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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]]
* ''[[Enneaplonic]]'', {126/125, 40353607/40310784} → [[Tritrizo clan #Enneaplonic|Tritrizo clan]]


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.  
Line 44: Line 45:


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 9/10 9/10 0 0 }}, {{monzo| 17/10 7/10 0 0 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 9/10 9/10 0 0 }}, {{monzo| 17/10 7/10 0 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 3
: [[Eigenmonzo]]s (unchanged intervals): 2, 3
Line 61: Line 62:
POTE generator: ~6/5 = 310.144
POTE generator: ~6/5 = 310.144


Vals: {{Val list| 27e, 31, 58, 89 }}
Optimal GPV sequence: {{Val list| 27e, 31, 58, 89 }}


Badness: 0.016842
Badness: 0.016842
Line 74: Line 75:
POTE generator: ~6/5 = 310.276
POTE generator: ~6/5 = 310.276


Vals: {{Val list| 27e, 31, 58 }}
Optimal GPV sequence: {{Val list| 27e, 31, 58 }}


Badness: 0.017125
Badness: 0.017125
Line 87: Line 88:
POTE generator: ~6/5 = 310.381
POTE generator: ~6/5 = 310.381


Vals: {{Val list| 27e, 31f, 58f }}
Optimal GPV sequence: {{Val list| 27e, 31f, 58f }}


Badness: 0.027568
Badness: 0.027568
Line 100: Line 101:
POTE generator: ~6/5 = 309.804
POTE generator: ~6/5 = 309.804


Vals: {{Val list| 27eff, 31 }}
Optimal GPV sequence: {{Val list| 27eff, 31 }}


Badness: 0.029868
Badness: 0.029868
Line 113: Line 114:
POTE generator: ~6/5 = 309.737
POTE generator: ~6/5 = 309.737


Vals: {{Val list| 27, 31 }}
Optimal GPV sequence: {{Val list| 27, 31 }}


Badness: 0.033434
Badness: 0.033434
Line 126: Line 127:
POTE generator: ~6/5 = 310.853
POTE generator: ~6/5 = 310.853


Vals: {{Val list| 4, 23bc, 27e }}
Optimal GPV sequence: {{Val list| 4, 23bc, 27e }}


Badness: 0.048687
Badness: 0.048687
Line 192: Line 193:
Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.


Vals: {{Val list| 15, 31, 46, 77, 262cdee, 339cdeee }}
Optimal GPV sequence: {{Val list| 15, 31, 46, 77, 262cdee, 339cdeee }}


Badness: 0.016687
Badness: 0.016687
Line 205: Line 206:
POTE generator: ~21/20 = 78.219
POTE generator: ~21/20 = 78.219


Vals: {{Val list| 15, 31f, 46 }}
Optimal GPV sequence: {{Val list| 15, 31f, 46 }}


Badness: 0.023461
Badness: 0.023461
Line 218: Line 219:
POTE generator: ~21/20 = 77.709
POTE generator: ~21/20 = 77.709


Vals: {{Val list| 15, 31, 77ff, 108eff, 139efff }}
Optimal GPV sequence: {{Val list| 15, 31, 77ff, 108eff, 139efff }}


Badness: 0.021328
Badness: 0.021328
Line 231: Line 232:
POTE generator: ~21/20 = 77.958
POTE generator: ~21/20 = 77.958


Vals: {{Val list| 15f, 31, 46, 77 }}
Optimal GPV sequence: {{Val list| 15f, 31, 46, 77 }}


Badness: 0.020665
Badness: 0.020665
Line 244: Line 245:
POTE generator: ~21/20 = 77.839
POTE generator: ~21/20 = 77.839


Vals: {{Val list| 16, 30, 46, 62, 108ef }}
Optimal GPV sequence: {{Val list| 16, 30, 46, 62, 108ef }}


Badness: 0.032749
Badness: 0.032749
Line 257: Line 258:
POTE generator: ~40/39 = 39.044
POTE generator: ~40/39 = 39.044


Vals: {{Val list| 30, 31, 61, 92f, 123f }}
Optimal GPV sequence: {{Val list| 30, 31, 61, 92f, 123f }}


