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The '''vulture family''' of [[temperament]]s [[tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} = 10485760000/10460353203, a small [[5-limit]] comma of 4.2 [[cent]]s.
{{Interwiki
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{{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]]).


Temperaments discussed elsewhere includes [[Landscape microtemperaments #Terture|terture]].
== Vulture ==
The generator of the vulture temperament is a grave fourth of [[320/243]], that is, a [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Four of these make a [[3/1|perfect twelfth]]. Its [[ploidacot]] is alpha-tetracot. It is a member of the [[syntonic–diatonic equivalence continuum]] with {{nowrap| ''n'' {{=}} 4 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of four syntonic commas. It is also in the [[schismic–Mercator equivalence continuum]] with {{nowrap|''n'' {{=}} 4}}, so unless [[53edo]] is used as a tuning, the [[schisma]] is always observed.  


== Vulture ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 10485760000/10460353203
[[Comma list]]: 10485760000/10460353203


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


Mapping generators: ~2, ~320/243
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9430{{c}}, ~320/243 = 475.5200{{c}}
: [[error map]]: {{val| -0.057 +0.125 -0.051 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5396{{c}}
: error map: {{val| 0.000 +0.203 +0.018 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~320/243 = 475.5426
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}


{{Val list|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, 3500b, 3823b, 4146b, 4469b }}
[[Badness]] (Sintel): 0.972


[[Badness]]: 0.041431
=== Overview to extensions ===
Temperaments discussed elsewhere include [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, terture, condor, eagle, and turkey.


== Septimal vulture ==
== Septimal vulture ==
The vulture temperament can be described as the 53 & 217 temperament, tempering out the ragisma, 4375/4374 and the [[garischisma]], {{monzo| 25 -14 0 -1 }} = 33554432/33480783 aside from the vulture comma. [[270edo]] is a good tuning for this temperament, with generator 107/270, and [[mos scale]]s of 3, 5, 8, 13, 18, 23, 28, 33, 38, 43, 48, or 53 notes are available.
Septimal vulture can be described as the {{nowrap| 53 & 270 }} microtemperament, tempering out the [[ragisma]], 4375/4374 and the [[garischisma]], 33554432/33480783 ({{monzo| 25 -14 0 -1 }}) aside from the vulture comma. [[270edo]] is an excellent tuning for this temperament, with generator 107\270. Other compatible tunings include [[217edo]] and [[323edo]]. The harmonic 7 is found at -14 fifths or {{nowrap| (-14) × 4 {{=}} -56 }} generator steps, so the smallest [[mos scale]] that includes it is the 58-note one, though for larger scope of harmony, you could try the 111- or 164-note one. For a much simpler mapping of 7 at the cost of higher error, you could try [[Buzzardsmic clan #Septimal buzzard|buzzard]].
 
It can be extended to the 11-limit by identifying a stack of four [[5/4]]'s as [[11/9]], tempering out [[5632/5625]], and to the 13-limit by identifying the hemitwelfth as [[26/15]], tempering out [[676/675]]. Furthermore, the generator of vulture is very close to [[25/19]]; a stack of three generator steps octave-reduced thus represents its fifth complement, [[57/50]]. This corresponds to tempering out [[1216/1215]] with the effect of equating the schisma with [[513/512]] and [[361/360]] in addition to many 11- and 13-limit commas. 270edo remains an excellent tuning in all cases.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 25: Line 40:
[[Comma list]]: 4375/4374, 33554432/33480783
[[Comma list]]: 4375/4374, 33554432/33480783


[[Mapping]]: [{{val| 1 0 -6 25 }}, {{val| 0 4 21 -56 }}]
{{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }}


{{Multival|legend=1| 4 21 -56 24 -100 -189 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9050{{c}}, ~320/243 = 475.5135{{c}}
: [[error map]]: {{val| -0.095 +0.099 +0.039 +0.044 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5515{{c}}
: error map: {{val| 0.000 +0.251 +0.267 +0.292 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~320/243 = 475.5511
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863, 1133, 1996d }}


