Starling temperaments: Difference between revisions

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This page discusses some of the rank ~~two~~ temperaments tempering out [[126/125]], the starling comma or septimal semicomma. 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.

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

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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.~~

~~Subgroup: 2.3.5.7~~

~~[[Comma list]]: 126/125, 16807/16384~~

~~[[POTE generator]]: ~21/20 = 73.044~~

~~[[Mapping]]: [{{val| 5 1 7 14 }}, {{val| 0 3 2 0 }}]~~

~~{{Val list|legend=1| 15, 35, 50, 65, 115d }}~~

~~[[Badness]]: 0.1073~~

~~== 11-limit ==~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 126/125, 245/242, 385/384~~

~~POTE generator: ~21/20 = 73.264~~

~~Mapping: [{{val| 5 1 7 14 15 }}, {{val| 0 3 2 0 1 }}]~~

~~{{Val list|legend=1| 15, 35, 50, 65, 115d }}~~

= Casablanca =

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-This page discusses some of the rank two temperaments tempering out [[126/125]], the starling comma or septimal semicomma. 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.
+This page discusses some of the rank-2 temperaments tempering out [[126/125]], the starling comma or septimal semicomma. 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.
-Temperaments discussed else where are [[Father family #Pater|pater]], [[Dicot family #Flat|flat]], [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Diminished|diminished]], [[Kleismic family #Keemun|keemun]], [[Augmented family #Augene|augene]], [[Meantone family #Septimal meantone|septimal meantone]], [[Pelogic family #Mavila|mavila]], [[Sensipent family #Sensi|sensi]], [[Shibboleth family #Gilead|gilead]], [[Magic family #Muggles|muggles]], [[Diaschismic family #Diaschismic|diaschismic]], [[Tetracot family #Wollemia|wollemia]], [[Schismatic family #Grackle|grackle]] and [[Würschmidt family #Worschmidt|worschmidt]].
+Temperaments discussed else where are [[Father family #Pater|pater]], [[Dicot family #Flat|flat]], [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Diminished|diminished]], [[Kleismic family #Keemun|keemun]], [[Augmented family #Augene|augene]], [[Meantone family #Septimal meantone|septimal meantone]], [[Pelogic family #Mavila|mavila]], [[Sensipent family #Sensi|sensi]], [[Shibboleth family #Gilead|gilead]], [[Magic family #Muggles|muggles]], [[Diaschismic family #Diaschismic|diaschismic]], [[Tetracot family #Wollemia|wollemia]], [[Cloudy clan #Coblack|coblack]], [[Schismatic family #Grackle|grackle]] and [[Würschmidt family #Worschmidt|worschmidt]].
 = Myna =
@@ Line 328: / Line 328: @@
 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.
-Subgroup: 2.3.5.7
-[[Comma list]]: 126/125, 16807/16384
-[[POTE generator]]: ~21/20 = 73.044
-[[Mapping]]: [{{val| 5 1 7 14 }}, {{val| 0 3 2 0 }}]
-{{Val list|legend=1| 15, 35, 50, 65, 115d }}
-[[Badness]]: 0.1073
-== 11-limit ==
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 245/242, 385/384
-POTE generator: ~21/20 = 73.264
-Mapping: [{{val| 5 1 7 14 15 }}, {{val| 0 3 2 0 1 }}]
-{{Val list|legend=1| 15, 35, 50, 65, 115d }}
 = Casablanca =