Badness: 0.047059
Badness: 0.047059
Line 270: Line 271:
POTE generator: ~45/44 = 38.921
POTE generator: ~45/44 = 38.921


Vals: {{Val list| 31, 92e, 123, 154, 185 }}
Optimal GPV sequence: {{Val list| 31, 92e, 123, 154, 185 }}


Badness: 0.061275
Badness: 0.061275
Line 283: Line 284:
POTE generator: ~45/44 = 38.948
POTE generator: ~45/44 = 38.948


Vals: {{Val list| 31, 92e, 123, 154 }}
Optimal GPV sequence: {{Val list| 31, 92e, 123, 154 }}


Badness: 0.057919
Badness: 0.057919
Line 296: Line 297:
POTE generator: ~40/39 = 38.993
POTE generator: ~40/39 = 38.993


Vals: {{Val list| 31, 92ef, 123f }}
Optimal GPV sequence: {{Val list| 31, 92ef, 123f }}


Badness: 0.057931
Badness: 0.057931
Line 330: Line 331:
POTE generator: ~16/11 = 657.923
POTE generator: ~16/11 = 657.923


Vals: {{Val list| 11b, 20b, 31 }}
Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}


Badness: 0.067291
Badness: 0.067291
Line 343: Line 344:
POTE generator: ~16/11 = 657.854
POTE generator: ~16/11 = 657.854


Vals: {{Val list| 11b, 20b, 31 }}
Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}


=== Marrakesh ===
=== Marrakesh ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 355: Line 355:
POTE generator: ~22/15 = 657.791
POTE generator: ~22/15 = 657.791


Vals: {{Val list| 31, 73, 104c, 135c }}
Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c }}


Badness: 0.040539
Badness: 0.040539
Line 368: Line 368:
POTE generator: ~22/15 = 657.756
POTE generator: ~22/15 = 657.756


Vals: {{Val list| 31, 73, 104c, 135c, 239ccf }}
Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c, 239ccf }}


Badness: 0.040774
Badness: 0.040774
Line 381: Line 381:
POTE generator: ~22/15 = 657.700
POTE generator: ~22/15 = 657.700


Vals: {{Val list| 31, 104cff, 135cff }}
Optimal GPV sequence: {{Val list| 31, 104cff, 135cff }}


Badness: 0.041395
Badness: 0.041395
Line 432: Line 432:
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.


Vals: {{Val list| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
Optimal GPV sequence: {{Val list| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}


Badness: 0.025621
Badness: 0.025621
Line 445: Line 445:
POTE generator: ~11/10 = 154.478
POTE generator: ~11/10 = 154.478


Vals: {{Val list| 8d, 23de, 31, 70f, 101ff }}
Optimal GPV sequence: {{Val list| 8d, 23de, 31, 70f, 101ff }}


Badness: 0.023323
Badness: 0.023323
Line 475: Line 475:
POTE generator: ~11/8 = 558.620
POTE generator: ~11/8 = 558.620


Vals: {{Val list| 15, 43, 58 }}
Optimal GPV sequence: {{Val list| 15, 43, 58 }}


Badness: 0.033078
Badness: 0.033078
Line 488: Line 488:
POTE generator: ~11/8 = 558.589
POTE generator: ~11/8 = 558.589


Vals: {{Val list| 15, 43, 58 }}
Optimal GPV sequence: {{Val list| 15, 43, 58 }}


Badness: 0.022838
Badness: 0.022838
Line 501: Line 501:
POTE generator: ~11/8 = 558.509
POTE generator: ~11/8 = 558.509


Vals: {{Val list| 15, 43, 58 }}
Optimal GPV sequence: {{Val list| 15, 43, 58 }}


Badness: 0.022293
Badness: 0.022293
Line 514: Line 514:
POTE generator: ~11/8 = 558.504
POTE generator: ~11/8 = 558.504


Vals: {{Val list| 15, 43, 58h }}
Optimal GPV sequence: {{Val list| 15, 43, 58h }}


Badness: 0.018938
Badness: 0.018938
Line 527: Line 527:
POTE generator: ~11/8 = 558.522
POTE generator: ~11/8 = 558.522


