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Starling temperaments: Difference between revisions

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* [[Würschmidt family #Worschmidt|worschmidt]] ({126/125, 33075/32768}, würschmidt family)
* [[Würschmidt family #Worschmidt|worschmidt]] ({126/125, 33075/32768}, würschmidt 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]] 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.  


== Myna ==
== Myna ==
Line 343: Line 343:
{{Val list|legend=1| 8, 23, 31, 70, 101, 132c, 233c, 365bcc }}
{{Val list|legend=1| 8, 23, 31, 70, 101, 132c, 233c, 365bcc }}


[[Badness]]: 0.4665
[[Badness]]: 0.466493


=== 7-limit ===
=== 7-limit ===
Line 368: Line 368:
{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


[[Badness]]: 0.0504
[[Badness]]: 0.050389


=== 11-limit ===
=== 11-limit ===
Line 388: Line 388:
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.


{{Val list|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
Vals: {{Val list| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}


Badness: 0.0256
Badness: 0.025621


=== 13-limit ===
=== 13-limit ===
Line 401: Line 401:
POTE generator: ~11/10 = 154.478
POTE generator: ~11/10 = 154.478


{{Val list|legend=1| 8d, 23de, 31, 70f, 101ff }}
Vals: {{Val list| 8d, 23de, 31, 70f, 101ff }}


Badness: 0.0233
Badness: 0.023323


== Thuja ==
== Thuja ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 126/125, 65536/64827
[[Comma list]]: 126/125, 65536/64827


Mapping: [{{val| 1 8 5 -2 }}, {{val| 0 -12 -5 9 }}]
[[Mapping]]: [{{val| 1 -4 0 7 }}, {{val| 0 12 5 -9 }}]


{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}
{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}


POTE generator: ~175/128 = 558.605
[[POTE generator]]: ~175/128 = 558.605


{{Val list|legend=1| 15, 43, 58 }}
{{Val list|legend=1| 15, 43, 58 }}


Badness: 0.0884
[[Badness]]: 0.088441


=== 11-limit ===
=== 11-limit ===
Line 425: Line 425:
Comma list: 126/125, 176/175, 1344/1331
Comma list: 126/125, 176/175, 1344/1331


Mapping: [{{val| 1 8 5 -2 4 }}, {{val| 0 -12 -5 9 -1 }}]
Mapping: [{{val| 1 -4 0 7 3 }}, {{val| 0 12 5 -9 1 }}]


POTE generator: ~11/8 = 558.620
POTE generator: ~11/8 = 558.620


{{Val list|legend=1| 15, 43, 58 }}
Vals: {{Val list| 15, 43, 58 }}


Badness: 0.0331
Badness: 0.033078


=== 13-limit ===
=== 13-limit ===
Line 438: Line 438:
Comma list: 126/125, 144/143, 176/175, 364/363
Comma list: 126/125, 144/143, 176/175, 364/363


Mapping: [{{val| 1 8 5 -2 4 16 }}, {{val| 0 -12 -5 9 -1 -23 }}]
Mapping: [{{val| 1 -4 0 7 3 -7 }}, {{val| 0 12 5 -9 1 23 }}]


POTE generator: ~11/8 = 558.589
POTE generator: ~11/8 = 558.589


{{Val list|legend=1| 15, 43, 58 }}
Vals: {{Val list| 15, 43, 58 }}
 
Badness: 0.022838
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 126/125, 144/143, 176/175, 221/220, 256/255
 
Mapping: [{{val| 1 -4 0 7 3 -7 12 }}, {{val| 0 12 5 -9 1 23 -17 }}]
 
POTE generator: ~11/8 = 558.509
 
Vals: {{Val list| 15, 43, 58 }}
 
Badness: 0.022293
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220
 
Mapping: [{{val| 1 -4 0 7 3 -7 12 1 }}, {{val| 0 12 5 -9 1 23 -17 7 }}]
 
POTE generator: ~11/8 = 558.504
 
Vals: {{Val list| 15, 43, 58h }}
 
Badness: 0.018938
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230
 
Mapping: [{{val| 1 -4 0 7 3 -7 12 1 5 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 }}]
 
POTE generator: ~11/8 = 558.522
 
Vals: {{Val list| 15, 43, 58hi }}


Badness: 0.0228
Badness: 0.016581


=== 29-limit ===
=== 29-limit ===
Line 455: Line 494:
POTE generator: ~11/8 = 558.520
POTE generator: ~11/8 = 558.520


