Xenharmonic Wiki

Vulture family

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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 that is the amount by which a stack of four syntonic commas falls short of the 256/243 Pythagorean limma. As their defining feature, vulture temperaments split the interval 3/1 into four segments (identified in the 5-limit as 320/243).

Temperaments discussed elsewhere include terture and buzzard. Considered below are septimal vulture, 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 = 1200.0000 ¢, ~320/243 = 475.5351 ¢
error map: ⟨0.0000 +0.1855 -0.0758]
  • POTE: ~2 = 1200.0000 ¢, ~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
  • Sintel: 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]]

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

Condor

Subgroup: 2.3.5.7

Comma list: 10976/10935, 40353607/40000000

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

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~81/56 = 641.4791 ¢

Optimal ET sequence: 58, 159, 217

Badness (Smith): 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 = 1200.0000 ¢, 81/56 = 641.4822 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~81/56 = 641.4797 ¢

Optimal ET sequence: 58, 159, 217

Badness (Smith): 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 = 1200.0000 ¢, ~81/56 = 641.4794 ¢

Optimal ET sequence: 58, 159, 217

Badness (Smith): 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

Optimal tuning (POTE): ~177147/125440 = 600.000 ¢, ~28/27 = 62.229 ¢

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

Badness (Smith): 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 = 600.000 ¢, ~28/27 = 62.224 ¢

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

Badness (Smith): 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 = 600.000 ¢, ~28/27 = 62.220 ¢

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

Badness (Smith): 0.016282

Turkey

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 5250987/5242880

Mapping: [⟨1 8 36 0], ⟨0 -16 -84 7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~1715/1296 = 481.120 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~33/25 = 481.120 ¢

Optimal ET sequence: 212, 429

Badness (Smith): 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 = 1200.000 ¢, ~33/25 = 481.118 ¢

Optimal ET sequence: 212, 217, 429

Badness (Smith): 0.043787

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