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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 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. |
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| 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]], [[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]], [[Schismatic family #Grackle|grackle]] and [[Würschmidt family #Worschmidt|worschmidt]]. |
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| = Myna = | | = Myna = |
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| {{Val list|legend=1| 4, …, 23bc, 27e }} | | {{Val list|legend=1| 4, …, 23bc, 27e }} |
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| Badness: 0.0487
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| = Sensi =
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| {{main|Sensi}}
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| {{see also|Sensipent family #Sensi}}
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| Sensi tempers out [[686/675]], [[245/243]] and [[4375/4374]] in addition to [[126/125]], and can be described as the 19&27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."
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| == 7-limit ==
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| Subgroup: 2.3.5.7
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| [[Comma list]]: 126/125, 245/243
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| [[Mapping]]: [{{val| 1 6 8 11 }}, {{val| 0 -7 -9 -13 }}]
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| Mapping generators: ~2, ~14/9
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| {{Multival|legend=1| 7 9 13 -2 1 5 }}
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| [[POTE generator]]: ~9/7 = 443.383
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| [[Minimax tuning]]:
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| * [[7-odd-limit]]
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| : [{{monzo| 1 0 0 0 }}, {{monzo| 1/13 0 0 7/13 }}, {{monzo| 5/13 0 0 9/13 }}, {{monzo| 0 0 0 1 }}]
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| : [[Eigenmonzo]]s: 2, 7
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| * [[9-odd-limit]]
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| : [{{monzo| 1 0 0 0 }}, {{monzo| 2/5 14/5 -7/5 0 }}, {{monzo| 4/5 18/5 -9/5 0 }}, {{monzo| 3/5 26/5 -13/5 0 }}]
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| : [[Eigenmonzo]]s: 2, 9/5
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| [[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.
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| {{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
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| [[Badness]]: 0.0256
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| === Sensation ===
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| Subgroup: 2.3.5.7.13
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| Comma list: 91/90, 126/125, 169/168
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| Sval mapping: [{{val| 1 6 8 11 10 }}, {{val| 0 -7 -9 -13 -10 }}]
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| Gencom mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}]
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| Gencom: [2 9/7; 91/90 126/125 169/168]
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| POTE generator: ~9/7 = 443.322
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| {{Val list|legend=1| 19, 27, 46, 111de, 157de }}
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| == Sensor ==
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| Subgroup: 2.3.5.7.11
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| Comma list: 126/125, 245/243, 385/384
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| Mapping: [{{val| 1 6 8 11 -6 }}, {{val| 0 -7 -9 -13 15 }}]
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| POTE generator: ~9/7 = 443.294
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| {{Val list|legend=1| 19, 27, 46, 111d, 157d, 268cdd }}
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| Badness: 0.0379
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 91/90, 126/125, 169/168, 385/384
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| Mapping: [{{val| 1 6 8 11 -6 10 }}, {{val| 0 -7 -9 -13 15 -10 }}]
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| POTE generator: ~9/7 = 443.321
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| {{Val list|legend=1| 19, 27, 46, 111df, 157df }}
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| Badness: 0.0256
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| == Sensis ==
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| Subgroup: 2.3.5.7.11
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| Comma list: 56/55, 100/99, 245/243
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| Mapping: [{{val| 1 6 8 11 6 }}, {{val| 0 -7 -9 -13 -4 }}]
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| POTE generator: ~9/7 = 443.962
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| {{Val list|legend=1| 19, 27e, 73ee }}
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| Badness: 0.0287
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 56/55, 78/77, 91/90, 100/99
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| Mapping: [{{val| 1 6 8 11 6 10 }}, {{val| 0 -7 -9 -13 -4 -10 }}]
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| POTE generator: ~9/7 = 443.945
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| {{Val list|legend=1| 19, 27e, 46e, 73ee }}
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| Badness: 0.0200
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| == Sensus ==
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| Subgroup: 2.3.5.7.11
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| Comma list: 126/125, 176/175, 245/243
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| Mapping: [{{val| 1 6 8 11 23 }}, {{val| 0 -7 -9 -13 -31 }}]
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| POTE generator: ~9/7 = 443.626
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| {{Val list|legend=1| 19e, 27e, 46, 119c, 165c }}
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| Badness: 0.0295
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 91/90, 126/125, 169/168, 352/351
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| Mapping: [{{val| 1 6 8 11 23 10 }}, {{val| 0 -7 -9 -13 -31 -10 }}]
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| POTE generator: ~9/7 = 443.559
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| {{Val list|legend=1| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
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| Badness: 0.0208
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| == Sensa ==
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| Subgroup: 2.3.5.7.11
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| Comma list: 55/54, 77/75, 99/98
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| Mapping: [{{val| 1 6 8 11 11 }}, {{val| 0 -7 -9 -13 -12 }}]
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| POTE generator: ~9/7 = 443.518
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| {{Val list|legend=1| 19e, 27, 46ee }}
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| Badness: 0.0368
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 55/54, 66/65, 77/75, 143/140
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| Mapping: [{{val| 1 6 8 11 11 10 }}, {{val| 0 -7 -9 -13 -12 -11 }}]
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| POTE generator: ~9/7 = 443.506
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| {{Val list|legend=1| 19e, 27, 46ee }}
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| Badness: 0.0233
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| == Hemisensi ==
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| Subgroup: 2.3.5.7.11
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| Comma list: 126/125, 243/242, 245/242
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| Mapping: [{{val| 1 13 17 24 32 }}, {{val| 0 -14 -18 -26 -35 }}]
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| POTE generator: ~25/22 = 221.605
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| {{Val list|legend=1| 27e, 65, 157de, 222cde }}
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| Badness: 0.0487 | | Badness: 0.0487 |