Revision as of 07:55, 11 April 2021

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 pater, flat, opossum, diminished, keemun, augene, septimal meantone, mavila, gilead, muggles, diaschismic, wollemia, grackle and worschmidt.

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. 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: [⟨1 9 9], ⟨0 -10 -9]]

POTE generator: ~6/5 = 310.140

@@ Line 6: / Line 6: @@
 {{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&amp;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<sup>1/10</sup> 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&amp;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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
 == 5-limit (mynic) ==
@@ Line 31: / Line 31: @@
 Mapping generators: ~2, ~5/3
+{{Multival|legend=1| 10 9 7 -9 -17 -9 }}
 [[POTE generator]]: ~6/5 = 310.146
@@ Line 431: / Line 433: @@
 = Alicorn =
-{{see also|Unicorn family #Alicorn}}
+{{see also| Unicorn family #Alicorn }}
+Subgroup: 2.3.5.7
-Commas: 126/125, 10976/10935
+[[Comma list]]: 126/125, 10976/10935
-POTE generator: ~28/27 = 62.278
+[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -8 -13 -23 }}]
-Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
+{{Multival|legend=1| 8 13 23 2 14 17 }}
-Wedgie: &lt;&lt;8 13 23 2 14 17||
+[[POTE generator]]: ~28/27 = 62.278
-EDOs: {{EDOs|19, 39d, 58, 77, 135c}}
+{{Val list|legend=1| 19, 39d, 58, 77, 135c }}
-Badness: 0.0409
+[[Badness]]: 0.0409
 == 11-limit ==
-Commas: 126/125, 540/539, 896/891
+Subgroup: 2.3.5.7.11
+Comma list: 126/125, 540/539, 896/891
+Mapping: [{{val| 1 2 3 4 3 }}, {{val| 0 -8 -13 -23 9 }}]
 POTE generator: ~28/27 = 62.101
-Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
+{{Val list|legend=1| 19, 39d, 58 }}
-EDOs: {{EDOs|19, 39d, 58}}
 Badness: 0.0392
 === 13-limit ===
-Commas: 126/125, 144/143, 196/195, 676/675
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 144/143, 196/195, 676/675
+Mapping: [{{val| 1 2 3 4 3 5 }}, {{val| 0 -8 -13 -23 9 -25 }}]
 POTE generator: ~28/27 = 62.119
-Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
+{{Val list|legend=1| 19, 39df, 58 }}
-EDOs: {{EDOs|19, 39df, 58}}
 Badness: 0.0237
 == Camahueto ==
-Commas: 126/125, 10976/10935, 385/384
+Subgroup: 2.3.5.7.11
+Comma list: 126/125, 385/384, 10976/10935
+Mapping: [{{val| 1 2 3 4 2 }}, {{val| 0 -8 -13 -23 28 }}]
 POTE generator: ~28/27 = 62.431
-Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
+{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
-EDOs: {{EDOs|19, 58e, 77, 96d, 173d}}
 Badness: 0.0659
 === 13-limit ===
-Commas: 126/125, 196/195, 385/384, 676/675
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 196/195, 385/384, 676/675
+Mapping: [{{val| 1 2 3 4 2 5 }}, {{val| 0 -8 -13 -23 28 -25 }}]
 POTE generator: ~28/27 = 62.434
-Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
+{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
-EDOs: {{EDOs|19, 58e, 77, 96d, 173d}}
 Badness: 0.0362
@@ Line 492: / Line 508: @@
 {{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.
+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.
+Subgroup: 2.3.5.7
-Commas: 126/125, 16807/16384
+[[Comma list]]: 126/125, 16807/16384
-POTE generator: ~21/20 = 73.044
+[[POTE generator]]: ~21/20 = 73.044
-Map: [&lt;5 1 7 14|, &lt;0 3 2 0|]
+[[Mapping]]: [{{val| 5 1 7 14 }}, {{val| 0 3 2 0 }}]
-EDOs: {{EDOs|15, 35, 50, 65, 115d}}
+{{Val list|legend=1| 15, 35, 50, 65, 115d }}
-Badness: 0.1073
+[[Badness]]: 0.1073
-==11-limit==
+== 11-limit ==
-Commas: 126/125, 245/242, 385/384
+Subgroup: 2.3.5.7.11
