Vulture family: Difference between revisions

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=== Overview to extensions ===

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

Strong 7-limit extensions include septimal vulture (53 & 270), which tempers out [[4375/4374]], and buzzard (53 & 58), which tempers out [[1728/1715]].

Terture adds [[250047/250000]] with a 1/3-octave period. Condor adds [[10976/10935]] and splits the generator in three. Eagle adds [[2401/2400]]. Turkey adds [[4802000/4782969]]. Those split the generator in four.

Most of these temperaments extend naturally to the 11-limit via tempering out the [[symbiotic comma]] and [[seascape comma]], from 10485760000/10460353203 = (19712/19683)<sup>2</sup>/(160083/160000), and to the 13-limit via identifying the hemitwelfth as [[~]][[26/15]], tempering out the [[island comma]].

Buzzard is discussed in its own clan (→ [[Buzzardsmic clan #Septimal buzzard]]). The rest are considered below.

== Septimal vulture ==

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

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

Optimal tunings:

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Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374

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

Optimal tunings:

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

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

{{Mapping|legend=0| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}

Optimal tunings:

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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

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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:

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

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

Optimal tunings:

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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:

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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:

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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 }}

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

Optimal tunings:

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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 }}

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

Optimal tunings:

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Comma list: 441/440, 4000/3993, 10976/10935

~~Mapping:~~ {{~~mapping~~| 1 -4 -27 -20 -24 | 0 12 63 49 59 }}

Optimal tunings:

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Comma list: 364/363, 441/440, 676/675, 10976/10935

~~Mapping:~~ {{~~mapping~~| 1 -4 -27 -20 -24 -34 | 0 12 63 49 59 81 }}

Optimal tunings:

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Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619

~~Mapping:~~ {{~~mapping~~| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}

{{Mapping|legend=0| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}

Optimal tunings:

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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~~]].

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 edo tunings include [[212edo]], [[270edo]] and [[328edo]] in the 212h, 270 and 328h [[val]]s, respectively.

[[Subgroup]]: 2.3.5.7

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Comma list: 2401/2400, 9801/9800, 19712/19683

~~Mapping:~~ {{~~mapping~~| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}

Optimal tunings:

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Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647

~~Mapping:~~ {{~~mapping~~| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}

Optimal tunings:

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Badness (Sintel): 0.673

=== 2.3.5.7.11.13.19 subgroup ===

Subgroup: 2.3.5.7.11.13.19

Comma list: 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539

Optimal tunings:

* WE: ~99/70 = 599.9810{{c}} ~28/27 = 62.2195{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.2223{{c}}

Badness (Sintel): 0.609

== Turkey ==

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Comma list: 19712/19683, 42875/42768, 160083/160000

~~Mapping:~~ {{~~mapping~~| 1 -8 -48 7 -87 | 0 16 84 -7 151 }}

Optimal tunings:

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Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104

~~Mapping:~~ {{~~mapping~~| 1 -8 -48 7 -87 -61 | 0 16 84 -7 151 108 }}

Optimal tunings:

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Badness (Sintel): 1.81

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

[[Category:Vulture family| ]]

[[Category:Temperament families]]

[[Category:Catalogs of rank-2 temperaments]]

