Starling temperaments: Difference between revisions
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Since (6/5)<sup>3</sup> = 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. | Since (6/5)<sup>3</sup> = 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. 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&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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits. | ||
== 5-limit (mynic) == | === 5-limit (mynic) === | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 40: | Line 40: | ||
[[Badness]]: 0.249965 | [[Badness]]: 0.249965 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 62: | Line 62: | ||
[[Badness]]: 0.027044 | [[Badness]]: 0.027044 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 75: | Line 75: | ||
Badness: 0.016842 | Badness: 0.016842 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 88: | Line 88: | ||
Badness: 0.017125 | Badness: 0.017125 | ||
=== Minah === | ==== Minah ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 101: | Line 101: | ||
Badness: 0.027568 | Badness: 0.027568 | ||
=== Maneh === | ==== Maneh ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 114: | Line 114: | ||
Badness: 0.029868 | Badness: 0.029868 | ||
== Myno == | === Myno === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 127: | Line 127: | ||
Badness: 0.033434 | Badness: 0.033434 | ||
== Coleto == | === Coleto === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 140: | Line 140: | ||
Badness: 0.048687 | Badness: 0.048687 | ||
= Valentine = | == Valentine == | ||
{{main| Valentine }} | {{main| Valentine }} | ||
{{see also| Gamelismic clan #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)<sup>1/9</sup> 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; | 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)<sup>1/9</sup> 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)<sup>1/10</sup>. | ||
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. | 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. | ||
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[[Badness]]: 0.122765 | [[Badness]]: 0.122765 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 185: | Line 185: | ||
[[Badness]]: 0.031056 | [[Badness]]: 0.031056 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.032749 | Badness: 0.032749 | ||
= Casablanca = | == Casablanca == | ||
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. | 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. | ||
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[[Badness]]: 0.101191 | [[Badness]]: 0.101191 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 291: | Line 291: | ||
Badness: 0.067291 | Badness: 0.067291 | ||
== Marrakesh == | === Marrakesh === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 304: | Line 304: | ||
Badness: 0.040539 | Badness: 0.040539 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 317: | Line 317: | ||
Badness: 0.040774 | Badness: 0.040774 | ||
=== Murakuc === | ==== Murakuc ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 330: | Line 330: | ||
Badness: 0.041395 | Badness: 0.041395 | ||
= Nusecond = | == 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. | 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|31EDO]] can be used as a tuning, or [[132edo|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 | Subgroup: 2.3.5 | ||
| Line 345: | Line 345: | ||
[[Badness]]: 0.4665 | [[Badness]]: 0.4665 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 370: | Line 370: | ||
[[Badness]]: 0.0504 | [[Badness]]: 0.0504 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 392: | Line 392: | ||
Badness: 0.0256 | Badness: 0.0256 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 405: | Line 405: | ||
Badness: 0.0233 | Badness: 0.0233 | ||
= Thuja = | == Thuja == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 420: | Line 420: | ||
Badness: 0.0884 | Badness: 0.0884 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 433: | Line 433: | ||
Badness: 0.0331 | Badness: 0.0331 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 446: | Line 446: | ||
Badness: 0.0228 | Badness: 0.0228 | ||
== 29-limit == | === 29-limit === | ||
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. | 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. | ||
| Line 457: | Line 457: | ||
{{Val list|legend=1| 43, 58hi }} | {{Val list|legend=1| 43, 58hi }} | ||
= Cypress = | == Cypress == | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 470: | Line 470: | ||
[[Badness]]: 0.8166 | [[Badness]]: 0.8166 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 485: | Line 485: | ||
[[Badness]]: 0.0998 | [[Badness]]: 0.0998 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 498: | Line 498: | ||
Badness: 0.0427 | Badness: 0.0427 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 511: | Line 511: | ||
Badness: 0.0378 | Badness: 0.0378 | ||
= Bisemidim = | == Bisemidim == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 526: | Line 526: | ||
[[Badness]]: 0.0978 | [[Badness]]: 0.0978 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 539: | Line 539: | ||
Badness: 0.0412 | Badness: 0.0412 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 552: | Line 552: | ||
Badness: 0.0239 | Badness: 0.0239 | ||
= Vines = | == Vines == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 565: | Line 565: | ||
Badness: 0.0780 | Badness: 0.0780 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 578: | Line 578: | ||
Badness: 0.0445 | Badness: 0.0445 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 591: | Line 591: | ||
Badness: 0.0297 | Badness: 0.0297 | ||
= Kumonga = | == Kumonga == | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 604: | Line 604: | ||
[[Badness]]: 0.7296 | [[Badness]]: 0.7296 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 619: | Line 619: | ||
[[Badness]]: 0.0875 | [[Badness]]: 0.0875 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 632: | Line 632: | ||
Badness: 0.0433 | Badness: 0.0433 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 645: | Line 645: | ||
Badness: 0.0289 | Badness: 0.0289 | ||
= Amigo = | == Amigo == | ||
{{see also| Sensamagic clan #Magus }} | {{see also| Sensamagic clan #Magus }} | ||
| Line 660: | Line 660: | ||
[[Badness]]: 0.1109 | [[Badness]]: 0.1109 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 673: | Line 673: | ||
Badness: 0.0434 | Badness: 0.0434 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 686: | Line 686: | ||
Badness: 0.0307 | Badness: 0.0307 | ||
= Oolong = | == Oolong == | ||
{{main|Oolong}} | {{main|Oolong}} | ||
| Line 701: | Line 701: | ||
[[Badness]]: 0.9428 | [[Badness]]: 0.9428 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 714: | Line 714: | ||
[[Badness]]: 0.0735 | [[Badness]]: 0.0735 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 727: | Line 727: | ||
Badness: 0.0569 | Badness: 0.0569 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||