{{Val list|legend=1| 53, 164, 217, 270, 593, 863, 1133 }}
[[Badness]] (Sintel): 0.936
 
[[Badness]]: 0.036985


=== 11-limit ===
=== 11-limit ===
Line 40: Line 57:
Comma list: 4375/4374, 5632/5625, 41503/41472
Comma list: 4375/4374, 5632/5625, 41503/41472


Mapping: [{{val| 1 0 -6 25 -33 }}, {{val| 0 4 21 -56 92 }}]
Mapping: {{mapping| 1 0 -6 25 -33 | 0 4 21 -56 92 }}


Optimal tuning (POTE): ~2 = 1\1, ~320/243 = 475.5567
Optimal tunings:
* WE: ~2 = 1199.9392{{c}}, ~320/243 = 475.5326{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5655{{c}}


Optimal GPV sequence: {{Val list| 53, 217, 270 }}
{{Optimal ET sequence|legend=0| 53, 217, 270, 2107c, 2377bc }}


Badness: 0.031907
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 53: Line 72:
Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374
Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374


Mapping: [{{val| 1 0 -6 25 -33 -7 }}, {{val| 0 4 21 -56 92 27 }}]
Mapping: {{mapping| 1 0 -6 25 -33 -7 | 0 4 21 -56 92 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~320/243 = 475.5572
Optimal tunings:
* WE: ~2 = 1199.9695{{c}}, ~154/117 = 475.5451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~154/117 = 475.5571{{c}}


Optimal GPV sequence: {{Val list| 53, 217, 270 }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}


Badness: 0.018758
Badness (Sintel): 0.775


==== 17-limit ====
==== 2.3.5.7.11.13.19 subgroup ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.19


Comma list: 676/675, 936/935, 1001/1000, 1225/1224, 4096/4095
Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728


Mapping: [{{val| 1 0 -6 25 -33 -7 35 }}, {{val| 0 4 21 -56 92 27 -78 }}]
Mapping: {{mapping| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}


Optimal tuning (POTE): ~2 = 1\1, ~112/85 = 475.5617
Optimal tunings:
* WE: ~2 = 1199.9636{{c}}, ~25/19 = 475.5426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/19 = 475.5569{{c}}


Optimal GPV sequence: {{Val list| 53, 217, 270, 487, 757g }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}


Badness: 0.020103
Badness (Sintel): 0.579
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 676/675, 936/935, 1001/1000, 1216/1215, 1225/1224, 1540/1539
 
Mapping: [{{val| 1 0 -6 25 -33 -7 35 -12 }}, {{val| 0 4 21 -56 92 27 -78 41 }}]
 
Optimal tuning (POTE): ~2 = 1\1, ~25/19 = 475.5615
 
Optimal GPV sequence: {{Val list| 53, 217, 270, 487, 757g }}
 
Badness: 0.013850


=== Semivulture ===
=== Semivulture ===
Line 92: Line 102:
Comma list: 3025/3024, 4375/4374, 33554432/33480783
Comma list: 3025/3024, 4375/4374, 33554432/33480783


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


Optimal tuning (POTE): ~99/70 = 1\2, ~320/243 = 475.550
Optimal tunings:
* WE: ~99/70 = 599.9594{{c}}, ~320/243 = 475.5174{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5501{{c}}


Optimal GPV sequence: {{Val list| 106, 164, 270, 916, 1186, 1456 }}
{{Optimal ET sequence|legend=0| 106, 164, 270, 916, 1186, 1456 }}


Badness: 0.040799
Badness (Sintel): 1.35


==== 13-limit ====
==== 13-limit ====
Line 107: Line 118:
Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374
Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374


Mapping: [{{val| 2 0 -12 50 41 -14 }}, {{val| 0 4 21 -56 -43 27 }}]
Mapping: {{mapping| 2 0 -12 50 41 -14 | 0 4 21 -56 -43 27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~320/243 = 475.553


