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Vulture family: Difference between revisions

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{{interwiki
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| en = Vulture family
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{{Technical data page}}
{{Technical data page}}
The '''vulture family''' of [[temperament]]s [[tempering out|tempers out]] the [[vulture comma]] ({{monzo|legend=1| 24 -21 4 }}, [[ratio]]: 10 485 760 000 / 10 460 353 203), a small [[5-limit]] comma of 4.2 [[cent]]s that is the amount by which a stack of four [[syntonic comma]]s falls short of the [[256/243]] Pythagorean limma. As their defining feature, vulture temperaments split the interval [[3/1]] into four segments (identified in the 5-limit as [[320/243]]).
The '''vulture family''' of [[temperament]]s [[tempering out|tempers out]] the [[vulture comma]] ({{monzo|legend=1| 24 -21 4 }}, [[ratio]]: 10 485 760 000 / 10 460 353 203), a small [[5-limit]] comma of 4.2 [[cent]]s that is the amount by which a stack of four [[syntonic comma]]s falls short of the [[256/243]] Pythagorean limma. As their defining feature, vulture temperaments split the interval [[3/1]] into four segments (identified in the 5-limit as [[320/243]]).
Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]] and [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, condor, eagle, and turkey.


== Vulture ==
== Vulture ==
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{{Mapping|legend=1| 1 0 -6 | 0 4 21 }}
{{Mapping|legend=1| 1 0 -6 | 0 4 21 }}
: mapping generators: ~2, ~320/243
: mapping generators: ~2, ~320/243


Line 30: Line 27:


[[Badness]] (Sintel): 0.972
[[Badness]] (Sintel): 0.972
=== Overview to extensions ===
Strong 7-limit extensions include septimal vulture (53 & 270), which tempers out [[4375/4374]], and buzzard (53 & 58), which tempers out [[1728/1715]].
Terture adds [[250047/250000]] with a 1/3-octave period. Condor adds [[10976/10935]] and splits the generator in three. Eagle adds [[2401/2400]]. Turkey adds [[4802000/4782969]]. Those split the generator in four.
Most of these temperaments extend naturally to the 11-limit via tempering out the [[symbiotic comma]] and [[seascape comma]], from 10485760000/10460353203 = (19712/19683)<sup>2</sup>/(160083/160000), and to the 13-limit via identifying the hemitwelfth as [[~]][[26/15]], tempering out the [[island comma]].
Buzzard is discussed in its own clan (→ [[Buzzardsmic clan #Septimal buzzard]]). The rest are considered below.


== Septimal vulture ==
== Septimal vulture ==
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Comma list: 4375/4374, 5632/5625, 41503/41472
Comma list: 4375/4374, 5632/5625, 41503/41472


Mapping: {{mapping| 1 0 -6 25 -33 | 0 4 21 -56 92 }}
{{Mapping|legend=0| 1 0 -6 25 -33 | 0 4 21 -56 92 }}


Optimal tunings:  
Optimal tunings:  
Line 72: Line 78:
Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374
Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374


Mapping: {{mapping| 1 0 -6 25 -33 -7 | 0 4 21 -56 92 27 }}
{{Mapping|legend=0| 1 0 -6 25 -33 -7 | 0 4 21 -56 92 27 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9695{{c}}, ~320/243 = 475.5451{{c}}
* WE: ~2 = 1199.9695{{c}}, ~154/117 = 475.5451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~154/117 = 475.5571{{c}}


{{Optimal ET sequence|legend=0| 53, 217, 270 }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}
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Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728
Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728


Mapping: {{mapping| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}
{{Mapping|legend=0| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 3025/3024, 4375/4374, 33554432/33480783
Comma list: 3025/3024, 4375/4374, 33554432/33480783


Mapping: {{mapping| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
{{Mapping|legend=0| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
: mapping generators: ~99/70, ~320/243
: mapping generators: ~99/70, ~320/243


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Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374
Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374


Mapping: {{mapping| 2 0 -12 50 41 -14 | 0 4 21 -56 -43 27 }}
{{Mapping|legend=0| 2 0 -12 50 41 -14 | 0 4 21 -56 -43 27 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 1.47
Badness (Sintel): 1.47
== Terture ==
Named by [[Xenllium]] in 2021, terture tempers out 250047/250000, the [[landscape comma]], and may be described as the {{nowrap| 111 & 159 }} temperament, with a [[ploidacot]] signature of triploid gamma-tetracot.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, 359661568/358722675
{{Mapping|legend=1| 3 0 -18 -32 | 0 4 21 34 }}
: mapping generators: ~63/50, ~320/243
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9723{{c}}, ~320/243 = 475.5221{{c}} (~392/375 = 75.5499{{c}})
: [[error map]]: {{val| -0.083 +0.134 +0.151 -0.185 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5519{{c}} (~392/375 = 75.5519{{c}})
: error map: {{val| 0.000 +0.253 +0.276 -0.061 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}
[[Badness]] (Sintel): 2.21
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 19712/19683, 102487/102400
{{Mapping|legend=0| 3 0 -18 -32 8 | 0 4 21 34 2 }}
Optimal tunings:
* WE: ~63/50 = 399.9902{{c}}, ~320/243 = 475.5383{{c}} (~392/375 = 75.5481{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5490{{c}} (~392/375 = 75.5490{{c}})
{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
Badness (Sintel): 0.969
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
{{Mapping|legend=0| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}
Optimal tunings:
* WE: ~63/50 = 399.9958{{c}}, ~154/117 = 475.5485{{c}} (~117/112 = 75.5527{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~154/117 = 475.5531{{c}} (~117/112 = 75.5531{{c}})
{{Optimal ET sequence|legend=0| 111, 159, 270 }}
Badness (Sintel): 0.771
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
{{Mapping|legend=0| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}
Optimal tunings:
* WE: ~34/27 = 399.9664{{c}}, ~112/85 = 475.5198{{c}} (~117/112 = 75.5534{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~117/112 = 75.5568{{c}})
{{Optimal ET sequence|legend=0| 111, 159, 270 }}
Badness (Sintel): 0.953
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
{{Mapping|legend=0| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}
Optimal tunings:
* WE: ~34/27 = 399.9665{{c}}, ~112/85 = 475.5198{{c}} (~95/91 = 75.5533{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~95/91 = 75.5568{{c}})
{{Optimal ET sequence|legend=0| 111, 159, 270 }}
Badness (Sintel): 0.846
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215
{{Mapping|legend=0| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}
Optimal tunings:
* WE: ~34/27 = 400.0026{{c}}, ~112/85 = 475.5510{{c}} (~24/23 = 75.5485{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5482{{c}} (~24/23 = 75.5482{{c}})
{{Optimal ET sequence|legend=0| 111, 159, 270 }}
Badness (Sintel): 1.07