+Comma list: 126/125, 245/242, 385/384
 POTE generator: ~21/20 = 73.264
-Map: [&lt;5 1 7 14 15|, &lt;0 3 2 0 1|]
+Mapping: [{{val| 5 1 7 14 15 }}, {{val| 0 3 2 0 1 }}]
-EDOs: {{EDOs|15, 35, 50, 65, 115d}}
+{{Val list|legend=1| 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, &lt;&lt;19 14 4 -22 -47 -30||, or as 31&amp;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 as 31&amp;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
+Subgroup: 2.3.5.7
-POTE generator: ~35/24 = 657.818
+[[Comma list]]: 126/125, 589824/588245
-Map: [&lt;1 12 10 5|, &lt;0 -19 -14 -4|]
+[[Mapping]]: [{{val| 1 12 10 5 }}, {{val| 0 -19 -14 -4 }}]
-EDOs: {{EDOs|11b, 20b, 31, 104c, 135c, 166c}}
+{{Multival|legend=1| 19 14 4 -22 -47 -30 }}
-Badness: 0.1012
+[[POTE generator]]: ~35/24 = 657.818
-==11-limit==
+{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
-Commas: 126/125, 385/384, 2420/2401
+[[Badness]]: 0.1012
+== 11-limit ==
+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
-Map: [&lt;1 12 10 5 4|, |0 -19 -14 -4 -1&gt;]
+{{Val list|legend=1| 11b, 20b, 31 }}
-EDOs: {{EDOs|11b, 20b, 31}}
 Badness: 0.0623
 == Marrakesh ==
-Commas: 126/125, 176/175, 14641/14580
+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
-Map: [&lt;1 12 10 5 21|, |0 -19 -14 -4 -32&gt;]
+{{Val list|legend=1| 31, 73, 104c, 135c }}
-EDOs: {{EDOs|31, 73, 104c, 135c}}
 Badness: 0.0405
 === 13-limit ===
-Commas: 126/125, 176/175, 196/195, 14641/14580
+Subgroup: 2.3.5.7.11.13
+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
-Map: [&lt;1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25&gt;]
+{{Val list|legend=1| 31, 73, 104c, 135c, 239ccf }}
-EDOs: {{EDOs|31, 73, 104c, 135c, 239ccf}}
 Badness: 0.0408
 === Murakuc ===
-Commas: 126/125, 144/143, 176/175, 1540/1521
+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
-Map: [&lt;1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6&gt;]
+{{Val list|legend=1| 31, 104cf, 135cf, 166c }}
-EDOs: {{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&amp;70, or in terms of its wedgie as &lt;&lt;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&amp;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.
 == 5-limit ==
-Comma: 51018336/48828125
-POTE generator: ~3125/2916 = 154.523
+Subgroup: 2.3.5
-Map: [&lt;1 3 4|, &lt;0 -11 -13|]
+[[Comma list]]: 51018336/48828125
-EDOs: {{EDOs|8, 23, 31, 70, 101, 132c, 233c, 365bcc}}
+[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -11 -13 }}]
-Badness: 0.4665
+[[POTE generator]]: ~3125/2916 = 154.523
-==7-limit==
+{{Val list|legend=1| 8, 23, 31, 70, 101, 132c, 233c, 365bcc }}
-[[Comma]]s: 126/125, 2430/2401
--limit minimax
+[[Badness]]: 0.4665
-[|1 0 0 0&gt;, |-5/13 0 11/13 0&gt;, |0 0 1 0&gt;, |-3/13 0 17/13 0&gt;]
+== 7-limit ==
-[[Eigenmonzo]]s: 2, 5
+Subgroup: 2.3.5.7
--limit minimax
+[[Comma list]]: 126/125, 2430/2401
-[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]
+[[Mapping]]: [{{val| 1 3 4 5 }}, {{val| 0 -11 -13 -17 }}]
-[[Eigenmonzo]]s: 2, 3
+Mapping generators: ~2, ~49/45
-[[POTE_tuning|POTE generator]]: 154.579
+{{Multival|legend=1| 11 13 17 -5 -4 3 }}
-Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]
+[[POTE generator]]: ~49/45 = 154.579
-[[Generator]]s: 2, 49/45
+[[Minimax tuning]]:
+* [[7-odd-limit]]
+: [{{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
+* [[9-odd-limit]]
+: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]
+: [[Eigenmonzo]]s: 2, 3
-EDOs: {{EDOs|8d, 23d, 31, 101, 132c, 163c}}
+{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
-Badness: 0.0504
+[[Badness]]: 0.0504
-==11-limit==
