11 -13|]
Map: [<1 3 4|, <0 -11 -13|]


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


Badness: 0.4665
Badness: 0.4665
Line 483: Line 482:
[[Generator]]s: 2, 49/45
[[Generator]]s: 2, 49/45


EDOs: 7, 8, 31, 101, 132, 163
EDOs: {{EDOs|8d, 23d, 31, 101, 132c, 163c}}


Badness: 0.0504
Badness: 0.0504
Line 506: Line 505:
[[Generator]]s: 2, 11/10
[[Generator]]s: 2, 11/10


EDOs: 7, 8, 31, 101, 194
EDOs: {{EDOs|8d, 23de, 31, 101, 132ce, 163ce, 194cee}}


Badness: 0.0256
Badness: 0.0256


==13-limit==
==13-limit==
Commas: 66/65 99/98 121/120 126/125
Commas: 66/65, 99/98, 121/120, 126/125


POTE generator: ~11/10 = 154.478
POTE generator: ~11/10 = 154.478
Line 517: Line 516:
Map: [<1 3 4 5 5 5|, <0 -11 -13 -17 -12 -10|]
Map: [<1 3 4 5 5 5|, <0 -11 -13 -17 -12 -10|]


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


Badness: 0.0233
Badness: 0.0233
Line 530: Line 529:
Wedgie: <<12 5 -9 -20 -48 -35||
Wedgie: <<12 5 -9 -20 -48 -35||


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


Badness: 0.0884
Badness: 0.0884
Line 541: Line 540:
Map: [<1 8 5 -2 4|, <0 -12 -5 9 -1|]
Map: [<1 8 5 -2 4|, <0 -12 -5 9 -1|]


EDOs: 13, 15, 28, 43, 58
EDOs: {{EDOs|15, 43, 58}}


Badness: 0.0331
Badness: 0.0331
Line 552: Line 551:
Map: [<1 8 5 -2 4 16|, <0 -12 -5 9 -1 -23|]
Map: [<1 8 5 -2 4 16|, <0 -12 -5 9 -1 -23|]


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


Badness: 0.0228
Badness: 0.0228
Line 561: Line 560:
Map: [<1 -4 0 7 3 -7 12 1 5 3|, <0 12 5 -9 1 23 -17 7 -1 4|]
Map: [<1 -4 0 7 3 -7 12 1 5 3|, <0 12 5 -9 1 23 -17 7 -1 4|]


EDOs: 43, 58
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.)
(''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=
= Cypress =
== 5-limit ==
Comma: 258280326/244140625
Comma: 258280326/244140625


Line 572: Line 572:
Map: [<1 7 10|, <0 -12 -17|]
Map: [<1 7 10|, <0 -12 -17|]


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


Badness: 0.8166
Badness: 0.8166
Line 585: Line 585:
Wedgie: <<12 17 27 -1 9 15||
Wedgie: <<12 17 27 -1 9 15||


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


Badness: 0.0998
Badness: 0.0998
Line 596: Line 596:
Map: [<1 7 10 15 17|, <0 -12 -17 -27 -30|]
Map: [<1 7 10 15 17|, <0 -12 -17 -27 -30|]


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


Badness: 0.0427
Badness: 0.0427


==13-limit==
== 13-limit ==
Commas: 66/65, 99/98. 126/125, 243/242
Commas: 66/65, 99/98. 126/125, 243/242


Line 607: Line 607:
Map: [<1 7 10 15 17 15|, <0 -12 -17 -27 -30 -25|]
Map: [<1 7 10 15 17 15|, <0 -12 -17 -27 -30 -25|]


EDOs: 31
EDOs: {{EDOs|11cdeef, 20cdef, 31}}


Badness: 0.0378
Badness: 0.0378


=Bisemidim=
= Bisemidim =
Commas: 126/125, 118098/117649
Commas: 126/125, 118098/117649


Line 620: Line 620:
Wedgie: <<18 22 30 -7 -3 8||
Wedgie: <<18 22 30 -7 -3 8||


EDOs: 50, 58, 108, 166c, 408c
EDOs: {{EDOs|50, 58, 108, 166c, 408ccc}}


Badness: 0.0978
Badness: 0.0978


==11-limit==
== 11-limit ==
Commas: 126/125, 540/539, 1344/1331
Commas: 126/125, 540/539, 1344/1331