== Condor ==
== Condor ==
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Comma list: 441/440, 4000/3993, 10976/10935
Comma list: 441/440, 4000/3993, 10976/10935


Mapping: {{mapping| 1 -4 -27 -20 -24 | 0 12 63 49 59 }}
{{Mapping|legend=0| 1 -4 -27 -20 -24 | 0 12 63 49 59 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 364/363, 441/440, 676/675, 10976/10935
Comma list: 364/363, 441/440, 676/675, 10976/10935


Mapping: {{mapping| 1 -4 -27 -20 -24 -34 | 0 12 63 49 59 81 }}
{{Mapping|legend=0| 1 -4 -27 -20 -24 -34 | 0 12 63 49 59 81 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619
Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619


Mapping: {{mapping| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}
{{Mapping|legend=0| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 194: Line 295:


== Eagle ==
== Eagle ==
Eagle tempers out [[2401/2400]] and may be described as the {{nowrap| 58 & 270 }} temperament. It has a semi-octave period and a generator of ~28/27, four of which make a hemifourth which may be identified with 15/13, and two of those make a perfect fourth; its ploidacot thus is diploid wau-octacot. Compatible tunings include [[212edo]], [[270edo]], and [[328edo]].  
Eagle tempers out [[2401/2400]] and may be described as the {{nowrap| 58 & 270 }} temperament. It has a semi-octave period and a generator of [[~]][[28/27]], four of which make a hemifourth which may be identified with [[15/13]], and two of those make a perfect fourth; its ploidacot thus is diploid wau-octacot. Compatible edo tunings include [[212edo]], [[270edo]] and [[328edo]] in the 212h, 270 and 328h [[val]]s, respectively.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 218: Line 319:
Comma list: 2401/2400, 9801/9800, 19712/19683
Comma list: 2401/2400, 9801/9800, 19712/19683


Mapping: {{mapping| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}
{{Mapping|legend=0| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}


Optimal tunings:  
Optimal tunings:  
Line 233: Line 334:
Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647
Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647


Mapping: {{mapping| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}
{{Mapping|legend=0| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}


Optimal tunings:  
Optimal tunings:  
Line 242: Line 343:


Badness (Sintel): 0.673
Badness (Sintel): 0.673
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
Comma list: 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539
{{Mapping|legend=0| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 -82 }}
Optimal tunings:
* WE: ~99/70 = 599.9810{{c}} ~28/27 = 62.2195{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.2223{{c}}
{{Optimal ET sequence|legend=0| 58h, 154cefhh, 212h, 270 }}
Badness (Sintel): 0.609


== Turkey ==
== Turkey ==
{{Redirect|Turkey|information about Turkish music|Middle-Eastern music}}
Named by [[Xenllium]] in 2021, turkey may be described as the {{nowrap| 212 & 217 }} temperament. It is generated by a fifth sharp of just, close to 3\5 but on the flat side thereof, which can be interpreted as [[50/33]] in the 11-limit. Sixteen generators minus nine octaves make a perfect fifth; its ploidacot is thus theta-16-cot. [[429edo]] may be recommended as a tuning.  
Named by [[Xenllium]] in 2021, turkey may be described as the {{nowrap| 212 & 217 }} temperament. It is generated by a fifth sharp of just, close to 3\5 but on the flat side thereof, which can be interpreted as [[50/33]] in the 11-limit. Sixteen generators minus nine octaves make a perfect fifth; its ploidacot is thus theta-16-cot. [[429edo]] may be recommended as a tuning.  


Line 268: Line 386:
Comma list: 19712/19683, 42875/42768, 160083/160000
Comma list: 19712/19683, 42875/42768, 160083/160000


Mapping: {{mapping| 1 -8 -48 7 -87 | 0 16 84 -7 151 }}
{{Mapping|legend=0| 1 -8 -48 7 -87 | 0 16 84 -7 151 }}


Optimal tunings:  
Optimal tunings:  
Line 283: Line 401:
Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104
Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104


Mapping: {{mapping| 1 -8 -48 7 -87 -61 | 0 16 84 -7 151 108 }}
{{Mapping|legend=0| 1 -8 -48 7 -87 -61 | 0 16 84 -7 151 108 }}


Optimal tunings:  
Optimal tunings:  
Line 293: Line 411:
Badness (Sintel): 1.81
Badness (Sintel): 1.81


[[Category:Vulture family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Vulture family| ]] <!-- main article
[[Category:Rank 2]]
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