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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]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
Line 26: Line 27:
* ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]]
* ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnu family #Vishnean|Vishnu 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]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]]


Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, supersensi, and cobalt, sorted by increasing [[badness]].  
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.  
Line 69: Line 71:
Comma list: 126/125, 176/175, 243/242
Comma list: 126/125, 176/175, 243/242


Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}
{{Mapping|legend=0| 1 -1 0 1 -3 | 0 10 9 7 25 }}


Optimal tunings:  
Optimal tunings:  
Line 84: Line 86:
Comma list: 126/125, 144/143, 176/175, 196/195
Comma list: 126/125, 144/143, 176/175, 196/195


Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
{{Mapping|legend=0| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 99: Line 101:
Comma list: 78/77, 91/90, 126/125, 176/175
Comma list: 78/77, 91/90, 126/125, 176/175


Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}
{{Mapping|legend=0| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}


Optimal tunings:  
Optimal tunings:  
Line 114: Line 116:
Comma list: 66/65, 105/104, 126/125, 243/242
Comma list: 66/65, 105/104, 126/125, 243/242


Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}
{{Mapping|legend=0| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}


Optimal tunings:  
Optimal tunings:  
Line 129: Line 131:
Comma list: 99/98, 126/125, 385/384
Comma list: 99/98, 126/125, 385/384


Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}
{{Mapping|legend=0| 1 -1 0 1 5 | 0 10 9 7 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 144: Line 146:
Comma list: 56/55, 100/99, 1728/1715
Comma list: 56/55, 100/99, 1728/1715


Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}
{{Mapping|legend=0| 1 -1 0 1 4 | 0 10 9 7 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 157: Line 159:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''


Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 70 }}. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.  
Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as the {{nowrap| 31 & 39d }} temperament. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.  


[[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.
[[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.
Line 191: Line 193:
Comma list: 99/98, 121/120, 126/125
Comma list: 99/98, 121/120, 126/125


Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}
{{Mapping|legend=0| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}


Optimal tunings:  
Optimal tunings:  
Line 213: Line 215:
Comma list: 66/65, 99/98, 121/120, 126/125
Comma list: 66/65, 99/98, 121/120, 126/125


Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}
{{Mapping|legend=0| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}


Optimal tunings:  
Optimal tunings:  
Line 249: Line 251:
Comma list: 126/125, 176/175, 26411/26244
Comma list: 126/125, 176/175, 26411/26244


Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}
{{Mapping|legend=0| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}


Optimal tunings:  
Optimal tunings:  
Line 264: Line 266:
Comma list: 126/125, 176/175, 196/195, 13013/12960
Comma list: 126/125, 176/175, 196/195, 13013/12960


Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}
{{Mapping|legend=0| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}


Optimal tunings:  
Optimal tunings:  
Line 301: Line 303:
Comma list: 126/125, 385/384, 2401/2376
Comma list: 126/125, 385/384, 2401/2376


Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}
{{Mapping|legend=0| 2 -1 1 3 9 | 0 8 7 5 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 316: Line 318:
Comma list: 126/125, 196/195, 364/363, 385/384
Comma list: 126/125, 196/195, 364/363, 385/384


Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}
{{Mapping|legend=0| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 354: Line 356:
Comma list: 126/125, 540/539, 16384/16335
Comma list: 126/125, 540/539, 16384/16335


Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
{{Mapping|legend=0| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}


Optimal tunings:  
Optimal tunings:  
Line 369: Line 371:
Comma list: 126/125, 169/168, 540/539, 729/728
Comma list: 126/125, 169/168, 540/539, 729/728


Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}


Optimal tunings:  
Optimal tunings:  
Line 384: Line 386:
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539


Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}


Optimal tunings:  
Optimal tunings:  
Line 399: Line 401:
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539


Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}


Optimal tunings:  
Optimal tunings:  
Line 414: Line 416:
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230


Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}
{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}


Optimal tunings:  
Optimal tunings:  
Line 449: Line 451:
Comma list: 126/125, 176/175, 864/847
Comma list: 126/125, 176/175, 864/847


Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}
{{Mapping|legend=0| 1 -9 -5 2 -12 | 0 13 9 1 19 }}


Optimal tunings:  
Optimal tunings:  
Line 464: Line 466:
Comma list: 78/77, 126/125, 144/143, 176/175
Comma list: 78/77, 126/125, 144/143, 176/175


Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}
{{Mapping|legend=0| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}


Optimal tunings:  
Optimal tunings:  
Line 473: Line 475:


Badness (Sintel): 1.19
Badness (Sintel): 1.19
== Bisemidim ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 118098/117649
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
[[Badness]] (Sintel): 2.47
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 126/125, 540/539, 1344/1331
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
Badness (Sintel): 1.36
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 196/195, 364/363
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
Badness (Sintel): 0.987


== Cypress ==
== Cypress ==
Line 548: Line 500:
Comma list: 99/98, 126/125, 243/242
Comma list: 99/98, 126/125, 243/242


Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
{{Mapping|legend=0| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}


Optimal tunings:  
Optimal tunings:  
Line 563: Line 515:
Comma list: 66/65, 99/98, 126/125, 243/242
Comma list: 66/65, 99/98, 126/125, 243/242


Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
{{Mapping|legend=0| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}


Optimal tunings:  
Optimal tunings:  
Line 606: Line 558:
Comma list: 126/125, 385/384, 2420/2401
Comma list: 126/125, 385/384, 2420/2401


Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}
{{Mapping|legend=0| 1 -7 -4 1 3 | 0 19 14 4 1 }}


Optimal tunings:  
Optimal tunings:  
Line 621: Line 573:
Comma list: 126/125, 196/195, 385/384, 2420/2401
Comma list: 126/125, 196/195, 385/384, 2420/2401


Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}
{{Mapping|legend=0| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}


Optimal tunings:  
Optimal tunings:  
Line 636: Line 588:
Comma list: 126/125, 176/175, 14641/14580
Comma list: 126/125, 176/175, 14641/14580


Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}
{{Mapping|legend=0| 1 -7 -4 1 -11 | 0 19 14 4 32 }}


Optimal tunings:  
Optimal tunings:  
Line 651: Line 603:
Comma list: 126/125, 176/175, 196/195, 14641/14580
Comma list: 126/125, 176/175, 196/195, 14641/14580


Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}
{{Mapping|legend=0| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 666: Line 618:
Comma list: 126/125, 144/143, 176/175, 1540/1521
Comma list: 126/125, 144/143, 176/175, 1540/1521


Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}
{{Mapping|legend=0| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}


Optimal tunings:  
Optimal tunings:  
Line 701: Line 653:
Comma list: 126/125, 176/175, 16384/16335
Comma list: 126/125, 176/175, 16384/16335


Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}
{{Mapping|legend=0| 1 -2 2 9 9 | 0 11 1 -19 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 716: Line 668:
Comma list: 126/125, 169/168, 176/175, 364/363
Comma list: 126/125, 169/168, 176/175, 364/363


Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}
{{Mapping|legend=0| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 751: Line 703:
Comma list: 99/98, 126/125, 864/847
Comma list: 99/98, 126/125, 864/847


Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
{{Mapping|legend=0| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}


Optimal tunings:  
Optimal tunings:  
Line 766: Line 718:
Comma list: 78/77, 99/98, 126/125, 144/143
Comma list: 78/77, 99/98, 126/125, 144/143


Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
{{Mapping|legend=0| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}


Optimal tunings:  
Optimal tunings:  
Line 781: 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: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
{{Mapping|legend=0| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}


Optimal tunings:  
Optimal tunings:  
Line 820: Line 772:
Comma list: 126/125, 540/539, 21609/21296
Comma list: 126/125, 540/539, 21609/21296


Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}
{{Mapping|legend=0| 27 0 20 33 8 | 0 1 1 1 2 }}


Optimal tunings:  
Optimal tunings:  
Line 835: Line 787:
Comma list: 126/125, 144/143, 196/195, 21609/21296
Comma list: 126/125, 144/143, 196/195, 21609/21296


Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}
{{Mapping|legend=0| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}


Optimal tunings:  
Optimal tunings:  
Line 850: 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: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}
{{Mapping|legend=0| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}


Optimal tunings:  
Optimal tunings:  
Line 865: 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: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}
{{Mapping|legend=0| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}


Optimal tunings:  
Optimal tunings:  
Line 880: 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: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}
{{Mapping|legend=0| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}


Optimal tunings:  
Optimal tunings:  
Line 895: 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: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}
{{Mapping|legend=0| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}


Optimal tunings:  
Optimal tunings:  
Line 910: Line 862:
Comma list: 126/125, 169/168, 540/539, 975/968
Comma list: 126/125, 169/168, 540/539, 975/968


Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}
{{Mapping|legend=0| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}


Optimal tunings:  
Optimal tunings:  
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