Starling temperaments: Difference between revisions
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* ''[[Vishnean]]'', {126/125, 540225/524288} → [[Vishnuzmic family #Vishnean|Vishnuzmic family]] | * ''[[Vishnean]]'', {126/125, 540225/524288} → [[Vishnuzmic family #Vishnean|Vishnuzmic family]] | ||
* ''[[Ditonic]]'', {126/125, 8751645/8388608} → [[Ditonmic family #Ditonic|Ditonmic family]] | * ''[[Ditonic]]'', {126/125, 8751645/8388608} → [[Ditonmic family #Ditonic|Ditonmic family]] | ||
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|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|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. | ||
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Badness: 0.035582 | Badness: 0.035582 | ||
== Cobalt == | |||
The name of ''cobalt temperament'' comes from Cobalt, the 27th element. | |||
Cobalt (27 & 81) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]]. | |||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 126/125, 40353607/40310784 | |||
[[Mapping]]: [{{val|27 43 63 76}}, {{val|0 -1 -1 -1}}] | |||
[[POTE generator]]: ~3/2 = 701.244 | |||
{{Val list|legend=1| 27, 81, 108, 135c, 243c }} | |||
[[Badness]]: 0.173308 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 126/125, 540/539, 21609/21296 | |||
Mapping: [{{val|27 43 63 76 94}}, {{val|0 -1 -1 -1 -2}}] | |||
POTE generator: ~3/2 = 700.001 | |||
Optimal GPV sequence: {{Val list| 27e, 81, 108 }} | |||
Badness: 0.078060 | |||
==== Cobaltand ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 126/125, 144/143, 196/195, 21609/21296 | |||
Mapping: [{{val|27 43 63 76 94 100}}, {{val|0 -1 -1 -1 -2 0}}] | |||
POTE generator: ~3/2 = 700.867 | |||
Optimal GPV sequence: {{Val list| 27e, 81, 108, 243ceef }} | |||
Badness: 0.057145 | |||
==== Cobalat ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 126/125, 169/168, 540/539, 975/968 | |||
Mapping: [{{val|27 43 63 76 94 100}}, {{val|0 -1 -1 -1 -2 -1}}] | |||
POTE generator: ~3/2 = 699.179 | |||
Optimal GPV sequence: {{Val list| 27e, 54bdef, 81f, 108f }} | |||
Badness: 0.052732 | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||