Vulture family: Difference between revisions

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The '''vulture family''' of [[temperament|temperament]]s [[tempering_out|tempers out]] the [[vulture comma]], |24 -21 4> = 10485760000/10460353203, a small [[5-limit]] comma of size 4.1998 [[cent]]s. Aside from vulture itself, the family contains [[Hemifamity_temperaments#Buzzard|buzzard]], considered elsewhere, and condor.
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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.  


[[Comma|Comma]]: 10485760000/10460353203
Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]] and [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, condor, eagle, and turkey.


POTE generator: ~320/243 = 475.5426
== 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.  


Map: [<1 0 -6|, <0 4 21|]
[[Subgroup]]: 2.3.5


EDOs: [[5edo|5]], [[48edo|48]], [[53edo|53]], [[164edo|164]], [[217edo|217]], [[270edo|270]], [[323edo|323]], [[2531edo|2531]]
[[Comma list]]: 10485760000/10460353203


Badness: 0.0414
{{Mapping|legend=1| 1 0 -6 | 0 4 21 }}


=Vulture=
: mapping generators: ~2, ~320/243
Commas: 4375/4374, 33554432/33480783


POTE generator: ~320/243 = 475.5511
[[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 }}


Map: [<1 0 -6 25|, <0 4 21 -56|]
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}


EDOs: 53, 164, 217, 270, [[593edo|593]], [[863edo|863]], [[1133edo|1133]]
[[Badness]]:
* Smith: 0.041431
* Dirichlet: 0.972


Badness: 0.0370
== Septimal vulture ==
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]].  


==11-limit==
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.
Commas: 4375/4374, 5632/5625, 41503/41472


POTE generator: ~320/243 = 475.5567
[[Subgroup]]: 2.3.5.7


Map: [<1 0 -6 25 -33|, <0 4 21 -56 92|]
[[Comma list]]: 4375/4374, 33554432/33480783


EDOs: 53, 217, 270
{{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }}


Bad: 0.0319
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~320/243 = 475.5528
: [[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863, 1133 }}
Commas: 676/675, 1001/1000, 4375/4374, 4225/4224


POTE generator: ~320/243 = 475.5572
[[Badness]] (Smith): 0.036985


Map: [<1 0 -6 25 -33 -7|, <0 4 21 -56 92 27|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 53, 217, 270
Comma list: 4375/4374, 5632/5625, 41503/41472


Badness: 0.0188
Mapping: {{mapping| 1 0 -6 25 -33 | 0 4 21 -56 92 }}


=Condor=
Optimal tunings:
Commas: 10976/10935, 40353607/40000000
* CTE: ~2 = 1200.0000, ~320/243 = 475.5558
* POTE: ~2 = 1200.0000, ~320/243 = 475.5567


==11-limit==
{{Optimal ET sequence|legend=0| 53, 217, 270, 2107c, 2377bc }}
Commas: 441/440, 4000/3993, 10976/10935


POTE generator: ~81/56 = 641.4822
Badness (Smith): 0.031907


Map: [<1 8 36 29 35|, <0 -12 -63 -49 -59|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 58, 159, 217
Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374


Badness: 0.0484
Mapping: {{mapping| 1 0 -6 25 -33 -7 | 0 4 21 -56 92 27 }}


==13-limit==
Optimal tunings:
Commas: 364/363, 441/440, 676/675, 10976/10935
* CTE: ~2 = 1200.0000, ~320/243 = 475.5566
* POTE: ~2 = 1200.0000, ~320/243 = 475.5572


POTE generator: ~81/56 = 641.4797
{{Optimal ET sequence|legend=0| 53, 217, 270 }}


Map: [<1 8 36 29 35 47|, <0 -12 -63 -49 -59 -81|]
Badness (Smith): 0.018758


EDOs: 58, 159, 217
==== 2.3.5.7.11.13.19 subgroup ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0255
Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728


[[Category:family]]
Mapping: {{mapping| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}
[[Category:temperament]]
 
[[Category:vulture]]
Optimal tunings:
* CTE: ~2 = 1200.0000, ~25/19 = 475.5561
* CWE: ~2 = 1200.0000, , ~25/19 = 475.5569
 
{{Optimal ET sequence|legend=0| 53, 217, 270 }}
 
Badness (Smith): 0.00704
 
=== Semivulture ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 33554432/33480783
 
Mapping: {{mapping| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
 
: mapping generators: ~99/70, ~320/243
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~320/243 = 475.5523
* POTE: ~99/70 = 600.0000, ~320/243 = 475.5496
 
{{Optimal ET sequence|legend=0| 106, 164, 270, 916, 1186, 1456 }}
 
Badness (Smith): 0.040799
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374
 
Mapping: {{mapping| 2 0 -12 50 41 -14 | 0 4 21 -56 -43 27 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~320/243 = 475.5540
* POTE: ~99/70 = 600.0000, ~320/243 = 475.553
 
{{Optimal ET sequence|legend=0| 106, 164, 270 }}
 
Badness (Smith): 0.035458
 
== Condor ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 40353607/40000000
 
{{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791
 
{{Optimal ET sequence|legend=1| 58, 159, 217 }}
 
[[Badness]]: 0.154715
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4000/3993, 10976/10935
 
Mapping: {{mapping| 1 8 36 29 35 | 0 -12 -63 -49 -59 }}
 
Optimal tuning (POTE): ~2 = 1\1, 81/56 = 641.4822
 
{{Optimal ET sequence|legend=1| 58, 101cd, 159, 217 }}
 
Badness: 0.048401
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 10976/10935
 
