Vulture family: Difference between revisions

← Older edit

VisualWikitext

@@ Line 5: / Line 5: @@
 | ja =
 }}
-{{Technical data page}}<br><br>
+{{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, 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.
 == Vulture ==
@@ Line 43: / Line 43: @@
 {{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }}
-{{Multival|legend=1| 4 21 -56 24 -100 -189 }}
 [[Optimal tuning]]s:
@@ Line 132: / Line 130: @@
 Badness (Smith): 0.035458
-== Buzzard ==
-{{Main| Buzzard }}
-{{See also| No-fives subgroup temperaments #Buzzard }}
-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.
-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.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 1728/1715, 5120/5103
-{{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }}
-{{Multival|legend=1| 4 21 -3 24 -16 -66 }}
-[[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~21/16 = 475.555
-: [[error map]]: {{val| 0.000 +0.263 +0.333 +4.510 }}
-* [[POTE]]: ~2 = 1200.000, ~21/16 = 475.636
-: error map: {{val| 0.000 +0.589 +2.045 +4.266 }}
-{{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}
-[[Badness]] (Smith): 0.047963
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 176/175, 540/539, 5120/5103
-Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }}
-Wedgie: {{multival| 4 21 -3 39 24 -16 48 -66 18 120 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.625
-* POTE: ~2 = 1200.000, ~21/16 = 475.700
-{{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}
-Badness (Smith): 0.034484
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 176/175, 351/350, 540/539, 676/675
-Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}
-Wedgie: {{multival| 4 21 -3 39 27 24 -16 48 28 -66 18 -15 120 87 -51 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.615
-* POTE: ~2 = 1200.000, ~21/16 = 475.697
-{{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}
-Badness (Smith): 0.018842
-==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
-Mapping: {{mapping| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.638
-* POTE: ~2 = 1200.000, ~21/16 = 475.692
-{{Optimal ET sequence|legend=0| 53, 58, 111 }}
-Badness (Smith): 0.018403
-==== 19-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
-Mapping: {{mapping| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.617
-* POTE: ~2 = 1200.000, ~21/16 = 475.679
-{{Optimal ET sequence|legend=0| 53, 58h, 111 }}
-Badness (Smith): 0.015649
-=== Buteo ===
-Subgroup: 2.3.5.7.11
-Comma list: 99/98, 385/384, 2200/2187
-Mapping: {{mapping| 1 0 -6 4 9 | 0 4 21 -3 -14 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.454
-* POTE: ~2 = 1200.000, ~21/16 = 475.436
-{{Optimal ET sequence|legend=0| 5, 48, 53 }}
-Badness (Smith): 0.060238
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 99/98, 275/273, 385/384, 572/567
-Mapping: {{mapping| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.495
-* POTE: ~2 = 1200.000, ~21/16 = 475.464
-{{Optimal ET sequence|legend=0| 5, 48f, 53 }}
-Badness (Smith): 0.039854
 == Condor ==
@@ Line 259: / Line 137: @@
 {{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }}
-{{Multival|legend=1| 12 63 49 72 44 -63 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791
@@ Line 315: / Line 191: @@
 : mapping generators: ~177147/125440, ~28/27
-{{Multival|legend=1|16 84 46 96 28 -129}}
 [[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229
@@ Line 356: / Line 230: @@
 {{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }}
-{{Multival|legend=1|16 84 -7 96 -56 -252}}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120
@@ Line 392: / Line 264: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Vulture family| ]] <!-- main article -->
 [[Category:Vulture| ]] <!-- key article -->
 [[Category:Rank 2]]