{{Val list|legend=1| 43, 58hi }}
Vals: {{Val list| 15, 43, 58hi }}
 
Badness: 0.013762


== Cypress ==
== Cypress ==
Line 468: Line 509:
{{Val list|legend=1| 11c, 20c, 31, 113c, 144c, 175c, 381bcc }}
{{Val list|legend=1| 11c, 20c, 31, 113c, 144c, 175c, 381bcc }}


[[Badness]]: 0.8166
[[Badness]]: 0.816635


=== 7-limit ===
=== 7-limit ===
Line 483: Line 524:
{{Val list|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}
{{Val list|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}


[[Badness]]: 0.0998
[[Badness]]: 0.099801


=== 11-limit ===
=== 11-limit ===
Line 494: Line 535:
POTE generator: ~15/11 = 541.772
POTE generator: ~15/11 = 541.772


{{Val list|legend=1| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
Vals: {{Val list| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}


Badness: 0.0427
Badness: 0.042719


=== 13-limit ===
=== 13-limit ===
Line 507: Line 548:
POTE generator: ~15/11 = 541.778
POTE generator: ~15/11 = 541.778


{{Val list|legend=1| 11cdeef, 20cdef, 31 }}
Vals: {{Val list| 11cdeef, 20cdef, 31 }}


Badness: 0.0378
Badness: 0.037849


== Bisemidim ==
== Bisemidim ==
Line 524: Line 565:
{{Val list|legend=1| 50, 58, 108, 166c, 408ccc }}
{{Val list|legend=1| 50, 58, 108, 166c, 408ccc }}


[[Badness]]: 0.0978
[[Badness]]: 0.097786


=== 11-limit ===
=== 11-limit ===
Line 535: Line 576:
POTE generator: ~35/27 = 455.373
POTE generator: ~35/27 = 455.373


{{Val list|legend=1| 50, 58, 108, 166ce, 224cee }}
Vals: {{Val list| 50, 58, 108, 166ce, 224cee }}


Badness: 0.0412
Badness: 0.041190


=== 13-limit ===
=== 13-limit ===
Line 548: Line 589:
POTE generator: ~35/27 = 455.347
POTE generator: ~35/27 = 455.347


{{Val list|legend=1| 50, 58, 166cef, 224ceeff }}
Vals: {{Val list| 50, 58, 166cef, 224ceeff }}


Badness: 0.0239
Badness: 0.023877


== Vines ==
== Vines ==
Line 563: Line 604:
{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}
{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}


Badness: 0.0780
[[Badness]]: 0.078049


=== 11-limit ===
=== 11-limit ===
Line 574: Line 615:
POTE generator: ~6/5 = 312.601
POTE generator: ~6/5 = 312.601


{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}
Vals: {{Val list| 42, 46, 96d, 142d, 238dd }}


Badness: 0.0445
Badness: 0.044499


=== 13-limit ===
=== 13-limit ===
Line 587: Line 628:
POTE generator: ~6/5 = 312.564
POTE generator: ~6/5 = 312.564


{{Val list|legend=1| 42, 46, 96d, 238ddf }}
Vals: {{Val list| 42, 46, 96d, 238ddf }}


Badness: 0.0297
Badness: 0.029693


== Kumonga ==
== Kumonga ==
Line 602: Line 643:
{{Val list|legend=1| 16, 27, 43, 70, 183cc }}
{{Val list|legend=1| 16, 27, 43, 70, 183cc }}


[[Badness]]: 0.7296
[[Badness]]: 0.729609


=== 7-limit ===
=== 7-limit ===
Line 617: Line 658:
{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}
{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}


[[Badness]]: 0.0875
[[Badness]]: 0.087500


=== 11-limit ===
=== 11-limit ===
Line 628: Line 669:
POTE generator: ~8/7 = 222.898
POTE generator: ~8/7 = 222.898


{{Val list|legend=1| 16, 27e, 43, 70e }}
Vals: {{Val list| 16, 27e, 43, 70e }}


Badness: 0.0433
Badness: 0.043336


=== 13-limit ===
=== 13-limit ===
Line 641: Line 682:
POTE generator: ~8/7 = 222.961
POTE generator: ~8/7 = 222.961


{{Val list|legend=1| 16, 27e, 43, 70e, 113cdee }}
Vals: {{Val list| 16, 27e, 43, 70e, 113cdee }}


Badness: 0.0289
Badness: 0.028920


== Amigo ==
== Amigo ==
Line 652: Line 693:
[[Comma list]]: 126/125, 2097152/2083725
[[Comma list]]: 126/125, 2097152/2083725


[[Mapping]]: [{{val| 1 9 3 -10 }}, {{val| 0 -11 -1 19 }}]
[[Mapping]]: [{{val| 1 -2 2 9 }}, {{val| 0 11 1 -19 }}]
 