Line 631: Line 631:
Map: [<2 1 2 2 5|, <0 9 11 15 8|]
Map: [<2 1 2 2 5|, <0 9 11 15 8|]


EDOs: 50, 58, 108, 166ce, 224ce
EDOs: {{EDOs|50, 58, 108, 166ce, 224cee}}


Badness: 0.0412
Badness: 0.0412


==13-limit==
== 13-limit ==
Commas: 126/125, 144/143, 196/195, 364/363
Commas: 126/125, 144/143, 196/195, 364/363


Line 642: Line 642:
Map: [<2 1 2 2 5 5|, <0 9 11 15 8 10|]
Map: [<2 1 2 2 5 5|, <0 9 11 15 8 10|]


EDOs: 50, 58, 166cef, 224cef
EDOs: {{EDOs|50, 58, 166cef, 224ceeff}}


Badness: 0.0239
Badness: 0.0239


=Vines=
= Vines =
Commas: 126/125, 84035/82944
Commas: 126/125, 84035/82944


Line 653: Line 653:
Map: [<2 7 8 8|, <0 -8 -7 -5|]
Map: [<2 7 8 8|, <0 -8 -7 -5|]


EDOs: 4, 42, 46, 96d, 142d, 238d
EDOs: {{EDOs|42, 46, 96d, 142d, 238dd}}


Badness: 0.0780
Badness: 0.0780
Line 664: Line 664:
Map: [<2 7 8 8 5|, <0 -8 -7 -5 4|]
Map: [<2 7 8 8 5|, <0 -8 -7 -5 4|]


EDOs: 4, 42, 46, 96d, 142d, 238d
EDOs: {{EDOs|42, 46, 96d, 142d, 238dd}}


Badness: 0.0445
Badness: 0.0445
Line 675: Line 675:
Map: [<2 7 8 8 5 5|, <0 -8 -7 -5 4 5|]
Map: [<2 7 8 8 5 5|, <0 -8 -7 -5 4 5|]


EDOs: 4, 42, 46, 96d, 238df
EDOs: {{EDOs|42, 46, 96d, 238ddf}}


Badness: 0.0297
Badness: 0.0297


=Kumonga=
= Kumonga =
== 5-limit ==
Comma: 1289945088/1220703125
Comma: 1289945088/1220703125


Line 686: Line 687:
Map: [<1 4 4|, <0 -13 -9|]
Map: [<1 4 4|, <0 -13 -9|]


EDOs: 16, 27, 43, 70, 183c
EDOs: {{EDOs|16, 27, 43, 70, 183cc}}


Badness: 0.7296
Badness: 0.7296


==7-limit==
== 7-limit ==
Commas: 126/125, 12288/12005
Commas: 126/125, 12288/12005


Line 699: Line 700:
Wedgie: <<13 9 1 -16 -35 -23||
Wedgie: <<13 9 1 -16 -35 -23||


EDOs: 16, 27, 43, 70, 167cd
EDOs: {{EDOs|16, 27, 43, 70, 167ccdd}}


Badness: 0.0875
Badness: 0.0875


==11-limit==
== 11-limit ==
Commas: 126/125, 176/175, 864/847
Commas: 126/125, 176/175, 864/847


Line 710: Line 711:
Map: [<1 4 4 3 7|, <0 -13 -9 -1 -19|]
Map: [<1 4 4 3 7|, <0 -13 -9 -1 -19|]


EDOs: 16, 27e, 43, 70e
EDOs: {{EDOs|16, 27e, 43, 70e}}


Badness: 0.0433
Badness: 0.0433


==13-limit==
== 13-limit ==
Commas: 78/77, 126/125, 144/143, 176/175
Commas: 78/77, 126/125, 144/143, 176/175


Line 721: Line 722:
Map: [<1 4 4 3 7 5|, <0 -13 -9 -1 -19 -7|]
Map: [<1 4 4 3 7 5|, <0 -13 -9 -1 -19 -7|]


