Vulture family: Difference between revisions

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== 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. 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.

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

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[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.~~000~~, ~320/243 = 475.5351

* [[CTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5351{{c}}

: [[error map]]: {{val| 0.0000 +0.1855 -0.0758 }}

* [[POTE]]: ~2 = 1200.~~000~~, ~320/243 = 475.5426

* [[POTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5426{{c}}

: error map: {{val| 0.0000 +0.2154 +0.0811 }}

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[[Badness]]:

* Smith: 0.041431

* ~~Dirichlet~~: 0.972

* Sintel: 0.972

== Septimal vulture ==

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[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.0000, ~320/243 = 475.5528

* [[CTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5528{{c}}

: [[error map]]: {{val| 0.0000 +0.2561 +0.2945 +0.2188 }}

* [[POTE]]: ~2 = 1200.0000, ~320/243 = 475.5511

* [[POTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5511{{c}}

: error map: {{val| 0.0000 +0.2495 +0.2601 +0.3106 }}

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Optimal tunings:

* CTE: ~2 = 1200.0000, ~320/243 = 475.5558

* CTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5558{{c}}

* POTE: ~2 = 1200.0000, ~320/243 = 475.5567

* POTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5567{{c}}

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Optimal tunings:

* CTE: ~2 = 1200.0000, ~320/243 = 475.5566

* CTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5566{{c}}

* POTE: ~2 = 1200.0000, ~320/243 = 475.5572

* POTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5572{{c}}

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Optimal tunings:

* CTE: ~2 = 1200.0000, ~25/19 = 475.5561

* CTE: ~2 = 1200.0000{{c}}, ~25/19 = 475.5561{{c}}

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

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

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Mapping: {{mapping| 2 0 -12 50 41 | 0 4 21 -56 -43 }}

: mapping generators: ~99/70, ~320/243

Optimal tunings:

* CTE: ~99/70 = 600.0000, ~320/243 = 475.5523

* CTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5523{{c}}

* POTE: ~99/70 = 600.0000, ~320/243 = 475.5496

* POTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5496{{c}}

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

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Optimal tunings:

* CTE: ~99/70 = 600.0000, ~320/243 = 475.5540

* CTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5540{{c}}

* POTE: ~99/70 = 600.0000, ~320/243 = 475.553

* POTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.553{{c}}

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[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~81/56 = 641.4791

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~81/56 = 641.4791{{c}}

[[Badness]]: 0.154715

[[Badness]] (Smith): 0.154715

=== 11-limit ===

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Mapping: {{mapping| 1 8 36 29 35 | 0 -12 -63 -49 -59 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, 81/56 = 641.4822

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, 81/56 = 641.4822{{c}}

Badness: 0.048401

Badness (Smith): 0.048401

=== 13-limit ===

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Mapping: {{mapping| 1 8 36 29 35 47 | 0 -12 -63 -49 -59 -81 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~81/56 = 641.4797

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~81/56 = 641.4797{{c}}

Badness: 0.025469

Badness (Smith): 0.025469

=== 17-limit ===

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Mapping: {{mapping| 1 8 36 29 35 47 -5 | 0 -12 -63 -49 -59 -81 17 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~81/56 = 641.4794

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~81/56 = 641.4794{{c}}

Badness: 0.021984

Badness (Smith): 0.021984

== Eagle ==

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: mapping generators: ~177147/125440, ~28/27

[[Optimal tuning]] ([[POTE]]): ~177147/125440 = ~~1\2~~, ~28/27 = 62.229

[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 600.000{{c}}, ~28/27 = 62.229{{c}}

{{Optimal ET sequence|legend=1| 58, 154c, 212, 270, 752, 1022, 1292, 2854b }}

[[Badness]]: 0.059498

[[Badness]] (Smith): 0.059498

=== 11-limit ===

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

Optimal tuning (POTE): ~99/70 = ~~1\2~~, ~28/27 = 62.224

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~28/27 = 62.224{{c}}

Badness: 0.024885

Badness (Smith): 0.024885

=== 13-limit ===

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

Optimal tuning (POTE): ~99/70 = ~~1\2~~, ~28/27 = 62.220

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~28/27 = 62.220{{c}}

Badness: 0.016282

Badness (Smith): 0.016282

== Turkey ==

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[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~1715/1296 = 481.120

