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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. | | 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. |
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| Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]]. Considered below are septimal vulture, buzzard, condor, eagle, and turkey. | | Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]] and [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, condor, eagle, and turkey. |
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| == Vulture == | | == Vulture == |
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| Badness (Smith): 0.035458 | | Badness (Smith): 0.035458 |
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| == Buzzard ==
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| {{Main| Buzzard }}
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| {{See also| No-fives subgroup temperaments #Buzzard }}
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| Buzzard is the main extension to vulture of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~[[21/16]], but is more of a full 13-limit system in its own right. It is most naturally described as 53 & 58 (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, [[111edo]] is a great tuning for it. [[mos scale]]s of 3, 5, 8, 13, 18, 23, 28, 33, 38, 43, 48 or 53 notes are available.
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| Its 13-limit [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[847/845|S11/S13]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. [[Hemifamity]] leverages it by splitting [[36/35]] into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]]. Then the vanish of the orwellisma implies [[49/48]], the large septimal diesis, is equated with 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.
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| [[Subgroup]]: 2.3.5.7
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| [[Comma list]]: 1728/1715, 5120/5103
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| {{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }}
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| {{Multival|legend=1| 4 21 -3 24 -16 -66 }}
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| [[Optimal tuning]]s:
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| * [[CTE]]: ~2 = 1200.000, ~21/16 = 475.555
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| : [[error map]]: {{val| 0.000 +0.263 +0.333 +4.510 }}
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| * [[POTE]]: ~2 = 1200.000, ~21/16 = 475.636
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| : error map: {{val| 0.000 +0.589 +2.045 +4.266 }}
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| {{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}
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| [[Badness]] (Smith): 0.047963
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| === 11-limit ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 176/175, 540/539, 5120/5103
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| Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }}
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| Wedgie: {{multival| 4 21 -3 39 24 -16 48 -66 18 120 }}
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| Optimal tunings:
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| * CTE: ~2 = 1200.000, ~21/16 = 475.625
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| * POTE: ~2 = 1200.000, ~21/16 = 475.700
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| {{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}
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| Badness (Smith): 0.034484
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 176/175, 351/350, 540/539, 676/675
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| Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}
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| Wedgie: {{multival| 4 21 -3 39 27 24 -16 48 28 -66 18 -15 120 87 -51 }}
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| Optimal tunings:
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| * CTE: ~2 = 1200.000, ~21/16 = 475.615
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| * POTE: ~2 = 1200.000, ~21/16 = 475.697
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| {{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}
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| Badness (Smith): 0.018842
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| ==== 17-limit ====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
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| Mapping: {{mapping| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}
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| Optimal tunings:
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| * CTE: ~2 = 1200.000, ~21/16 = 475.638
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| * POTE: ~2 = 1200.000, ~21/16 = 475.692
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| {{Optimal ET sequence|legend=0| 53, 58, 111 }}
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| Badness (Smith): 0.018403
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| ==== 19-limit ====
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| Subgroup: 2.3.5.7.11.13.17.19
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| Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
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| Mapping: {{mapping| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}
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| Optimal tunings:
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| * CTE: ~2 = 1200.000, ~21/16 = 475.617
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| * POTE: ~2 = 1200.000, ~21/16 = 475.679
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| {{Optimal ET sequence|legend=0| 53, 58h, 111 }}
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| Badness (Smith): 0.015649
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| === Buteo ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 99/98, 385/384, 2200/2187
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| Mapping: {{mapping| 1 0 -6 4 9 | 0 4 21 -3 -14 }}
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| Optimal tunings:
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| * CTE: ~2 = 1200.000, ~21/16 = 475.454
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| * POTE: ~2 = 1200.000, ~21/16 = 475.436
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| {{Optimal ET sequence|legend=0| 5, 48, 53 }}
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| Badness (Smith): 0.060238
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 99/98, 275/273, 385/384, 572/567
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| Mapping: {{mapping| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}
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| Optimal tunings:
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| * CTE: ~2 = 1200.000, ~21/16 = 475.495
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| * POTE: ~2 = 1200.000, ~21/16 = 475.464
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| {{Optimal ET sequence|legend=0| 5, 48f, 53 }}
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| Badness (Smith): 0.039854
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| == Condor == | | == Condor == |