Optimal GPV sequence: {{Val list| 106, 164, 270 }}
Optimal tunings:  
* WE: ~99/70 = 599.9859{{c}}, ~320/243 = 475.5423{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5536{{c}}


Badness: 0.035458
{{Optimal ET sequence|legend=0| 106, 164, 270 }}


== Buzzard ==
Badness (Sintel): 1.47
{{Main| Buzzard }}


''Buzzard'' is an alternative lower complexity extension to vulture, but more of a full 13-limit system in its own right. It can be described as 53 & 58. As one might expect, 111edo is a great tuning for it.  
== 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 5120/5103
[[Comma list]]: 250047/250000, 359661568/358722675
 
[[Mapping]]: [{{val| 1 0 -6 4 }}, {{val| 0 4 21 -3 }}]


{{Multival|legend=1| 4 21 -3 24 -16 -66 }}
{{Mapping|legend=1| 3 0 -18 -32 | 0 4 21 34 }}
: mapping generators: ~63/50, ~320/243


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.636
[[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 }}


{{Val list|legend=1| 5, 43c, 48, 53, 111, 164d, 275d }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


[[Badness]]: 0.047963
[[Badness]] (Sintel): 2.21


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 5120/5103
Comma list: 3025/3024, 19712/19683, 102487/102400


Mapping: [{{val| 1 0 -6 4 -12 }}, {{val| 0 4 21 -3 39 }}]
Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.700
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 GPV sequence: {{Val list| 53, 58, 111, 280cd, 391cd }}
{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}


Badness: 0.034484
Badness (Sintel): 0.969


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 351/350, 540/539, 676/675
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976


Mapping: [{{val| 1 0 -6 4 -12 -7 }}, {{val| 0 4 21 -3 39 27 }}]
Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.697
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 GPV sequence: {{Val list| 53, 58, 111, 280cdf, 391cdf }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.018842
Badness (Sintel): 0.771


==== 17-limit ====
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913


Mapping: [{{val| 1 0 -6 4 -12 -7 14 }}, {{val| 0 4 21 -3 39 27 -25 }}]
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.692
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 GPV sequence: {{Val list| 53, 58, 111, 321cdfg }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.018403
Badness (Sintel): 0.953


==== 19-limit ====
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615


Mapping: [{{val| 1 0 -6 4 -12 -7 14 -12 }}, {{val| 0 4 21 -3 39 27 -25 41 }}]
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.679
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 GPV sequence: {{Val list| 53, 58h, 111 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.015649
Badness (Sintel): 0.846


=== Buteo ===
=== 23-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 99/98, 385/384, 2200/2187
Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215


Mapping: [{{val| 1 0 -6 4 9 }}, {{val| 0 4 21 -3 -14 }}]
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.436
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 GPV sequence: {{Val list| 5, 48, 53 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.060238
Badness (Sintel): 1.07


==== 13-limit ====
== Condor ==
Subgroup: 2.3.5.7.11.13
Condor tempers out [[10976/10935]] and may be described as the {{nowrap| 58 & 159 }} temperament. The generator represents the [[112/81|septimal diminished fifth (112/81)]], and three minus an octave make vulture's generator of ~320/243. The ploidacot for this temperament is epsilon-dodecacot. [[217edo]] is an excellent tuning for this temperament.  


Comma list: 99/98, 275/273, 385/384, 572/567
Mapping: [{{val| 1 0 -6 4 9 -7 }}, {{val| 0 4 21 -3 -14 27 }}]
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.464
Optimal GPV sequence: {{Val list| 5, 48f, 53 }}
Badness: 0.039854
== Condor ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 40353607/40000000
[[Comma list]]: 10976/10935, 40353607/40000000


[[Mapping]]: [{{Val| 1 8 36 29 }}, {{Val| 0 -12 -63 -49 }}]
{{Mapping|legend=1| 1 -4 -27 -20 | 0 12 63 49 }}
 
: mapping generators: ~2, ~112/81
{{Multival|legend=1| 12 63 49 72 44 -63 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0142{{c}}, ~112/81 = 558.5276{{c}}
: [[error map]]: {{val| +0.014 +0.319 +0.539 -1.260 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~112/81 = 558.5212{{c}}
: error map: {{val| 0.000 +0.300 +0.523 -1.287 }}


