Starling temperaments: Difference between revisions

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=Starling comma=

This page discusses some of the rank two temperaments tempering out [[~~126/125|~~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|~~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 temperament=

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

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Badness: 0.0362

=Coblack temperament=

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.

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

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

=Casablanca temperament=

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 __FORCETOC__
 =Starling comma=
-This page discusses some of the rank two temperaments tempering out [[126/125|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|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 temperament=
 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 &lt;&lt;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
@@ Line 355: / Line 355: @@
 Badness: 0.0362
+=Coblack temperament=
+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.
+Commas: 126/125, 16807/16384
+POTE generator: ~21/20 = 73.044
+Map: [&lt;5 1 7 14|, &lt;0 3 2 0|]
+EDOs: 15, 35, 50, 65
+Badness: 0.1073
+==11-limit==
+Commas: 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|]
+EDOs: 15, 35, 50, 65
 =Casablanca temperament=