+== 11-limit ==
-[[Comma]]s: 99/98, 121/120, 126/125
--limit minimax
+Subgroup: 2.3.5.7.11
-[|1 0 0 0 0&gt;, |19/10 11/5 0 0 -11/10&gt;,
+Comma list: 99/98, 121/120, 126/125
-|27/10 13/5 0 0 -13/10&gt;, |33/10 17/5 0 0 -17/10&gt;,
-|19/5 12/5 0 0 -6/5&gt;<nowiki>]</nowiki>
-[[Eigenmonzo]]s: 2, 11/9
+Mapping: [{{val| 1 3 4 5 5 }}, {{val| 0 -11 -13 -17 -12 }}]
-[[POTE_tuning|POTE generator]]: ~11/10 = 154.645
+Mapping generators: ~2, ~11/10
-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.
+POTE generator: ~11/10 = 154.645
-Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]
+Minimax tuning:
+* 11-odd-limit
+: [{{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 }}]
+: Eigenmonzos: 2, 11/9
-[[Generator]]s: 2, 11/10
+Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
-EDOs: {{EDOs|8d, 23de, 31, 101, 132ce, 163ce, 194cee}}
+{{Val list|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
 Badness: 0.0256
-==13-limit==
+== 13-limit ==
-Commas: 66/65, 99/98, 121/120, 126/125
+Subgroup: 2.3.5.7.11.13
+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
-Map: [&lt;1 3 4 5 5 5|, &lt;0 -11 -13 -17 -12 -10|]
+{{Val list|legend=1| 8d, 23de, 31, 70f, 101ff }}
+Badness: 0.0233
-EDOs: {{EDOs|8d, 23de, 31, 70f, 101ff}}
+= Thuja =
+Comma list: 126/125, 65536/64827
-Badness: 0.0233
+Mapping: [{{val| 1 8 5 -2 }}, {{val| 0 -12 -5 9 }}]
-=Thuja=
+{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}
-Commas: 126/125, 65536/64827
 POTE generator: ~175/128 = 558.605
-Map: [&lt;1 8 5 -2|, &lt;0 -12 -5 9|]
+{{Val list|legend=1| 15, 43, 58 }}
-Wedgie: &lt;&lt;12 5 -9 -20 -48 -35||
+Badness: 0.0884
-EDOs: {{EDOs|15, 43, 58}}
+== 11-limit ==
-Badness: 0.0884
+Subgroup: 2.3.5.7.11
+Comma list: 126/125, 176/175, 1344/1331
-==11-limit==
+Mapping: [{{val| 1 8 5 -2 4 }}, {{val| 0 -12 -5 9 -1 }}]
-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|]
+{{Val list|legend=1| 15, 43, 58 }}
-EDOs: {{EDOs|15, 43, 58}}
+Badness: 0.0331
-Badness: 0.0331
+== 13-limit ==
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 144/143, 176/175, 364/363
-==13-limit==
+Mapping: [{{val| 1 8 5 -2 4 16 }}, {{val| 0 -12 -5 9 -1 -23 }}]
-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|]
+{{Val list|legend=1| 15, 43, 58 }}
+Badness: 0.0228
-EDOs: {{EDOs|15, 43, 58}}
+== 29-limit ==
-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.
-==29-limit==
+Subgroup: 2.3.5.7.11.13.17.19.23.29
-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|]
+Mapping: [{{val| 1 -4 0 7 3 -7 12 1 5 3 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 4 }}]
-EDOs: {{EDOs|43, 58hi}}
+POTE generator: ~11/8 = 558.520
-(''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.)
+{{Val list|legend=1| 43, 58hi }}
 = Cypress =

Starling temperaments: Difference between revisions

Revision as of 07:55, 11 April 2021

Myna

5-limit (mynic)

7-limit

11-limit

13-limit

Minah

Maneh

Myno

Coleto

Sensi

7-limit

Sensation

Sensor

13-limit

Sensis

13-limit

Sensus

13-limit

Sensa

13-limit

Hemisensi

Valentine

5-limit

7-limit

11-limit

Dwynwen

Lupercalia

Valentino

Semivalentine

Alicorn

11-limit

13-limit

Camahueto

13-limit

Coblack

11-limit

Casablanca

11-limit

Marrakesh

13-limit

Murakuc

Nusecond

5-limit

7-limit

11-limit

13-limit

Thuja

11-limit

13-limit

29-limit

Cypress

5-limit

7-limit

11-limit

13-limit

Bisemidim

11-limit

13-limit

Vines

11-limit

13-limit

Kumonga

5-limit

7-limit

11-limit

13-limit

Amigo

11-limit

13-limit

Oolong

5-limit

7-limit

11-limit

13-limit

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