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]]: | {{Technical data page}} | ||
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]]. Considered below are septimal vulture | Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]] and [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, condor, eagle, and turkey. | ||
== 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 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 | [[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]] | [[Badness]]: | ||
* Smith: 0.041431 | |||
* Dirichlet: 0.972 | |||
== Septimal vulture == | == 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. | 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 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }} | {{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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Badness (Smith): 0.018758 | Badness (Smith): 0.018758 | ||
==== | ==== 2.3.5.7.11.13.19 subgroup ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 676/675 | Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728 | ||
Mapping: {{mapping| 1 0 -6 25 -33 -7 | 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. | * CTE: ~2 = 1200.0000, ~25/19 = 475.5561 | ||
* | * CWE: ~2 = 1200.0000, , ~25/19 = 475.5569 | ||
{{Optimal ET sequence|legend=0| 53, 217, 270 | {{Optimal ET sequence|legend=0| 53, 217, 270 }} | ||
Badness (Smith): 0. | Badness (Smith): 0.00704 | ||
=== Semivulture === | === Semivulture === | ||
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Badness (Smith): 0.035458 | Badness (Smith): 0.035458 | ||
== Condor == | == Condor == | ||
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{{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }} | {{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791 | ||
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: mapping generators: ~177147/125440, ~28/27 | : mapping generators: ~177147/125440, ~28/27 | ||
[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229 | [[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229 | ||
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{{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }} | {{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120 | ||
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[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Vulture family| ]] <!-- main article --> | [[Category:Vulture family| ]] <!-- main article --> | ||
[[Category:Vulture| ]] <!-- key article --> | [[Category:Vulture| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||