{{Val list|legend=1| 58, 159, 217 }}
{{Optimal ET sequence|legend=1| 58, 159, 217 }}


[[Badness]]: 0.154715
[[Badness]] (Sintel): 3.92


=== 11-limit ===
=== 11-limit ===
Line 232: Line 248:
Comma list: 441/440, 4000/3993, 10976/10935
Comma list: 441/440, 4000/3993, 10976/10935


Mapping: [{{Val| 1 8 36 29 35 }}, {{Val| 0 -12 -63 -49 -59 }}]
Mapping: {{mapping| 1 -4 -27 -20 -24 | 0 12 63 49 59 }}


Optimal tuning (POTE): ~2 = 1\1, 81/56 = 641.4822
Optimal tunings:
* WE: ~2 = 1199.9730{{c}}, ~112/81 = 558.5052{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/81 = 558.5173{{c}}


Optimal GPV sequence: {{Val list| 58, 101cd, 159, 217 }}
{{Optimal ET sequence|legend=0| 58, 101cd, 159, 217, 376d }}


Badness: 0.048401
Badness (Sintel): 1.60


=== 13-limit ===
=== 13-limit ===
Line 245: Line 263:
Comma list: 364/363, 441/440, 676/675, 10976/10935
Comma list: 364/363, 441/440, 676/675, 10976/10935


Mapping: [{{val| 1 8 36 29 35 47 }}, {{val| 0 -12 -63 -49 -59 -81 }}]
Mapping: {{mapping| 1 -4 -27 -20 -24 -34 | 0 12 63 49 59 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4797
Optimal tunings:
* WE: ~2 = 1199.9649{{c}}, ~112/81 = 558.5040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/81 = 558.5197{{c}}


Optimal GPV sequence: {{Val list| 58, 159, 217 }}
{{Optimal ET sequence|legend=0| 58, 159, 217 }}


Badness: 0.025469
Badness (Sintel): 1.05


=== 17-limit ===
=== 17-limit ===
Line 258: Line 278:
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: [{{val| 1 8 36 29 35 47 -5 }}, {{val| 0 -12 -63 -49 -59 -81 17 }}]
Mapping: {{mapping| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4794
Optimal tunings:
* WE: ~2 = 1199.9594{{c}}, ~112/81 = 558.5017{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/81 = 558.5202{{c}}


Optimal GPV sequence: {{Val list| 58, 159, 217 }}
{{Optimal ET sequence|legend=0| 58, 159, 217 }}


Badness: 0.021984
Badness (Sintel): 1.12


== 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]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 10485760000/10460353203
[[Comma list]]: 2401/2400, 10485760000/10460353203


[[Mapping]]: [{{val| 2 4 9 8 }}, {{val| 0 -8 -42 -23 }}]
{{Mapping|legend=1| 2 4 9 8 | 0 -8 -42 -23 }}
 
: mapping generators: ~177147/125440, ~28/27
Mapping generators: ~177147/125440, ~28/27
 
{{Multival|legend=1|16 84 46 96 28 -129}}


[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229
[[Optimal tuning]]s:
* [[WE]]: ~177147/125440 = 599.9818{{c}}, ~28/27 = 62.2266{{c}}
: [[error map]]: {{val| -0.036 +0.159 +0.004 -0.184 }}
* [[CWE]]: ~177147/125440 = 600.0000{{c}}, ~28/27 = 62.2295{{c}}
: error map: {{val| 0.000 +0.209 +0.046 -0.105 }}


{{Val list|legend=1| 58, 154c, 212, 270, 752, 1022, 1292, 2854b }}
{{Optimal ET sequence|legend=1| 58, 154c, 212, 270, 752, 1022, 1292, 2854b }}


