Starling temperaments: Difference between revisions
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Temperaments discussed in families and clans are: | Temperaments discussed in families and clans are: | ||
* ''[[Brutus]]'' (+10/9) → [[Very low accuracy temperaments #Brutus|Very low accuracy temperaments]] | |||
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]] | * ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]] | ||
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]] | |||
* ''[[Mavling]]'' (+135/128) → [[Mavila family #Mavling|Mavila family]] | |||
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]] | |||
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]] | * ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]] | ||
* | * [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]] | ||
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]] | |||
* [[Opossum]] (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]] | |||
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]] | * [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]] | ||
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]] | |||
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]] | |||
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]] | |||
* ''[[Ripple]]'' (+405/392) → [[Ripple family #Septimal ripple|Ripple family]] | |||
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]] | * [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]] | ||
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]] | * [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]] | ||
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]] | * ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]] | ||
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]] | * ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]] | ||
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]] | |||
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]] | |||
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]] | * ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]] | ||
* ''[[ | * ''[[Paraguay]]'' (+12005/11664) → [[Parakleismic family #Paraguay|Parakleismic family]] | ||
* ''[[Vishnean]]'' (+540225/524288) → [[ | * ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]] | ||
* ''[[ | * ''[[Vishnean]]'' (+540225/524288) → [[Vishnu family #Vishnean|Vishnu family]] | ||
* ''[[ | * ''[[Bisemidim]]'' (+118098/117649) → [[Stearnsmic clan #Bisemidim|Stearnsmic clan]] | ||
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] | |||
Considered below are myna, nusecond, oolong, vines, kumonga, cypress | Considered below are myna, nusecond, oolong, vines, kumonga, cypress, casablanca, amigo, supersensi, and cobalt, sorted by increasing [[badness]]. | ||
Since {{nowrap|(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 {{nowrap|(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. | ||
== Myna == | == Myna == | ||
| Line 67: | Line 71: | ||
Comma list: 126/125, 176/175, 243/242 | Comma list: 126/125, 176/175, 243/242 | ||
{{Mapping|legend=0| 1 -1 0 1 -3 | 0 10 9 7 25 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 82: | Line 86: | ||
Comma list: 126/125, 144/143, 176/175, 196/195 | Comma list: 126/125, 144/143, 176/175, 196/195 | ||
{{Mapping|legend=0| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 97: | Line 101: | ||
Comma list: 78/77, 91/90, 126/125, 176/175 | Comma list: 78/77, 91/90, 126/125, 176/175 | ||
{{Mapping|legend=0| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 112: | Line 116: | ||
Comma list: 66/65, 105/104, 126/125, 243/242 | Comma list: 66/65, 105/104, 126/125, 243/242 | ||
{{Mapping|legend=0| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 127: | Line 131: | ||
Comma list: 99/98, 126/125, 385/384 | Comma list: 99/98, 126/125, 385/384 | ||
{{Mapping|legend=0| 1 -1 0 1 5 | 0 10 9 7 -6 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 142: | Line 146: | ||
Comma list: 56/55, 100/99, 1728/1715 | Comma list: 56/55, 100/99, 1728/1715 | ||
{{Mapping|legend=0| 1 -1 0 1 4 | 0 10 9 7 -2 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 189: | Line 193: | ||
Comma list: 99/98, 121/120, 126/125 | Comma list: 99/98, 121/120, 126/125 | ||
{{Mapping|legend=0| 1 -8 -9 -12 -7 | 0 11 13 17 12 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 211: | Line 215: | ||
Comma list: 66/65, 99/98, 121/120, 126/125 | Comma list: 66/65, 99/98, 121/120, 126/125 | ||
{{Mapping|legend=0| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 247: | Line 251: | ||
Comma list: 126/125, 176/175, 26411/26244 | Comma list: 126/125, 176/175, 26411/26244 | ||
{{Mapping|legend=0| 1 -11 -11 -12 -38 | 0 17 18 20 56 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 262: | Line 266: | ||
Comma list: 126/125, 176/175, 196/195, 13013/12960 | Comma list: 126/125, 176/175, 196/195, 13013/12960 | ||
{{Mapping|legend=0| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 299: | Line 303: | ||
Comma list: 126/125, 385/384, 2401/2376 | Comma list: 126/125, 385/384, 2401/2376 | ||
{{Mapping|legend=0| 2 -1 1 3 9 | 0 8 7 5 -4 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 314: | Line 318: | ||
Comma list: 126/125, 196/195, 364/363, 385/384 | Comma list: 126/125, 196/195, 364/363, 385/384 | ||
{{Mapping|legend=0| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 325: | Line 329: | ||
== Xenial == | == Xenial == | ||
{{Main| Xenial }} | |||
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].'' | : ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].'' | ||
Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 | Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 342: | Line 347: | ||
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }} | : error map: {{val| 0.000 -1.644 +3.162 -1.021 }} | ||
{{Optimal ET sequence|legend=1| 19, 70, 89 }} | {{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }} | ||
[[Badness]] (Sintel): 2.13 | [[Badness]] (Sintel): 2.13 | ||
| Line 351: | Line 356: | ||
Comma list: 126/125, 540/539, 16384/16335 | Comma list: 126/125, 540/539, 16384/16335 | ||
{{Mapping|legend=0| 1 -6 -12 -25 22 | 0 9 17 33 -22 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}} | ||