Vals: {{Val list| 15, 43, 58hi }}
Optimal GPV sequence: {{Val list| 15, 43, 58hi }}


Badness: 0.016581
Badness: 0.016581
Line 542: Line 542:
POTE generator: ~11/8 = 558.520
POTE generator: ~11/8 = 558.520


Vals: {{Val list| 15, 43, 58hi }}
Optimal GPV sequence: {{Val list| 15, 43, 58hi }}


Badness: 0.013762
Badness: 0.013762
Line 572: Line 572:
POTE generator: ~15/11 = 541.772
POTE generator: ~15/11 = 541.772


Vals: {{Val list| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
Optimal GPV sequence: {{Val list| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}


Badness: 0.042719
Badness: 0.042719
Line 585: Line 585:
POTE generator: ~15/11 = 541.778
POTE generator: ~15/11 = 541.778


Vals: {{Val list| 11cdeef, 20cdef, 31 }}
Optimal GPV sequence: {{Val list| 11cdeef, 20cdef, 31 }}


Badness: 0.037849
Badness: 0.037849
Line 613: Line 613:
POTE generator: ~35/27 = 455.373
POTE generator: ~35/27 = 455.373


Vals: {{Val list| 50, 58, 108, 166ce, 224cee }}
Optimal GPV sequence: {{Val list| 50, 58, 108, 166ce, 224cee }}


Badness: 0.041190
Badness: 0.041190
Line 626: Line 626:
POTE generator: ~35/27 = 455.347
POTE generator: ~35/27 = 455.347


Vals: {{Val list| 50, 58, 166cef, 224ceeff }}
Optimal GPV sequence: {{Val list| 50, 58, 166cef, 224ceeff }}


Badness: 0.023877
Badness: 0.023877
Line 654: Line 654:
POTE generator: ~6/5 = 312.601
POTE generator: ~6/5 = 312.601


Vals: {{Val list| 42, 46, 96d, 142d, 238dd }}
Optimal GPV sequence: {{Val list| 42, 46, 96d, 142d, 238dd }}


Badness: 0.044499
Badness: 0.044499
Line 667: Line 667:
POTE generator: ~6/5 = 312.564
POTE generator: ~6/5 = 312.564


Vals: {{Val list| 42, 46, 96d, 238ddf }}
Optimal GPV sequence: {{Val list| 42, 46, 96d, 238ddf }}


Badness: 0.029693
Badness: 0.029693
Line 697: Line 697:
POTE generator: ~8/7 = 222.898
POTE generator: ~8/7 = 222.898


Vals: {{Val list| 16, 27e, 43, 70e }}
Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e }}


Badness: 0.043336
Badness: 0.043336
Line 710: Line 710:
POTE generator: ~8/7 = 222.961
POTE generator: ~8/7 = 222.961


Vals: {{Val list| 16, 27e, 43, 70e, 113cdee }}
Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e, 113cdee }}


Badness: 0.028920
Badness: 0.028920
Line 740: Line 740:
POTE generator: ~5/4 = 391.075
POTE generator: ~5/4 = 391.075


Vals: {{Val list| 43, 46, 89, 135c, 224c }}
Optimal GPV sequence: {{Val list| 43, 46, 89, 135c, 224c }}


Badness: 0.043438
Badness: 0.043438
Line 753: Line 753:
POTE generator: ~5/4 = 391.073
POTE generator: ~5/4 = 391.073


Vals: {{Val list| 43, 46, 89, 135cf, 224cf }}
Optimal GPV sequence: {{Val list| 43, 46, 89, 135cf, 224cf }}


Badness: 0.030666
Badness: 0.030666
Line 784: Line 784:
POTE generator: ~6/5 = 311.587
POTE generator: ~6/5 = 311.587


Vals: {{Val list| 27e, 77, 104c, 181c }}
Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}


Badness: 0.056915
Badness: 0.056915
Line 797: Line 797:
POTE generator: ~6/5 = 311.591
POTE generator: ~6/5 = 311.591


Vals: {{Val list| 27e, 77, 104c, 181c }}
Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}


Badness: 0.035582
Badness: 0.035582
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"