[[Badness]]: 0.059498
[[Badness]] (Sintel): 1.51


=== 11-limit ===
=== 11-limit ===
Line 288: Line 313:
Comma list: 2401/2400, 9801/9800, 19712/19683
Comma list: 2401/2400, 9801/9800, 19712/19683


Mapping: [{{val| 2 4 9 8 12 }}, {{val| 0 -8 -42 -23 -49 }}]
Mapping: {{mapping| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.224
Optimal tunings:
* WE: ~99/70 = 599.9796{{c}} ~28/27 = 62.2218{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.2251{{c}}


Optimal GPV sequence: {{Val list| 58, 154ce, 212, 270 }}
{{Optimal ET sequence|legend=0| 58, 154ce, 212, 270 }}


Badness: 0.024885
Badness (Sintel): 0.823


=== 13-limit ===
=== 13-limit ===
Line 301: Line 328:
Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647
Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647


Mapping: [{{val| 2 4 9 8 12 13 }}, {{val| 0 -8 -42 -23 -49 -54 }}]
Mapping: {{mapping| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.220
Optimal tunings:
* WE: ~99/70 = 599.9763{{c}} ~28/27 = 62.2174{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.2211{{c}}


Optimal GPV sequence: {{Val list| 58, 154cef, 212, 270 }}
{{Optimal ET sequence|legend=0| 58, 154cef, 212, 270 }}


Badness: 0.016282
Badness (Sintel): 0.673


== Turkey ==
== Turkey ==
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.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4802000/4782969, 5250987/5242880
[[Comma list]]: 4802000/4782969, 5250987/5242880


[[Mapping]]: [{{val| 1 8 36 0 }}, {{val| 0 -16 -84 7 }}]
{{Mapping|legend=1| 1 -8 -48 7 | 0 16 84 -7 }}
 
: mapping generators: ~2, ~3592/1715
{{Multival|legend=1|16 84 -7 96 -56 -252}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1147{{c}}, ~3592/1715 = 718.9483{{c}}
: [[error map]]: {{val| +0.115 +0.300 -0.161 -0.661 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3592/1715 = 718.8806{{c}}
: error map: {{val| 0.000 +0.134 -0.345 -0.990 }}


{{Val list|legend=1| 5, 207c, 212, 429 }}
{{Optimal ET sequence|legend=1| 212, 429, 1070d }}


[[Badness]]: 0.210964
[[Badness]] (Sintel): 5.34


=== 11-limit ===
=== 11-limit ===
Line 329: Line 363:
Comma list: 19712/19683, 42875/42768, 160083/160000
Comma list: 19712/19683, 42875/42768, 160083/160000


Mapping: [{{val| 1 8 36 0 64 }}, {{val| 0 -16 -84 7 -151 }}]
Mapping: {{mapping| 1 -8 -48 7 -87 | 0 16 84 -7 151 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.120
Optimal tunings:
* WE: ~2 = 1200.1131{{c}} ~50/33 = 718.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/33 = 718.8808{{c}}


Optimal GPV sequence: {{Val list| 212, 429 }}
{{Optimal ET sequence|legend=0| 212, 429 }}


Badness: 0.079694
Badness (Sintel): 2.63


=== 13-limit ===
=== 13-limit ===
Line 342: Line 378:
Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104
Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104


Mapping: [{{val| 1 8 36 0 64 47 }}, {{val| 0 -16 -84 7 -151 -108 }}]
Mapping: {{mapping| 1 -8 -48 7 -87 -61 | 0 16 84 -7 151 108 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.118
Optimal tunings:
* WE: ~2 = 1200.1324{{c}} ~50/33 = 718.9608{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/33 = 718.8825{{c}}


Optimal GPV sequence: {{Val list| 212, 217, 429 }}
{{Optimal ET sequence|legend=0| 212, 217, 429 }}


Badness: 0.043787
Badness (Sintel): 1.81


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Vulture family| ]] <!-- main article -->
[[Category:Vulture family| ]] <!-- main article  
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 17:57, 24 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The vulture family of temperaments tempers out the vulture comma (monzo: [24 -21 4⟩, ratio: 10 485 760 000 / 10 460 353 203), a small 5-limit comma of 4.2 cents that is the amount by which a stack of four syntonic commas 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).