@@ Line 29: / Line 29: @@
 === Overview to extensions ===
-Temperaments discussed elsewhere include [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, terture, condor, eagle, and turkey.
+Strong 7-limit extensions include septimal vulture (53 & 270), which tempers out [[4375/4374]], and buzzard (53 & 58), which tempers out [[1728/1715]].
+Terture adds [[250047/250000]] with a 1/3-octave period. Condor adds [[10976/10935]] and splits the generator in three. Eagle adds [[2401/2400]]. Turkey adds [[4802000/4782969]]. Those split the generator in four.
+Most of these temperaments extend naturally to the 11-limit via tempering out the [[symbiotic comma]] and [[seascape comma]], from 10485760000/10460353203 = (19712/19683)<sup>2</sup>/(160083/160000), and to the 13-limit via identifying the hemitwelfth as [[~]][[26/15]], tempering out the [[island comma]].
+Buzzard is discussed in its own clan (→ [[Buzzardsmic clan #Septimal buzzard]]). The rest are considered below.
 == Septimal vulture ==
@@ Line 57: / Line 63: @@
 Comma list: 4375/4374, 5632/5625, 41503/41472
-Mapping: {{mapping| 1 0 -6 25 -33 | 0 4 21 -56 92 }}
+{{Mapping|legend=0| 1 0 -6 25 -33 | 0 4 21 -56 92 }}
 Optimal tunings:
@@ Line 72: / Line 78: @@
 Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374
-Mapping: {{mapping| 1 0 -6 25 -33 -7 | 0 4 21 -56 92 27 }}
+{{Mapping|legend=0| 1 0 -6 25 -33 -7 | 0 4 21 -56 92 27 }}
 Optimal tunings:
@@ Line 87: / Line 93: @@
 Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728
-Mapping: {{mapping| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}
+{{Mapping|legend=0| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}
 Optimal tunings:
@@ Line 102: / Line 108: @@
 Comma list: 3025/3024, 4375/4374, 33554432/33480783
-Mapping: {{mapping| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
+{{Mapping|legend=0| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
 : mapping generators: ~99/70, ~320/243
@@ Line 118: / Line 124: @@
 Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374
-Mapping: {{mapping| 2 0 -12 50 41 -14 | 0 4 21 -56 -43 27 }}
+{{Mapping|legend=0| 2 0 -12 50 41 -14 | 0 4 21 -56 -43 27 }}
 Optimal tunings:
@@ Line 153: / Line 159: @@
 Comma list: 3025/3024, 19712/19683, 102487/102400
-Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}
+{{Mapping|legend=0| 3 0 -18 -32 8 | 0 4 21 34 2 }}
 Optimal tunings:
@@ Line 168: / Line 174: @@
 Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
-Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}
+{{Mapping|legend=0| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}
 Optimal tunings:
@@ Line 183: / Line 189: @@
 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 }}
+{{Mapping|legend=0| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}
 Optimal tunings:
@@ Line 198: / Line 204: @@
 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 }}
+{{Mapping|legend=0| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}
 Optimal tunings:
@@ Line 213: / Line 219: @@
 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 }}
+{{Mapping|legend=0| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}
 Optimal tunings:
@@ Line 248: / Line 254: @@
 Comma list: 441/440, 4000/3993, 10976/10935
-Mapping: {{mapping| 1 -4 -27 -20 -24 | 0 12 63 49 59 }}
+{{Mapping|legend=0| 1 -4 -27 -20 -24 | 0 12 63 49 59 }}
 Optimal tunings:
@@ Line 263: / Line 269: @@
 Comma list: 364/363, 441/440, 676/675, 10976/10935
-Mapping: {{mapping| 1 -4 -27 -20 -24 -34 | 0 12 63 49 59 81 }}
+{{Mapping|legend=0| 1 -4 -27 -20 -24 -34 | 0 12 63 49 59 81 }}
 Optimal tunings:
@@ Line 278: / Line 284: @@
 Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619
-Mapping: {{mapping| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}
+{{Mapping|legend=0| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}
 Optimal tunings:
@@ Line 289: / Line 295: @@
 == 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]].
+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 edo tunings include [[212edo]], [[270edo]] and [[328edo]] in the 212h, 270 and 328h [[val]]s, respectively.
 [[Subgroup]]: 2.3.5.7
@@ Line 313: / Line 319: @@
 Comma list: 2401/2400, 9801/9800, 19712/19683
-Mapping: {{mapping| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}
+{{Mapping|legend=0| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}
 Optimal tunings:
@@ Line 328: / Line 334: @@
 Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647
-Mapping: {{mapping| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}
+{{Mapping|legend=0| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}
 Optimal tunings:
@@ Line 337: / Line 343: @@
 Badness (Sintel): 0.673
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
+Comma list: 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539
+{{Mapping|legend=0| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 -82 }}
+Optimal tunings:
+* WE: ~99/70 = 599.9810{{c}} ~28/27 = 62.2195{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.2223{{c}}
+{{Optimal ET sequence|legend=0| 58h, 154cefhh, 212h, 270 }}
+Badness (Sintel): 0.609
 == Turkey ==
@@ Line 363: / Line 384: @@
 Comma list: 19712/19683, 42875/42768, 160083/160000
-Mapping: {{mapping| 1 -8 -48 7 -87 | 0 16 84 -7 151 }}
+{{Mapping|legend=0| 1 -8 -48 7 -87 | 0 16 84 -7 151 }}
 Optimal tunings:
@@ Line 378: / Line 399: @@
 Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104
-Mapping: {{mapping| 1 -8 -48 7 -87 -61 | 0 16 84 -7 151 108 }}
+{{Mapping|legend=0| 1 -8 -48 7 -87 -61 | 0 16 84 -7 151 108 }}
 Optimal tunings:
@@ Line 388: / Line 409: @@
 Badness (Sintel): 1.81
-[[Category:Vulture family| ]] <!-- main article
+[[Category:Vulture family| ]] <!-- main article -->
 [[Category:Temperament families]]
 [[Category:Catalogs of rank-2 temperaments]]