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]]).

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

== Vulture ==

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: mapping generators: ~2, ~320/243

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[[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 ==

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 [[Buzzardsmic clan #Septimal buzzard|buzzard]].

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.

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Optimal tunings:

* WE: ~2 = 1199.9695{{c}}, ~~~320~~/~~243~~ = 475.5451{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~~~320~~/~~243~~ = 475.5571{{c}}

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

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Badness (Sintel): 1.47

== Terture ==

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 generators: ~63/50, ~320/243

[[Optimal tuning]]s:

* [[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 }}

[[Badness]] (Sintel): 2.21

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}

Optimal tunings:

* WE: ~63/50 = 399.9902{{c}}, ~320/243 = 475.5383{{c}} (~392/375 = 75.5481{{c}})

* CWE: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5490{{c}} (~392/375 = 75.5490{{c}})

{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}

Badness (Sintel): 0.969

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976

Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}

Optimal tunings:

* WE: ~63/50 = 399.9958{{c}}, ~154/117 = 475.5485{{c}} (~117/112 = 75.5527{{c}})

* CWE: ~63/50 = 400.0000{{c}}, ~154/117 = 475.5531{{c}} (~117/112 = 75.5531{{c}})

Badness (Sintel): 0.771

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913

Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}

Optimal tunings:

* WE: ~34/27 = 399.9664{{c}}, ~112/85 = 475.5198{{c}} (~117/112 = 75.5534{{c}})

* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~117/112 = 75.5568{{c}})

Badness (Sintel): 0.953

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615

Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}

Optimal tunings:

* WE: ~34/27 = 399.9665{{c}}, ~112/85 = 475.5198{{c}} (~95/91 = 75.5533{{c}})

* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~95/91 = 75.5568{{c}})

Badness (Sintel): 0.846

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215

Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}

Optimal tunings:

* WE: ~34/27 = 400.0026{{c}}, ~112/85 = 475.5510{{c}} (~24/23 = 75.5485{{c}})

* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5482{{c}} (~24/23 = 75.5482{{c}})

Badness (Sintel): 1.07

== Condor ==

Condor tempers out [[10976/10935]] and may be described as the {{nowrap| 58 & 159 }} temperament. The generator represents the [[112/81|septimal diminished fifth (112/81)]], and three minus an octave make vulture's generator of ~320/243. The ploidacot for this temperament is epsilon-dodecacot. [[217edo]] is an excellent tuning for this temperament.

[[Subgroup]]: 2.3.5.7

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== Eagle ==

Eagle tempers out [[2401/2400]] and may be described as the {{nowrap| 58 & 270 }} temperament. It has a semi-octave period and a generator of ~28/27, four of which make a hemifourth which may be identified with 15/13, and two of those make a perfect fourth; its ploidacot thus is diploid wau-octacot. Compatible tunings include [[212edo]], [[270edo]], and [[328edo]].

[[Subgroup]]: 2.3.5.7

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== Turkey ==

Named by [[Xenllium]] in 2021, turkey may be described as the {{nowrap| 212 & 217 }} temperament. It is generated by a fifth sharp of just, close to 3\5 but on the flat side thereof, which can be interpreted as [[50/33]] in the 11-limit. Sixteen generators minus nine octaves make a perfect fifth; its ploidacot is thus theta-16-cot. [[429edo]] may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

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[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

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

[[Category:Rank 2]]