Vulture

The generator of the vulture temperament is a grave fourth of 320/243, that is, a perfect fourth minus a syntonic comma. Four of these make a perfect twelfth. Its ploidacot is alpha-tetracot. It is a member of the syntonic–diatonic equivalence continuum with n = 4, so it equates a Pythagorean limma with a stack of four syntonic commas. It is also in the schismic–Mercator equivalence continuum with n = 4, so unless 53edo is used as a tuning, the schisma is always observed.

Subgroup: 2.3.5

Comma list: 10485760000/10460353203

Mapping: [⟨1 0 -6], ⟨0 4 21]]

mapping generators: ~2, ~320/243

Optimal tunings:

  • WE: ~2 = 1199.9430 ¢, ~320/243 = 475.5200 ¢
error map: ⟨-0.057 +0.125 -0.051]
  • CWE: ~2 = 1200.0000 ¢, ~320/243 = 475.5396 ¢
error map: ⟨0.000 +0.203 +0.018]

Optimal ET sequence: 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b

Badness (Sintel): 0.972

Overview to extensions

Temperaments discussed elsewhere include buzzard. Considered below are septimal vulture, terture, condor, eagle, and turkey.

Septimal vulture

Septimal vulture can be described as the 53 & 270 microtemperament, tempering out the ragisma, 4375/4374 and the garischisma, 33554432/33480783 ([25 -14 0 -1⟩) aside from the vulture comma. 270edo is an excellent tuning for this temperament, with generator 107\270. Other compatible tunings include 217edo and 323edo. The harmonic 7 is found at -14 fifths or (-14) × 4 = -56 generator steps, so the smallest mos scale that includes it is the 58-note one, though for larger scope of harmony, you could try the 111- or 164-note one. For a much simpler mapping of 7 at the cost of higher error, you could try buzzard.

It can be extended to the 11-limit by identifying a stack of four 5/4's as 11/9, tempering out 5632/5625, and to the 13-limit by identifying the hemitwelfth as 26/15, tempering out 676/675. Furthermore, the generator of vulture is very close to 25/19; a stack of three generator steps octave-reduced thus represents its fifth complement, 57/50. This corresponds to tempering out 1216/1215 with the effect of equating the schisma with 513/512 and 361/360 in addition to many 11- and 13-limit commas. 270edo remains an excellent tuning in all cases.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33554432/33480783

Mapping: [⟨1 0 -6 25], ⟨0 4 21 -56]]

Optimal tunings:

  • WE: ~2 = 1199.9050 ¢, ~320/243 = 475.5135 ¢
error map: ⟨-0.095 +0.099 +0.039 +0.044]
  • CWE: ~2 = 1200.0000 ¢, ~320/243 = 475.5515 ¢
error map: ⟨0.000 +0.251 +0.267 +0.292]

Optimal ET sequence: 53, 164, 217, 270, 593, 863, 1133, 1996d

Badness (Sintel): 0.936

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 41503/41472

Mapping: [⟨1 0 -6 25 -33], ⟨0 4 21 -56 92]]

Optimal tunings:

  • WE: ~2 = 1199.9392 ¢, ~320/243 = 475.5326 ¢
  • CWE: ~2 = 1200.0000 ¢, ~320/243 = 475.5655 ¢

Optimal ET sequence: 53, 217, 270, 2107c, 2377bc

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374

Mapping: [⟨1 0 -6 25 -33 -7], ⟨0 4 21 -56 92 27]]

Optimal tunings:

  • WE: ~2 = 1199.9695 ¢, ~154/117 = 475.5451 ¢
  • CWE: ~2 = 1200.0000 ¢, ~154/117 = 475.5571 ¢

Optimal ET sequence: 53, 217, 270

Badness (Sintel): 0.775

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728

Mapping: [⟨1 0 -6 25 -33 -7 -12], ⟨0 4 21 -56 92 27 41]]