EDOs: 16, 27e, 43, 70e, 113cde
EDOs: {{EDOs|16, 27e, 43, 70e, 113cdee}}


Badness: 0.0289
Badness: 0.0289


=Amigo=
= Amigo =
Commas: 126/125, 2097152/2083725
Commas: 126/125, 2097152/2083725


Line 732: Line 733:
Map: [<1 9 3 -10|, <0 -11 -1 19|]
Map: [<1 9 3 -10|, <0 -11 -1 19|]


EDOs: 43, 46, 89, 135c, 359c
EDOs: {{EDOs|43, 46, 89, 135c, 359cc}}


Badness: 0.1109
Badness: 0.1109


==11-limit==
== 11-limit ==
Commas: 126/125, 176/175, 16384/16335
Commas: 126/125, 176/175, 16384/16335


Line 743: Line 744:
Map: [<1 9 3 -10 -8|, <0 -11 -1 19 17|]
Map: [<1 9 3 -10 -8|, <0 -11 -1 19 17|]


EDOs: 43, 46, 89, 135c, 224c
EDOs: {{EDOs|43, 46, 89, 135c, 224c}}


Badness: 0.0434
Badness: 0.0434


==13-limit==
== 13-limit ==
Commas: 126/125, 169/168, 176/175, 364/363
Commas: 126/125, 169/168, 176/175, 364/363


Line 754: Line 755:
Map: [<1 9 3 -10 -8 1|, <0 -11 -1 19 17 4|]
Map: [<1 9 3 -10 -8 1|, <0 -11 -1 19 17 4|]


EDOs: 43, 46, 89, 135cf, 224cf
EDOs: {{EDOs|43, 46, 89, 135cf, 224cf}}


Badness: 0.0307
Badness: 0.0307


=Oolong=
= Oolong =
{{main|Oolong}}
{{main|Oolong}}
== 5-limit ==
== 5-limit ==
Line 767: Line 768:
Map: [<1 6 7|, <0 -17 -18|]
Map: [<1 6 7|, <0 -17 -18|]


EDOs: 23, 27, 50, 77
EDOs: {{EDOs|23, 27, 50, 77}}


Badness: 0.9428
Badness: 0.9428
Line 778: Line 779:
Map: [&lt;1 6 7 8|, &lt;0 -17 -18 -20|]
Map: [&lt;1 6 7 8|, &lt;0 -17 -18 -20|]


EDOs: 27, 50, 77
EDOs: {{EDOs|27, 50, 77}}


Badness: 0.0735
Badness: 0.0735
Line 789: Line 790:
Map: [<1 6 7 8 18|, <0 -17 -18 -20 -56|]
Map: [<1 6 7 8 18|, <0 -17 -18 -20 -56|]


EDOs: 27e, 77, 104c, 181c
EDOs: {{EDOs|27e, 77, 104c, 181c}}


Badness: 0.0569
Badness: 0.0569
Line 800: Line 801:
Map: [<1 6 7 8 18 5|, <0 -17 -18 -20 -56 -5|]
Map: [<1 6 7 8 18 5|, <0 -17 -18 -20 -56 -5|]


EDOs: 27e, 77, 104c, 181c
EDOs: {{EDOs|27e, 77, 104c, 181c}}


Badness: 0.0356
Badness: 0.0356
Line 809: Line 810:
[[Category:Myna]]
[[Category:Myna]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Todo:improve layout]]
[[Category:Todo:review]]

Revision as of 07:21, 26 October 2020

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

Main article: 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, 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, 325cc

Badness: 0.2500

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

Badness: 0.0270

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: 27e, 31, 58, 89

Badness: 0.0168

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: 27e, 31, 58

Badness: 0.0171

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

Badness: 0.0276

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: 27eff, 31

Badness: 0.0299

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

Badness: 0.0334

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

Badness: 0.0487

Myna Music by Igliashon Jones

Sensi

Main article: 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 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, 157d, 203cd, 249cdd, 295ccdd

Badness: 0.0256

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: 19, 27, 46, 111d, 157d

Badness: 0.0379

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: 19, 27, 46, 111df, 157df

Badness: 0.0256

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, 27e, 73ee

Badness: 0.0287

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, 27e, 46e, 73ee

Badness: 0.0200

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: 19e, 27e, 46, 119c, 165c

Badness: 0.0295

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: 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff

