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Vulture family: Difference between revisions

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m →Buzzard: improve S-expression-based comma list to be 13-limit
In a private discourse with Godtone, we decided vulture should be defined on 2.3.5.7.11.13.19
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== 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.  
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 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
[[Subgroup]]: 2.3.5
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{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}


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


Badness (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]].  


=== 2.3.5.19 ===
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.  
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]])<sup>2</sup> = 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 of the interpretation of 1 gen as ~25/19, 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. {{nowrap|* 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.
 
Subgroup: 2.3.5.19
 
Commas: 1216/1215, 64000000/63950067
 
{{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 }}
 
Badness (Dirichlet): 0.232
 
== 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 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]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness (Smith): 0.018758
Badness (Smith): 0.018758


==== 17-limit ====
==== 2.3.5.7.11.13.19 subgroup ====
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 tunings:
* CTE: ~2 = 1200.0000, ~112/85 = 475.5613
* POTE: ~2 = 1200.0000, ~112/85 = 475.5617
 
{{Optimal ET sequence|legend=0| 53, 217, 270, 487, 757g }}
 
Badness (Smith): 0.020103
 
==== 19-limit ====
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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~25/19 = 475.5606
* CTE: ~2 = 1200.0000, ~25/19 = 475.5561
* POTE: ~2 = 1200.0000, , ~25/19 = 475.5615
* CWE: ~2 = 1200.0000, , ~25/19 = 475.5569


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


Badness (Smith): 0.013850
Badness (Smith): 0.00704


=== Semivulture ===
=== Semivulture ===
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