Vulture family: Difference between revisions

| en = Vulture_family

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.  

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

[[Comma list]]: 10485760000/10460353203

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

: mapping generators: ~2, ~320/243

* [[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, …, 4469b }}

* Smith: 0.041431

* Dirichlet: 0.972

== 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]].

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|legend=1| 1 0 -6 25 | 0 4 21 -56 }}

 

: [[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 }}

 

[[Badness]] (Smith): 0.036985

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

Mapping: {{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|legend=0| 53, 217, 270, 2107c, 2377bc }}

 

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

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

 

 

Optimal tunings:  

* CTE: ~2 = 1200.0000, ~320/243 = 475.5566

* POTE: ~2 = 1200.0000, ~320/243 = 475.5572

 

 

 

 

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 }}

 

* CTE: ~2 = 1200.0000, ~25/19 = 475.5561

* CWE: ~2 = 1200.0000, , ~25/19 = 475.5569

 

{{Optimal ET sequence|legend=0| 53, 217, 270 }}

 

 

=== Semivulture ===

 

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

 

 

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

 

* 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

 

 

 

[[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 }}

 

 

=== 11-limit ===

 

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

 

 

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

 

 

Badness: 0.025469

 

Subgroup: 2.3.5.7.11.13.17

 

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

 

 

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

 

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

 

Badness: 0.021984

 

[[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

 

 

[[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 }}

 

 

 

 

=== 13-limit ===

 

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

 

 

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

 

 

 

 

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

 

 

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

 

 

 

=== 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

 

 

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

 

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

 

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