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-{{interwiki
+{{Interwiki
+| en = Vulture family
 | de = Vulture
-| en = Vulture_family
 | es =
 | ja =
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 {{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 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]]).
-Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]] and [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, condor, eagle, and turkey.
 == Vulture ==
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 {{Mapping|legend=1| 1 0 -6 | 0 4 21 }}
 : mapping generators: ~2, ~320/243
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 [[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 ==
-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 [[Buzzardsmic clan #Septimal buzzard|buzzard]].
+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.
@@ Line 75: / Line 75: @@
 Optimal tunings:
-* WE: ~2 = 1199.9695{{c}}, ~320/243 = 475.5451{{c}}
+* WE: ~2 = 1199.9695{{c}}, ~154/117 = 475.5451{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5571{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~154/117 = 475.5571{{c}}
 {{Optimal ET sequence|legend=0| 53, 217, 270 }}
@@ Line 127: / Line 127: @@
 Badness (Sintel): 1.47
+== Terture ==
+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
+[[Optimal tuning]]s:
+* [[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 }}
+{{Optimal ET sequence|legend=1| 111, 159, 270 }}
+[[Badness]] (Sintel): 2.21
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 3025/3024, 19712/19683, 102487/102400
+Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}
+Optimal tunings:
+* WE: ~63/50 = 399.9902{{c}}, ~320/243 = 475.5383{{c}} (~392/375 = 75.5481{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5490{{c}} (~392/375 = 75.5490{{c}})
+{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
+Badness (Sintel): 0.969
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
+Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}
+Optimal tunings:
+* WE: ~63/50 = 399.9958{{c}}, ~154/117 = 475.5485{{c}} (~117/112 = 75.5527{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~154/117 = 475.5531{{c}} (~117/112 = 75.5531{{c}})
+{{Optimal ET sequence|legend=0| 111, 159, 270 }}
+Badness (Sintel): 0.771
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
+Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}
+Optimal tunings:
+* WE: ~34/27 = 399.9664{{c}}, ~112/85 = 475.5198{{c}} (~117/112 = 75.5534{{c}})
+* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~117/112 = 75.5568{{c}})
+{{Optimal ET sequence|legend=0| 111, 159, 270 }}
+Badness (Sintel): 0.953
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
+Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}
+Optimal tunings:
+* WE: ~34/27 = 399.9665{{c}}, ~112/85 = 475.5198{{c}} (~95/91 = 75.5533{{c}})
+* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~95/91 = 75.5568{{c}})
+{{Optimal ET sequence|legend=0| 111, 159, 270 }}
+Badness (Sintel): 0.846
+=== 23-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215
+Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}
+Optimal tunings:
+* WE: ~34/27 = 400.0026{{c}}, ~112/85 = 475.5510{{c}} (~24/23 = 75.5485{{c}})
+* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5482{{c}} (~24/23 = 75.5482{{c}})
+{{Optimal ET sequence|legend=0| 111, 159, 270 }}
+Badness (Sintel): 1.07
 == Condor ==
+Condor tempers out [[10976/10935]] and may be described as the {{nowrap| 58 & 159 }} temperament. The generator represents the [[112/81|septimal diminished fifth (112/81)]], and three minus an octave make vulture's generator of ~320/243. The ploidacot for this temperament is epsilon-dodecacot. [[217edo]] is an excellent tuning for this temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 192: / Line 289: @@
 == Eagle ==
+Eagle tempers out [[2401/2400]] and may be described as the {{nowrap| 58 & 270 }} temperament. It has a semi-octave period and a generator of ~28/27, four of which make a hemifourth which may be identified with 15/13, and two of those make a perfect fourth; its ploidacot thus is diploid wau-octacot. Compatible tunings include [[212edo]], [[270edo]], and [[328edo]].
 [[Subgroup]]: 2.3.5.7
@@ Line 240: / Line 339: @@
 == Turkey ==
+Named by [[Xenllium]] in 2021, turkey may be described as the {{nowrap| 212 & 217 }} temperament. It is generated by a fifth sharp of just, close to 3\5 but on the flat side thereof, which can be interpreted as [[50/33]] in the 11-limit. Sixteen generators minus nine octaves make a perfect fifth; its ploidacot is thus theta-16-cot. [[429edo]] may be recommended as a tuning.
 [[Subgroup]]: 2.3.5.7
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 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Vulture family| ]] <!-- main article
 [[Category:Rank 2]]