Vulture family: Difference between revisions

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m →2.3.5.19: make respective & more proper and note the relevance of a couple of the more accurate tunings
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m →2.3.5.19: note how 2.3.5.19 vulture solves the problem of the interpretation of 3 gens in a more accurate way than buzzard
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=== 2.3.5.19 ===
=== 2.3.5.19 ===
It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])<sup>2</sup> = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. * Compared to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.
It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])<sup>2</sup> = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result, the 3 gen interval of ~226.6{{cent}} is interpreted as ([[3/2]])/([[25/19]]) = [[~]][[57/50]] which is tuned ~0.2{{cent}} flat. (Interpreting this interval as a damaged [[~]][[8/7]] leads to [[#Buzzard]].) Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. * Compared to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.


Subgroup: 2.3.5.19
Subgroup: 2.3.5.19

Revision as of 23:00, 1 February 2025

The vulture family of temperaments tempers out the vulture comma (monzo: [24 -21 4⟩, ratio: 10485760000/10460353203), 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.

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

Badness (Dirichlet): 0.972

2.3.5.19

It can be observed that the generator of vulture is very close to 25/19; this corresponds to tempering out 1216/1215 = (19/15)/(18/16)2 = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to octave equivalence) finds 19/16 at 41 generators so that 19/10 is found at 20 generators, 38/27 is found at 18, 19/15 is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with 27/16, which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result, the 3 gen interval of ~226.6 ¢ is interpreted as (3/2)/(25/19) = ~57/50 which is tuned ~0.2 ¢ flat. (Interpreting this interval as a damaged ~8/7 leads to #Buzzard.) Note that unless you are fine with the low accuracy* tuning offered by 53edo, you cannot temper out the schisma, nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and 513/512~361/360 (resp.) are observed. * Compared to what this microtemperament is capable of. This means that the step size of 270edo is especially ideal, being between 361/360 and 513/512, with 217edo exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = 111edo is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.

Subgroup: 2.3.5.19

Commas: 1216/1215, 64000000/63950067

Mapping: [⟨1 0 -6 -12], ⟨0 4 21 41]]

Optimal tuning (CTE): 2 = 1\1, ~25/19 = 475.542

error map: ⟨0.000 +0.214 +0.075 -0.278]

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

Badness (Dirichlet): 0.232

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 for 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.

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

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 936/935, 1001/1000, 1225/1224, 4096/4095

Mapping: [⟨1 0 -6 25 -33 -7 35], ⟨0 4 21 -56 92 27 -78]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~112/85 = 475.5613
  • POTE: ~2 = 1200.0000, ~112/85 = 475.5617

Optimal ET sequence: 53, 217, 270, 487, 757g

Badness (Smith): 0.020103

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 936/935, 1001/1000, 1216/1215, 1225/1224, 1540/1539

Mapping: [⟨1 0 -6 25 -33 -7 35 -12], ⟨0 4 21 -56 92 27 -78 41]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~25/19 = 475.5606
  • POTE: ~2 = 1200.0000, , ~25/19 = 475.5615

Optimal ET sequence: 53, 217, 270, 487, 757g

Badness (Smith): 0.013850

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 S-expression-based comma list is {S6/S7, S8/S9}, with the structure of its 7-limit implied by these 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

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