Optimal tunings:

  • WE: ~2 = 1199.9636 ¢, ~25/19 = 475.5426 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/19 = 475.5569 ¢

Optimal ET sequence: 53, 217, 270

Badness (Sintel): 0.579

Semivulture

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 -12 50 41], ⟨0 4 21 -56 -43]]

mapping generators: ~99/70, ~320/243

Optimal tunings:

  • WE: ~99/70 = 599.9594 ¢, ~320/243 = 475.5174 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~320/243 = 475.5501 ¢

Optimal ET sequence: 106, 164, 270, 916, 1186, 1456

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 -12 50 41 -14], ⟨0 4 21 -56 -43 27]]

Optimal tunings:

  • WE: ~99/70 = 599.9859 ¢, ~320/243 = 475.5423 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~320/243 = 475.5536 ¢

Optimal ET sequence: 106, 164, 270

Badness (Sintel): 1.47

Terture

Named by Xenllium in 2021, terture tempers out 250047/250000, the landscape comma, and may be described as the 111 & 159 temperament, with a ploidacot signature of triploid gamma-tetracot.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 359661568/358722675

Mapping: [⟨3 0 -18 -32], ⟨0 4 21 34]]

mapping generators: ~63/50, ~320/243

Optimal tunings:

  • WE: ~63/50 = 399.9723 ¢, ~320/243 = 475.5221 ¢ (~392/375 = 75.5499 ¢)
error map: ⟨-0.083 +0.134 +0.151 -0.185]
  • CWE: ~63/50 = 400.0000 ¢, ~320/243 = 475.5519 ¢ (~392/375 = 75.5519 ¢)
error map: ⟨0.000 +0.253 +0.276 -0.061]

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 2.21

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 19712/19683, 102487/102400

Mapping: [⟨3 0 -18 -32 8], ⟨0 4 21 34 2]]

Optimal tunings:

  • WE: ~63/50 = 399.9902 ¢, ~320/243 = 475.5383 ¢ (~392/375 = 75.5481 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~320/243 = 475.5490 ¢ (~392/375 = 75.5490 ¢)

Optimal ET sequence: 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: [⟨3 0 -18 -32 8 -21], ⟨0 4 21 34 2 27]]

Optimal tunings:

  • WE: ~63/50 = 399.9958 ¢, ~154/117 = 475.5485 ¢ (~117/112 = 75.5527 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~154/117 = 475.5531 ¢ (~117/112 = 75.5531 ¢)

Optimal ET sequence: 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: [⟨3 0 -18 -32 8 -21 -2], ⟨0 4 21 34 2 27 12]]

Optimal tunings:

  • WE: ~34/27 = 399.9664 ¢, ~112/85 = 475.5198 ¢ (~117/112 = 75.5534 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5568 ¢ (~117/112 = 75.5568 ¢)

Optimal ET sequence: 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: [⟨3 0 -18 -32 8 -21 -2 -36], ⟨0 4 21 34 2 27 12 41]]

Optimal tunings:

  • WE: ~34/27 = 399.9665 ¢, ~112/85 = 475.5198 ¢ (~95/91 = 75.5533 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5568 ¢ (~95/91 = 75.5568 ¢)

Optimal ET sequence: 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: [⟨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 ¢, ~112/85 = 475.5510 ¢ (~24/23 = 75.5485 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5482 ¢ (~24/23 = 75.5482 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 1.07

Condor

Condor tempers out 10976/10935 and may be described as the 58 & 159 temperament. The generator represents the septimal diminished fifth (112/81), and three minus an octave make vulture's generator of ~320/243. The ploidacot for this temperament is epsilon-dodecacot. 217edo is an excellent tuning for this temperament.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 40353607/40000000

Mapping: [⟨1 -4 -27 -20], ⟨0 12 63 49]]

mapping generators: ~2, ~112/81

Optimal tunings:

  • WE: ~2 = 1200.0142 ¢, ~112/81 = 558.5276 ¢
error map: ⟨+0.014 +0.319 +0.539 -1.260]
  • CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5212 ¢
error map: ⟨0.000 +0.300 +0.523 -1.287]