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1715/1296 = 481.120{{c}}

[[Badness]]: 0.210964

[[Badness]] (Smith): 0.210964

=== 11-limit ===

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Mapping: {{mapping| 1 8 36 0 64 | 0 -16 -84 7 -151 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~33/25 = 481.120

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~33/25 = 481.120{{c}}

Badness: 0.079694

Badness (Smith): 0.079694

=== 13-limit ===

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Mapping: {{mapping| 1 8 36 0 64 47 | 0 -16 -84 7 -151 -108 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~33/25 = 481.118

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~33/25 = 481.118{{c}}

Badness: 0.043787

Badness (Smith): 0.043787

[[Category:Temperament families]]

@@ Line 11: / Line 11: @@
 == 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. 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.
+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
@@ Line 22: / Line 22: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~320/243 = 475.5351
+* [[CTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5351{{c}}
 : [[error map]]: {{val| 0.0000 +0.1855 -0.0758 }}
-* [[POTE]]: ~2 = 1200.000, ~320/243 = 475.5426
+* [[POTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5426{{c}}
 : error map: {{val| 0.0000 +0.2154 +0.0811 }}
@@ Line 31: / Line 31: @@
 [[Badness]]:
 * Smith: 0.041431
-* Dirichlet: 0.972
+* Sintel: 0.972
 == Septimal vulture ==
@@ Line 45: / Line 45: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.0000, ~320/243 = 475.5528
+* [[CTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5528{{c}}
 : [[error map]]: {{val| 0.0000 +0.2561 +0.2945 +0.2188 }}
-* [[POTE]]: ~2 = 1200.0000, ~320/243 = 475.5511
+* [[POTE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5511{{c}}
 : error map: {{val| 0.0000 +0.2495 +0.2601 +0.3106 }}
@@ Line 62: / Line 62: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~320/243 = 475.5558
+* CTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5558{{c}}
-* POTE: ~2 = 1200.0000, ~320/243 = 475.5567
+* POTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5567{{c}}
 {{Optimal ET sequence|legend=0| 53, 217, 270, 2107c, 2377bc }}
@@ Line 77: / Line 77: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~320/243 = 475.5566
+* CTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5566{{c}}
-* POTE: ~2 = 1200.0000, ~320/243 = 475.5572
+* POTE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5572{{c}}
 {{Optimal ET sequence|legend=0| 53, 217, 270 }}
@@ Line 92: / Line 92: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000, ~25/19 = 475.5561
+* CTE: ~2 = 1200.0000{{c}}, ~25/19 = 475.5561{{c}}
-* CWE: ~2 = 1200.0000, , ~25/19 = 475.5569
+* CWE: ~2 = 1200.0000{{c}}, ~25/19 = 475.5569{{c}}
 {{Optimal ET sequence|legend=0| 53, 217, 270 }}
@@ Line 105: / Line 105: @@
 Mapping: {{mapping| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
 : mapping generators: ~99/70, ~320/243
 Optimal tunings:
-* CTE: ~99/70 = 600.0000, ~320/243 = 475.5523
+* CTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5523{{c}}
-* POTE: ~99/70 = 600.0000, ~320/243 = 475.5496
+* POTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5496{{c}}
 {{Optimal ET sequence|legend=0| 106, 164, 270, 916, 1186, 1456 }}
@@ Line 124: / Line 123: @@
 Optimal tunings:
-* CTE: ~99/70 = 600.0000, ~320/243 = 475.5540
+* CTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5540{{c}}
-* POTE: ~99/70 = 600.0000, ~320/243 = 475.553
+* POTE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.553{{c}}