Mapping: {{mapping| 1 8 36 29 35 47 | 0 -12 -63 -49 -59 -81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4797
 
{{Optimal ET sequence|legend=1| 58, 159, 217 }}
 
Badness: 0.025469
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619
 
Mapping: {{mapping| 1 8 36 29 35 47 -5 | 0 -12 -63 -49 -59 -81 17 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4794
 
{{Optimal ET sequence|legend=1| 58, 159, 217 }}
 
Badness: 0.021984
 
== Eagle ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 10485760000/10460353203
 
{{Mapping|legend=1| 2 4 9 8 | 0 -8 -42 -23 }}
 
: mapping generators: ~177147/125440, ~28/27
 
[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229
 
{{Optimal ET sequence|legend=1| 58, 154c, 212, 270, 752, 1022, 1292, 2854b }}
 
[[Badness]]: 0.059498
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 9801/9800, 19712/19683
 
Mapping: {{mapping| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.224
 
{{Optimal ET sequence|legend=1| 58, 154ce, 212, 270 }}
 
Badness: 0.024885
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647
 
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 ET sequence|legend=1| 58, 154cef, 212, 270 }}
 
Badness: 0.016282
 
== Turkey ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4802000/4782969, 5250987/5242880
 
{{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120
 
{{Optimal ET sequence|legend=1| 5, 207c, 212, 429 }}
 
[[Badness]]: 0.210964
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 19712/19683, 42875/42768, 160083/160000
 
Mapping: {{mapping| 1 8 36 0 64 | 0 -16 -84 7 -151 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.120
 
{{Optimal ET sequence|legend=1| 212, 429 }}
 
Badness: 0.079694
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104
 
Mapping: {{mapping| 1 8 36 0 64 47 | 0 -16 -84 7 -151 -108 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.118
 
{{Optimal ET sequence|legend=1| 212, 217, 429 }}
 
Badness: 0.043787
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Vulture family| ]] <!-- main article -->
[[Category:Vulture| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 00:36, 24 June 2025

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.

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

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:

  • CTE: ~2 = 1200.000, ~320/243 = 475.5351
error map: ⟨0.0000 +0.1855 -0.0758]
  • POTE: ~2 = 1200.000, ~320/243 = 475.5426
error map: ⟨0.0000 +0.2154 +0.0811]

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

Badness:

  • Smith: 0.041431
  • Dirichlet: 0.972

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 a good tuning for this temperament, with generator 107\270. The harmonic 7 is found at -14 fifths or (-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.

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:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5528
error map: ⟨0.0000 +0.2561 +0.2945 +0.2188]
  • POTE: ~2 = 1200.0000, ~320/243 = 475.5511
error map: ⟨0.0000 +0.2495 +0.2601 +0.3106]

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

Badness (Smith): 0.036985

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:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5558
  • POTE: ~2 = 1200.0000, ~320/243 = 475.5567

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

Badness (Smith): 0.031907

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:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5566
  • POTE: ~2 = 1200.0000, ~320/243 = 475.5572

Optimal ET sequence: 53, 217, 270

Badness (Smith): 0.018758

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.17.19

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:

  • CTE: ~2 = 1200.0000, ~25/19 = 475.5561
  • CWE: ~2 = 1200.0000, , ~25/19 = 475.5569

Optimal ET sequence: 53, 217, 270

Badness (Smith): 0.00704

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:

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

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

Badness (Smith): 0.040799

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:

  • CTE: ~99/70 = 600.0000, ~320/243 = 475.5540
  • POTE: ~99/70 = 600.0000, ~320/243 = 475.553

Optimal ET sequence: 106, 164, 270

Badness (Smith): 0.035458

Condor

Subgroup: 2.3.5.7

Comma list: 10976/10935, 40353607/40000000

Mapping: [⟨1 8 36 29], ⟨0 -12 -63 -49]]

Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4791

Optimal ET sequence: 58, 159, 217

Badness: 0.154715

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 8 36 29 35], ⟨0 -12 -63 -49 -59]]

Optimal tuning (POTE): ~2 = 1\1, 81/56 = 641.4822

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

Badness: 0.048401

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 8 36 29 35 47], ⟨0 -12 -63 -49 -59 -81]]

Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4797

Optimal ET sequence: 58, 159, 217

Badness: 0.025469

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 8 36 29 35 47 -5], ⟨0 -12 -63 -49 -59 -81 17]]

Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4794

Optimal ET sequence: 58, 159, 217

Badness: 0.021984

Eagle

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 tuning (POTE): ~177147/125440 = 1\2, ~28/27 = 62.229

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

Badness: 0.059498

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 tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.224

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

Badness: 0.024885

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 tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.220

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

Badness: 0.016282

Turkey

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 5250987/5242880

Mapping: [⟨1 8 36 0], ⟨0 -16 -84 7]]

Optimal tuning (POTE): ~2 = 1\1, ~1715/1296 = 481.120

Optimal ET sequence: 5, 207c, 212, 429

Badness: 0.210964

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 8 36 0 64], ⟨0 -16 -84 7 -151]]

Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.120

Optimal ET sequence: 212, 429

Badness: 0.079694

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 8 36 0 64 47], ⟨0 -16 -84 7 -151 -108]]

Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.118

Optimal ET sequence: 212, 217, 429

Badness: 0.043787

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