Optimal ET sequence: 58, 159, 217

Badness (Sintel): 3.92

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -4 -27 -20 -24], ⟨0 12 63 49 59]]

Optimal tunings:

  • WE: ~2 = 1199.9730 ¢, ~112/81 = 558.5052 ¢
  • CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5173 ¢

Optimal ET sequence: 58, 101cd, 159, 217, 376d

Badness (Sintel): 1.60

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -4 -27 -20 -24 -34], ⟨0 12 63 49 59 81]]

Optimal tunings:

  • WE: ~2 = 1199.9649 ¢, ~112/81 = 558.5040 ¢
  • CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5197 ¢

Optimal ET sequence: 58, 159, 217

Badness (Sintel): 1.05

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -4 -27 -20 -24 -34 12], ⟨0 12 63 49 59 81 -17]]

Optimal tunings:

  • WE: ~2 = 1199.9594 ¢, ~112/81 = 558.5017 ¢
  • CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5202 ¢

Optimal ET sequence: 58, 159, 217

Badness (Sintel): 1.12

Eagle

Eagle tempers out 2401/2400 and may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 10485760000/10460353203

Mapping: [⟨2 4 9 8], ⟨0 -8 -42 -23]]

mapping generators: ~177147/125440, ~28/27

Optimal tunings:

  • WE: ~177147/125440 = 599.9818 ¢, ~28/27 = 62.2266 ¢
error map: ⟨-0.036 +0.159 +0.004 -0.184]
  • CWE: ~177147/125440 = 600.0000 ¢, ~28/27 = 62.2295 ¢
error map: ⟨0.000 +0.209 +0.046 -0.105]

Optimal ET sequence: 58, 154c, 212, 270, 752, 1022, 1292, 2854b

Badness (Sintel): 1.51

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 19712/19683

Mapping: [⟨2 4 9 8 12], ⟨0 -8 -42 -23 -49]]

Optimal tunings:

  • WE: ~99/70 = 599.9796 ¢ ~28/27 = 62.2218 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~28/27 = 62.2251 ¢

Optimal ET sequence: 58, 154ce, 212, 270

Badness (Sintel): 0.823

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647

Mapping: [⟨2 4 9 8 12 13], ⟨0 -8 -42 -23 -49 -54]]

Optimal tunings:

  • WE: ~99/70 = 599.9763 ¢ ~28/27 = 62.2174 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~28/27 = 62.2211 ¢

Optimal ET sequence: 58, 154cef, 212, 270

Badness (Sintel): 0.673

Turkey

Named by Xenllium in 2021, turkey may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 5250987/5242880

Mapping: [⟨1 -8 -48 7], ⟨0 16 84 -7]]

mapping generators: ~2, ~3592/1715

Optimal tunings:

  • WE: ~2 = 1200.1147 ¢, ~3592/1715 = 718.9483 ¢
error map: ⟨+0.115 +0.300 -0.161 -0.661]
  • CWE: ~2 = 1200.0000 ¢, ~3592/1715 = 718.8806 ¢
error map: ⟨0.000 +0.134 -0.345 -0.990]

Optimal ET sequence: 212, 429, 1070d

Badness (Sintel): 5.34

11-limit

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 42875/42768, 160083/160000

Mapping: [⟨1 -8 -48 7 -87], ⟨0 16 84 -7 151]]

Optimal tunings:

  • WE: ~2 = 1200.1131 ¢ ~50/33 = 718.9478 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/33 = 718.8808 ¢

Optimal ET sequence: 212, 429

Badness (Sintel): 2.63

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104

Mapping: [⟨1 -8 -48 7 -87 -61], ⟨0 16 84 -7 151 108]]

Optimal tunings:

  • WE: ~2 = 1200.1324 ¢ ~50/33 = 718.9608 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/33 = 718.8825 ¢

Optimal ET sequence: 212, 217, 429

Badness (Sintel): 1.81

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