 {{Optimal ET sequence|legend=0| 106, 164, 270 }}
@@ Line 138: / Line 137: @@
 {{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 = 1200.0000{{c}}, ~81/56 = 641.4791{{c}}
 {{Optimal ET sequence|legend=1| 58, 159, 217 }}
-[[Badness]]: 0.154715
+[[Badness]] (Smith): 0.154715
 === 11-limit ===
@@ Line 151: / Line 150: @@
 Mapping: {{mapping| 1 8 36 29 35 | 0 -12 -63 -49 -59 }}
-Optimal tuning (POTE): ~2 = 1\1, 81/56 = 641.4822
+Optimal tuning (POTE): ~2 = 1200.0000{{c}}, 81/56 = 641.4822{{c}}
-{{Optimal ET sequence|legend=1| 58, 101cd, 159, 217 }}
+{{Optimal ET sequence|legend=0| 58, 101cd, 159, 217 }}
-Badness: 0.048401
+Badness (Smith): 0.048401
 === 13-limit ===
@@ Line 164: / Line 163: @@
 Mapping: {{mapping| 1 8 36 29 35 47 | 0 -12 -63 -49 -59 -81 }}
-Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4797
+Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~81/56 = 641.4797{{c}}
-{{Optimal ET sequence|legend=1| 58, 159, 217 }}
+{{Optimal ET sequence|legend=0| 58, 159, 217 }}
-Badness: 0.025469
+Badness (Smith): 0.025469
 === 17-limit ===
@@ Line 177: / Line 176: @@
 Mapping: {{mapping| 1 8 36 29 35 47 -5 | 0 -12 -63 -49 -59 -81 17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4794
+Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~81/56 = 641.4794{{c}}
-{{Optimal ET sequence|legend=1| 58, 159, 217 }}
+{{Optimal ET sequence|legend=0| 58, 159, 217 }}
-Badness: 0.021984
+Badness (Smith): 0.021984
 == Eagle ==
@@ Line 192: / Line 191: @@
 : mapping generators: ~177147/125440, ~28/27
-[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229
+[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 600.000{{c}}, ~28/27 = 62.229{{c}}
 {{Optimal ET sequence|legend=1| 58, 154c, 212, 270, 752, 1022, 1292, 2854b }}
-[[Badness]]: 0.059498
+[[Badness]] (Smith): 0.059498
 === 11-limit ===
@@ Line 205: / Line 204: @@
 Mapping: {{mapping| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.224
+Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~28/27 = 62.224{{c}}
-{{Optimal ET sequence|legend=1| 58, 154ce, 212, 270 }}
+{{Optimal ET sequence|legend=0| 58, 154ce, 212, 270 }}
-Badness: 0.024885
+Badness (Smith): 0.024885
 === 13-limit ===
@@ Line 218: / Line 217: @@
 Mapping: {{mapping| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.220
+Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~28/27 = 62.220{{c}}
-{{Optimal ET sequence|legend=1| 58, 154cef, 212, 270 }}
+{{Optimal ET sequence|legend=0| 58, 154cef, 212, 270 }}
-Badness: 0.016282
+Badness (Smith): 0.016282
 == Turkey ==
@@ Line 231: / Line 230: @@
 {{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 = 1200.000{{c}}, ~1715/1296 = 481.120{{c}}
 {{Optimal ET sequence|legend=1| 5, 207c, 212, 429 }}
-[[Badness]]: 0.210964
+[[Badness]] (Smith): 0.210964
 === 11-limit ===
@@ Line 244: / Line 243: @@
 Mapping: {{mapping| 1 8 36 0 64 | 0 -16 -84 7 -151 }}
-Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.120
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~33/25 = 481.120{{c}}
-{{Optimal ET sequence|legend=1| 212, 429 }}
+{{Optimal ET sequence|legend=0| 212, 429 }}
-Badness: 0.079694
+Badness (Smith): 0.079694
 === 13-limit ===
@@ Line 257: / Line 256: @@
 Mapping: {{mapping| 1 8 36 0 64 47 | 0 -16 -84 7 -151 -108 }}
-Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.118
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~33/25 = 481.118{{c}}
-{{Optimal ET sequence|legend=1| 212, 217, 429 }}
+{{Optimal ET sequence|legend=0| 212, 217, 429 }}
-Badness: 0.043787
+Badness (Smith): 0.043787
 [[Category:Temperament families]]