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m →Buzzard: the S6 = S8 * S9 equivalence deserves to be noted; the exposition you gave made it sound like it's not special when it is (it's nontrivial relative to the S-expression perspective of analysis) and is singlehandedly the cause of a lot of very elegant and efficient temperaments that relate {S5, S6, S7, S8, S9} to each-other in some way. sorry for the errors i forgot to check how it rendered. remove the tilde for 15/13 cuz all the other tildes were removed so assuming it's a mistake
 
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The '''vulture family''' of [[temperament]]s [[tempering out|tempers out]] the [[vulture comma]] ({{monzo|legend=1| 24 -21 4 }}, [[ratio]]: 10485760000/10460353203), a small [[5-limit]] comma of 4.2 [[cent]]s.
{{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.  


Temperaments discussed elsewhere includes [[Landscape microtemperaments #Terture|terture]].
Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]] and [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, condor, eagle, and turkey.  


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


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: mapping generators: ~2, ~320/243
: mapping generators: ~2, ~320/243


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~320/243 = 475.5426
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~320/243 = 475.5351
: [[error map]]: {{val| 0.0000 +0.1855 -0.0758 }}
* [[POTE]]: ~2 = 1200.000, ~320/243 = 475.5426
: error map: {{val| 0.0000 +0.2154 +0.0811 }}


{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, 3500b, 3823b, 4146b, 4469b }}
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}


[[Badness]]: 0.041431
[[Badness]]:
* Smith: 0.041431
* Dirichlet: 0.972


== 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 a good tuning for this temperament, with generator 107\270. The harmonic 7 is found at -14 fifths or {{nowrap| (-14) × 4 {{=}} -56 }} generator steps, so that 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|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
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{{Mapping|legend=1| 1 0 -6 25 | 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:
 
* [[CTE]]: ~2 = 1200.0000, ~320/243 = 475.5528
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~320/243 = 475.5511
: [[error map]]: {{val| 0.0000 +0.2561 +0.2945 +0.2188 }}
* [[POTE]]: ~2 = 1200.0000, ~320/243 = 475.5511
: error map: {{val| 0.0000 +0.2495 +0.2601 +0.3106 }}


{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863, 1133 }}
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863, 1133 }}


[[Badness]]: 0.036985
[[Badness]] (Smith): 0.036985


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 0 -6 25 -33 | 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:
* CTE: ~2 = 1200.0000, ~320/243 = 475.5558
* POTE: ~2 = 1200.0000, ~320/243 = 475.5567


{{Optimal ET sequence|legend=1| 53, 217, 270 }}
{{Optimal ET sequence|legend=0| 53, 217, 270, 2107c, 2377bc }}


Badness: 0.031907
Badness (Smith): 0.031907


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 0 -6 25 -33 -7 | 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:
 
* CTE: ~2 = 1200.0000, ~320/243 = 475.5566
{{Optimal ET sequence|legend=1| 53, 217, 270 }}
* POTE: ~2 = 1200.0000, ~320/243 = 475.5572
 
Badness: 0.018758
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 676/675, 936/935, 1001/1000, 1225/1224, 4096/4095
 
Mapping: {{mapping| 1 0 -6 25 -33 -7 35 | 0 4 21 -56 92 27 -78 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~112/85 = 475.5617


{{Optimal ET sequence|legend=1| 53, 217, 270, 487, 757g }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}


Badness: 0.020103
Badness (Smith): 0.018758


==== 19-limit ====
==== 2.3.5.7.11.13.19 subgroup ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


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


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


Optimal tuning (POTE): ~2 = 1\1, ~25/19 = 475.5615
Optimal tunings:
* CTE: ~2 = 1200.0000, ~25/19 = 475.5561
* CWE: ~2 = 1200.0000, , ~25/19 = 475.5569


{{Optimal ET sequence|legend=1| 53, 217, 270, 487, 757g }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}


Badness: 0.013850
Badness (Smith): 0.00704


=== Semivulture ===
=== Semivulture ===
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: 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:
* CTE: ~99/70 = 600.0000, ~320/243 = 475.5523
* POTE: ~99/70 = 600.0000, ~320/243 = 475.5496


{{Optimal ET sequence|legend=1| 106, 164, 270, 916, 1186, 1456 }}
{{Optimal ET sequence|legend=0| 106, 164, 270, 916, 1186, 1456 }}


Badness: 0.040799
Badness (Smith): 0.040799


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 2 0 -12 50 41 -14 | 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 tunings:
 
* CTE: ~99/70 = 600.0000, ~320/243 = 475.5540
{{Optimal ET sequence|legend=1| 106, 164, 270 }}
* POTE: ~99/70 = 600.0000, ~320/243 = 475.553
 
Badness: 0.035458
 
== Buzzard ==
{{Main| Buzzard }}
{{See also| No-fives subgroup temperaments #Buzzard }}
 
Buzzard is the main extension to vulture of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~[[21/16]], but is more of a full 13-limit system in its own right. It is most naturally described as 53 & 58 (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, 111edo is a great tuning for it.
 
Its S-expression-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]}, with the structure of its 7-limit being the consequences of these equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] * [[81/80|S9]] (which [[hemifamity]] leverages). That is, like in any hemifamity tuning, [[36/35]] is split into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]]. Then the vanish of the orwellisma implies [[49/48]], the large septimal diesis, is tuned to the same step as 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 5120/5103
 
{{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }}
 
{{Multival|legend=1| 4 21 -3 24 -16 -66 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.636
 
{{Optimal ET sequence|legend=1| 5, 43c, 48, 53, 111, 164d, 275d }}
 
[[Badness]]: 0.047963
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 5120/5103
 
Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }}


{{Multival|legend=1| 4 21 -3 39 24 -16 48 -66 18 120 }}
{{Optimal ET sequence|legend=0| 106, 164, 270 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.700
Badness (Smith): 0.035458
 
{{Optimal ET sequence|legend=1| 53, 58, 111, 280cd, 391cd }}
 
Badness: 0.034484
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 540/539, 676/675
 
Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}
 
{{Multival|legend=1| 4 21 -3 39 27 24 -16 48 28 -66 18 -15 120 87 -51 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.697
 
{{Optimal ET sequence|legend=1| 53, 58, 111, 280cdf, 391cdf }}
 
Badness: 0.018842
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
 
Mapping: {{mapping| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.692
 
{{Optimal ET sequence|legend=1| 53, 58, 111, 321cdfg }}
 
Badness: 0.018403
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
 
Mapping: {{mapping| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.679
 
{{Optimal ET sequence|legend=1| 53, 58h, 111 }}
 
Badness: 0.015649
 
=== Buteo ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 385/384, 2200/2187
 
Mapping: {{mapping| 1 0 -6 4 9 | 0 4 21 -3 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.436
 
{{Optimal ET sequence|legend=1| 5, 48, 53 }}
 
Badness: 0.060238
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 275/273, 385/384, 572/567
 
Mapping: {{mapping| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.464
 
{{Optimal ET sequence|legend=1| 5, 48f, 53 }}
 
Badness: 0.039854


== Condor ==
== Condor ==
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{{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }}
{{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }}
{{Multival|legend=1| 12 63 49 72 44 -63 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791
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: 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]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229
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{{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }}
{{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }}
{{Multival|legend=1|16 84 -7 96 -56 -252}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Vulture family| ]] <!-- main article -->
[[Category:Vulture family| ]] <!-- main article -->
[[Category:Vulture| ]] <!-- key article -->
[[Category:Vulture| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Vulture_family"