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011. | * CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 70, 89 }} | {{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }} | ||
Badness (Sintel): 2.31 | Badness (Sintel): 2.31 | ||
| Line 366: | Line 371: | ||
Comma list: 126/125, 169/168, 540/539, 729/728 | Comma list: 126/125, 169/168, 540/539, 729/728 | ||
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}} | ||
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011. | * CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 70, 89 }} | {{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }} | ||
Badness (Sintel): 1.98 | Badness (Sintel): 1.98 | ||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539 | |||
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}} | |||
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }} | |||
Badness (Sintel): 2.06 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539 | |||
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}} | |||
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }} | |||
Badness (Sintel): 2.03 | |||
=== 23-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19.23 | |||
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230 | |||
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}} | |||
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }} | |||
Badness (Sintel): 1.93 | |||
== Kumonga == | == Kumonga == | ||
| Line 401: | Line 451: | ||
Comma list: 126/125, 176/175, 864/847 | Comma list: 126/125, 176/175, 864/847 | ||
{{Mapping|legend=0| 1 -9 -5 2 -12 | 0 13 9 1 19 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 416: | Line 466: | ||
Comma list: 78/77, 126/125, 144/143, 176/175 | Comma list: 78/77, 126/125, 144/143, 176/175 | ||
{{Mapping|legend=0| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 425: | Line 475: | ||
Badness (Sintel): 1.19 | Badness (Sintel): 1.19 | ||
== Cypress == | == Cypress == | ||
| Line 500: | Line 500: | ||
Comma list: 99/98, 126/125, 243/242 | Comma list: 99/98, 126/125, 243/242 | ||
{{Mapping|legend=0| 1 -5 -7 -12 -13 | 0 12 17 27 30 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 515: | Line 515: | ||
Comma list: 66/65, 99/98, 126/125, 243/242 | Comma list: 66/65, 99/98, 126/125, 243/242 | ||
{{Mapping|legend=0| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 558: | Line 558: | ||
Comma list: 126/125, 385/384, 2420/2401 | Comma list: 126/125, 385/384, 2420/2401 | ||
{{Mapping|legend=0| 1 -7 -4 1 3 | 0 19 14 4 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 573: | Line 573: | ||
Comma list: 126/125, 196/195, 385/384, 2420/2401 | Comma list: 126/125, 196/195, 385/384, 2420/2401 | ||
{{Mapping|legend=0| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 588: | Line 588: | ||
Comma list: 126/125, 176/175, 14641/14580 | Comma list: 126/125, 176/175, 14641/14580 | ||
{{Mapping|legend=0| 1 -7 -4 1 -11 | 0 19 14 4 32 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 603: | Line 603: | ||
Comma list: 126/125, 176/175, 196/195, 14641/14580 | Comma list: 126/125, 176/175, 196/195, 14641/14580 | ||
{{Mapping|legend=0| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 618: | Line 618: | ||
Comma list: 126/125, 144/143, 176/175, 1540/1521 | Comma list: 126/125, 144/143, 176/175, 1540/1521 | ||
{{Mapping|legend=0| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 653: | Line 653: | ||
Comma list: 126/125, 176/175, 16384/16335 | Comma list: 126/125, 176/175, 16384/16335 | ||
{{Mapping|legend=0| 1 -2 2 9 9 | 0 11 1 -19 -17 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 668: | Line 668: | ||
Comma list: 126/125, 169/168, 176/175, 364/363 | Comma list: 126/125, 169/168, 176/175, 364/363 | ||
{{Mapping|legend=0| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 677: | Line 677: | ||
Badness (Sintel): 1.27 | Badness (Sintel): 1.27 | ||
== Supersensi == | == Supersensi == | ||
| Line 723: | Line 703: | ||
Comma list: 99/98, 126/125, 864/847 | Comma list: 99/98, 126/125, 864/847 | ||
{{Mapping|legend=0| 1 -4 -4 -5 -1 | 0 15 17 21 12 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 738: | Line 718: | ||
Comma list: 78/77, 99/98, 126/125, 144/143 | Comma list: 78/77, 99/98, 126/125, 144/143 | ||
{{Mapping|legend=0| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 753: | Line 733: | ||
Comma list: 78/77, 99/98, 120/119, 126/125, 144/143 | Comma list: 78/77, 99/98, 120/119, 126/125, 144/143 | ||
{{Mapping|legend=0| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 792: | Line 772: | ||
Comma list: 126/125, 540/539, 21609/21296 | Comma list: 126/125, 540/539, 21609/21296 | ||
{{Mapping|legend=0| 27 0 20 33 8 | 0 1 1 1 2 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 807: | Line 787: | ||
Comma list: 126/125, 144/143, 196/195, 21609/21296 | Comma list: 126/125, 144/143, 196/195, 21609/21296 | ||
{{Mapping|legend=0| 27 0 20 33 8 100 | 0 1 1 1 2 0 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 822: | Line 802: | ||
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445 | Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445 | ||
{{Mapping|legend=0| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 837: | Line 817: | ||
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968 | Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968 | ||
{{Mapping|legend=0| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 852: | Line 832: | ||
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968 | Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968 | ||
{{Mapping|legend=0| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 867: | Line 847: | ||
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083 | Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083 | ||
{{Mapping|legend=0| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 882: | Line 862: | ||
Comma list: 126/125, 169/168, 540/539, 975/968 | Comma list: 126/125, 169/168, 540/539, 975/968 | ||
{{Mapping|legend=0| 27 0 20 33 8 57 | 0 1 1 1 2 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 894: | Line 874: | ||
== References == | == References == | ||
[[Category:Starling temperaments| ]] <!-- main article --> | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||