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]]: 10485760000/10460353203), a small [[5-limit]] comma of 4.2 [[cent]]s.  

Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]]. Considered below are septimal vulture, buzzard, 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.  

[[Badness]] (Smith): 0.041431

 

=== 2.3.5.19 ===

It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])² = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result, the 3 gen interval of ~226.6{{cent}} is interpreted as ([[3/2]])/([[25/19]]) = [[~]][[57/50]] which is tuned ~0.2{{cent}} flat. (Interpreting this interval as a damaged [[~]][[8/7]] leads to [[#Buzzard]].) Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. * Compared to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.

 

 

{{Mapping|legend=1| 1 0 -6 -12 | 0 4 21 41 }}

 

[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~25/19 = 475.542

: [[error map]]: {{val| 0.000 +0.214 +0.075 -0.278  }}

 

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

 

 

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 for 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]].

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

 

 

 

 

 

 

==== 19-limit ====

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

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

* CTE: ~2 = 1200.0000, ~25/19 = 475.5606

* POTE: ~2 = 1200.0000, , ~25/19 = 475.5615

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

Badness (Smith): 0.013850