Vulture family

From Xenharmonic Wiki
Revision as of 13:41, 13 April 2025 by FloraC (talk | contribs) (Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}")
Jump to navigation Jump to search
This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The vulture family of temperaments tempers out the vulture comma (monzo: [24 -21 4⟩, ratio: 10 485 760 000 / 10 460 353 203), a small 5-limit comma of 4.2 cents.

Temperaments discussed elsewhere include terture. Considered below are septimal vulture, buzzard, condor, eagle, and turkey.

Vulture

The generator of the vulture temperament is a grave fourth of 320/243, that is, a perfect fourth minus a syntonic comma. Four of these make a perfect twelfth. Its ploidacot is alpha-tetracot. It is a member of the syntonic–diatonic equivalence continuum with n = 4, so it equates a Pythagorean limma with a stack of four syntonic commas. It is also in the schismic–Mercator equivalence continuum with n = 4, so unless 53edo is used as a tuning, the schisma is always observed.

Subgroup: 2.3.5

Comma list: 10485760000/10460353203

Mapping: [⟨1 0 -6], ⟨0 4 21]]

mapping generators: ~2, ~320/243

Optimal tunings:

  • CTE: ~2 = 1200.000, ~320/243 = 475.5351
error map: ⟨0.0000 +0.1855 -0.0758]
  • POTE: ~2 = 1200.000, ~320/243 = 475.5426
error map: ⟨0.0000 +0.2154 +0.0811]

Optimal ET sequence: 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b

Badness:

  • Smith: 0.041431
  • Dirichlet: 0.972

Septimal vulture

Septimal vulture can be described as the 53 & 270 microtemperament, tempering out the ragisma, 4375/4374 and the garischisma, 33554432/33480783 ([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 (-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 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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33554432/33480783

Mapping: [⟨1 0 -6 25], ⟨0 4 21 -56]]

Wedgie: ⟨⟨ 4 21 -56 24 -100 -189 ]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5528
error map: ⟨0.0000 +0.2561 +0.2945 +0.2188]
  • POTE: ~2 = 1200.0000, ~320/243 = 475.5511
error map: ⟨0.0000 +0.2495 +0.2601 +0.3106]

Optimal ET sequence: 53, 164, 217, 270, 593, 863, 1133

Badness (Smith): 0.036985

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -6 25 -33], ⟨0 4 21 -56 92]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5558
  • POTE: ~2 = 1200.0000, ~320/243 = 475.5567

Optimal ET sequence: 53, 217, 270, 2107c, 2377bc

Badness (Smith): 0.031907

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -6 25 -33 -7], ⟨0 4 21 -56 92 27]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5566
  • POTE: ~2 = 1200.0000, ~320/243 = 475.5572

Optimal ET sequence: 53, 217, 270

Badness (Smith): 0.018758

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 -6 25 -33 -7 -12], ⟨0 4 21 -56 92 27 41]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~25/19 = 475.5561
  • CWE: ~2 = 1200.0000, , ~25/19 = 475.5569

Optimal ET sequence: 53, 217, 270

Badness (Smith): 0.00704

Semivulture

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 33554432/33480783

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

Optimal ET sequence: 106, 164, 270, 916, 1186, 1456

Badness (Smith): 0.040799

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨2 0 -12 50 41 -14], ⟨0 4 21 -56 -43 27]]

Optimal tunings:

  • CTE: ~99/70 = 600.0000, ~320/243 = 475.5540
  • POTE: ~99/70 = 600.0000, ~320/243 = 475.553

Optimal ET sequence: 106, 164, 270

Badness (Smith): 0.035458

Buzzard

Main article: 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 scales 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 {S6/S7, S8/S9, S11/S13, S13/S15}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial JI equivalence S6 = S8 × 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: [⟨1 0 -6 4], ⟨0 4 21 -3]]

Wedgie: ⟨⟨ 4 21 -3 24 -16 -66 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/16 = 475.555
error map: ⟨0.000 +0.263 +0.333 +4.510]
  • POTE: ~2 = 1200.000, ~21/16 = 475.636
error map: ⟨0.000 +0.589 +2.045 +4.266]

Optimal ET sequence: 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: [⟨1 0 -6 4 -12], ⟨0 4 21 -3 39]]

Wedgie: ⟨⟨ 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: 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: [⟨1 0 -6 4 -12 -7], ⟨0 4 21 -3 39 27]]

Wedgie: ⟨⟨ 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: 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: [⟨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: 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: [⟨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: 53, 58h, 111

Badness (Smith): 0.015649

Buteo

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 2200/2187

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: 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: [⟨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: 5, 48f, 53

Badness (Smith): 0.039854

Condor

Subgroup: 2.3.5.7

Comma list: 10976/10935, 40353607/40000000

Mapping: [⟨1 8 36 29], ⟨0 -12 -63 -49]]

Wedgie: ⟨⟨ 12 63 49 72 44 -63 ]]

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

Optimal ET sequence: 58, 159, 217

Badness: 0.154715

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 8 36 29 35], ⟨0 -12 -63 -49 -59]]

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

Optimal ET sequence: 58, 101cd, 159, 217

Badness: 0.048401

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 8 36 29 35 47], ⟨0 -12 -63 -49 -59 -81]]

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

Optimal ET sequence: 58, 159, 217

Badness: 0.025469

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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 ET sequence: 58, 159, 217

Badness: 0.021984

Eagle

Subgroup: 2.3.5.7

Comma list: 2401/2400, 10485760000/10460353203

Mapping: [⟨2 4 9 8], ⟨0 -8 -42 -23]]

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

Wedgie: ⟨⟨ 16 84 46 96 28 -129 ]]

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

Optimal ET sequence: 58, 154c, 212, 270, 752, 1022, 1292, 2854b

Badness: 0.059498

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 4 9 8 12], ⟨0 -8 -42 -23 -49]]

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

Optimal ET sequence: 58, 154ce, 212, 270

Badness: 0.024885

13-limit

Subgroup: 2.3.5.7.11.13

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

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 ET sequence: 58, 154cef, 212, 270

Badness: 0.016282

Turkey

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 5250987/5242880

Mapping: [⟨1 8 36 0], ⟨0 -16 -84 7]]

Wedgie: ⟨⟨ 16 84 -7 96 -56 -252 ]]

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

Optimal ET sequence: 5, 207c, 212, 429

Badness: 0.210964

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 8 36 0 64], ⟨0 -16 -84 7 -151]]

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

Optimal ET sequence: 212, 429

Badness: 0.079694

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 8 36 0 64 47], ⟨0 -16 -84 7 -151 -108]]

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

Optimal ET sequence: 212, 217, 429

Badness: 0.043787

Retrieved from "https://en.xen.wiki/index.php?title=Vulture_family&oldid=191749"