{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}


[[POTE generator]]: ~5/4 = 391.094
[[POTE generator]]: ~5/4 = 391.094
Line 658: Line 701:
{{Val list|legend=1| 43, 46, 89, 135c, 359cc }}
{{Val list|legend=1| 43, 46, 89, 135c, 359cc }}


[[Badness]]: 0.1109
[[Badness]]: 0.110873


=== 11-limit ===
=== 11-limit ===
Line 665: Line 708:
Comma list: 126/125, 176/175, 16384/16335
Comma list: 126/125, 176/175, 16384/16335


Mapping: [{{val| 1 9 3 -10 -8 }}, {{val| 0 -11 -1 19 17 }}]
Mapping: [{{val| 1 -2 2 9 9 }}, {{val| 0 11 1 -19 -17 }}]


POTE generator: ~5/4 = 391.075
POTE generator: ~5/4 = 391.075


{{Val list|legend=1| 43, 46, 89, 135c, 224c }}
Vals: {{Val list| 43, 46, 89, 135c, 224c }}


Badness: 0.0434
Badness: 0.043438


=== 13-limit ===
=== 13-limit ===
Line 678: Line 721:
Comma list: 126/125, 169/168, 176/175, 364/363
Comma list: 126/125, 169/168, 176/175, 364/363


Mapping: [{{val| 1 9 3 -10 -8 1 }}, {{val| 0 -11 -1 19 17 4 }}]
Mapping: [{{val| 1 -2 2 9 9 5 }}, {{val| 0 11 1 -19 -17 -4 }}]


POTE generator: ~5/4 = 391.072
POTE generator: ~5/4 = 391.073


{{Val list|legend=1| 43, 46, 89, 135cf, 224cf }}
Vals: {{Val list| 43, 46, 89, 135cf, 224cf }}


Badness: 0.0307
Badness: 0.030666


== Oolong ==
== Oolong ==
Line 691: Line 734:
Subgroup: 2.3.5
Subgroup: 2.3.5


[[Comma list]]: {{monzo|11 18 -17}}
[[Comma list]]: {{monzo|11 18 -17}} = 793437161472/762939453125


[[Mapping]]: [{{val| 1 6 7 }}, {{val| 0 -17 -18 }}]
[[Mapping]]: [{{val| 1 6 7 }}, {{val| 0 -17 -18 }}]


[[POTE generator]]: ~6/5 = 311.6942
[[POTE generator]]: ~6/5 = 311.694


{{Val list|legend=1| 23, 27, 50, 77 }}
{{Val list|legend=1| 23, 27, 50, 77 }}


[[Badness]]: 0.9428
[[Badness]]: 0.942789


=== 7-limit ===
=== 7-limit ===
Line 708: Line 751:
[[Mapping]]: [{{val| 1 6 7 8 }}, {{val| 0 -17 -18 -20 }}]
[[Mapping]]: [{{val| 1 6 7 8 }}, {{val| 0 -17 -18 -20 }}]


[[POTE generator]]: ~6/5 = 311.6793
[[POTE generator]]: ~6/5 = 311.679


{{Val list|legend=1| 27, 50, 77 }}
{{Val list|legend=1| 27, 50, 77 }}


[[Badness]]: 0.0735
[[Badness]]: 0.073509


=== 11-limit ===
=== 11-limit ===
Line 721: Line 764:
Mapping: [{{val| 1 6 7 8 18 }}, {{val| 0 -17 -18 -20 -56 }}]
Mapping: [{{val| 1 6 7 8 18 }}, {{val| 0 -17 -18 -20 -56 }}]


POTE generator: ~6/5 = 311.5873
POTE generator: ~6/5 = 311.587


{{Val list|legend=1| 27e, 77, 104c, 181c }}
Vals: {{Val list| 27e, 77, 104c, 181c }}


Badness: 0.0569
Badness: 0.056915


=== 13-limit ===
=== 13-limit ===
Line 734: Line 777:
Mapping: [{{val| 1 6 7 8 18 5 }}, {{val| 0 -17 -18 -20 -56 -5 }}]
Mapping: [{{val| 1 6 7 8 18 5 }}, {{val| 0 -17 -18 -20 -56 -5 }}]


POTE generator: ~6/5 = 311.5908
POTE generator: ~6/5 = 311.591


{{Val list|legend=1| 27e, 77, 104c, 181c }}
Vals: {{Val list| 27e, 77, 104c, 181c }}


Badness: 0.0356
Badness: 0.035582


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"