Vulture family: Difference between revisions

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

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

* [[POTE]]: ~2 = 1200.000, ~320/243 = 475.5426

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

[[Badness]]:

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

[[Optimal tuning]]s:  

: 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

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 }}

* 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 }}

Badness (Smith): 0.031907

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness (Smith): 0.018758

Subgroup: 2.3.5.7.11.13.17.19

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 }}

Optimal tunings:  

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

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

Badness (Smith): 0.00704

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Vulture family| ]] <!-- main article -->

[[Category:Vulture| ]] <!-- key article -->

[[Category:Rank 2]]