Vulture family: Difference between revisions

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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 that is the amount by which a stack of four [[syntonic comma]]s falls short of the [[256/243]] Pythagorean limma. As their defining feature, vulture temperaments split the interval [[3/1]] into four segments (identified in the 5-limit as [[320/243]]).

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

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.9430{{c}}, ~320/243 = 475.5200{{c}}

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

[[Badness]] (Sintel): 0.972

=== Overview to extensions ===

Temperaments discussed elsewhere include [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, terture, condor, eagle, and turkey.

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 an excellent tuning for this temperament, with generator 107\270. Other compatible tunings include [[217edo]] and [[323edo]]. The harmonic 7 is found at -14 fifths or {{nowrap| (-14) × 4 {{=}} -56 }} generator steps, so 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 [[Buzzardsmic clan #Septimal 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 }}

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.9050{{c}}, ~320/243 = 475.5135{{c}}

: [[error map]]: {{val| -0.095 +0.099 +0.039 +0.044 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5515{{c}}

: error map: {{val| 0.000 +0.251 +0.267 +0.292 }}

 

{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863, 1133, 1996d }}

 

 

 

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

 

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

 

* WE: ~2 = 1199.9392{{c}}, ~320/243 = 475.5326{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5655{{c}}

 

{{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

 

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

 

* WE: ~2 = 1199.9695{{c}}, ~154/117 = 475.5451{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~154/117 = 475.5571{{c}}

 

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

 

 

==== 2.3.5.7.11.13.19 subgroup ====

 

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

 

 

* WE: ~2 = 1199.9636{{c}}, ~25/19 = 475.5426{{c}}

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

 

 

Badness (Sintel): 0.579

 

=== Semivulture ===

 

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

 

* WE: ~99/70 = 599.9594{{c}}, ~320/243 = 475.5174{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5501{{c}}

 

{{Optimal ET sequence|legend=0| 106, 164, 270, 916, 1186, 1456 }}

 

 

==== 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:  

* WE: ~99/70 = 599.9859{{c}}, ~320/243 = 475.5423{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5536{{c}}

 

 

Badness (Sintel): 1.47

 

Named by [[Xenllium]] in 2021, terture tempers out 250047/250000, the [[landscape comma]], and may be described as the {{nowrap| 111 & 159 }} temperament, with a [[ploidacot]] signature of triploid gamma-tetracot.

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 250047/250000, 359661568/358722675

 

{{Mapping|legend=1| 3 0 -18 -32 | 0 4 21 34 }}

: mapping generators: ~63/50, ~320/243

 

* [[WE]]: ~63/50 = 399.9723{{c}}, ~320/243 = 475.5221{{c}} (~392/375 = 75.5499{{c}})

: [[error map]]: {{val| -0.083 +0.134 +0.151 -0.185 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5519{{c}} (~392/375 = 75.5519{{c}})

: error map: {{val| 0.000 +0.253 +0.276 -0.061 }}

 

 

 

 

Comma list: 3025/3024, 19712/19683, 102487/102400