Badness: 0.0208

Valentine

Main article: 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 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

Minimax tuning:

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, 262cd

Badness: 0.0311

11-limit

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

Minimax tuning:

[|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

Badness: 0.0167

See also: Chords of valentine

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, 31f, 46

Badness: 0.0235

Lupercalia

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

POTE generator: ~21/20 = 77.709

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

EDOs: 15, 31, 108eff, 139efff

Badness: 0.0213

Valentino

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

POTE generator: ~21/20 = 77.958

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

EDOs: 15f, 31, 46, 77, 431ccdeeeef

Badness: 0.0207

Semivalentine

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

POTE generator: ~21/20 = 77.839

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

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

Badness: 0.0327

Alicorn

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||

EDOs: 19, 39d, 58, 77, 135c

Badness: 0.0409

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, 39d, 58

Badness: 0.0392

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, 39df, 58

Badness: 0.0237

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, 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: 19, 58e, 77, 96d, 173d

Badness: 0.0362

Coblack

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: 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: 15, 35, 50, 65, 115d

Casablanca

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: 11b, 20b, 31, 104c, 135c, 166c

Badness: 0.1012

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: 11b, 20b, 31

Badness: 0.0623

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: 31, 73, 104c, 135c

Badness: 0.0405

13-limit

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: 31, 73, 104c, 135c, 239ccf

Badness: 0.0408

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, 104cf, 135cf, 166c

Badness: 0.0414

Nusecond

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, 365bcc

Badness: 0.4665

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: 8d, 23d, 31, 101, 132c, 163c

Badness: 0.0504

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

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

Badness: 0.0233

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

Badness: 0.0884

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: 15, 43, 58

Badness: 0.0331

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

Badness: 0.0228

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, 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.)

Cypress

5-limit

Comma: 258280326/244140625

POTE generator: ~4374/3125 = 541.726

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

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

Badness: 0.8166

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: 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd

Badness: 0.0998

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: 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde

Badness: 0.0427

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: 11cdeef, 20cdef, 31

Badness: 0.0378

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, 408ccc

Badness: 0.0978

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, 224cee

Badness: 0.0412

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, 224ceeff

Badness: 0.0239

Vines

Commas: 126/125, 84035/82944

POTE generator: ~6/5 = 312.602

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

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

Badness: 0.0780

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: 42, 46, 96d, 142d, 238dd

Badness: 0.0445

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: 42, 46, 96d, 238ddf

Badness: 0.0297

Kumonga

5-limit

Comma: 1289945088/1220703125

POTE generator: ~144/125 = 222.912

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

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

Badness: 0.7296

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, 167ccdd

Badness: 0.0875

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

Badness: 0.0433

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, 113cdee

Badness: 0.0289

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, 359cc

Badness: 0.1109

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

Badness: 0.0434

13-limit

Commas: 126/125, 169/168, 176/175, 364/363

POTE generator: ~5/4 = 391.072

Map: [<1 9 3 -10 -8 1|, <0 -11 -1 19 17 4|]

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

Badness: 0.0307

Oolong

Main article: Oolong

5-limit

Comma: [11 18 -17>

POTE generator: ~6/5 = 311.6942

Map: [<1 6 7|, <0 -17 -18|]

EDOs: 23, 27, 50, 77

Badness: 0.9428

7-limit

Commas: 126/125, 117649/116640

POTE generator: ~6/5 = 311.6793

Map: [<1 6 7 8|, <0 -17 -18 -20|]

EDOs: 27, 50, 77

Badness: 0.0735

11-limit

Commas: 126/125, 176/175, 26411/26244

POTE generator: ~6/5 = 311.5873

Map: [<1 6 7 8 18|, <0 -17 -18 -20 -56|]

EDOs: 27e, 77, 104c, 181c

Badness: 0.0569

13-limit

Commas: 126/125, 176/175, 196/195, 13013/12960

POTE generator: ~6/5 = 311.5908

Map: [<1 6 7 8 18 5|, <0 -17 -18 -20 -56 -5|]

EDOs: 27e, 77, 104c, 181c

Badness: 0.0356

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