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m →Septimal vulture: mos scales listed for a septimal temperament should target all primes in the 7-limit, no? otherwise you may as well use buzzard, which should be noted due to how easy it is to mistakenly interpret the harmony in terms of the buzzard mapping
 
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The '''vulture family''' of [[temperament]]s [[tempering out|tempers out]] the [[vulture comma]] ({{monzo|legend=1| 24 -21 4 }}, [[ratio]]: 10485760000/10460353203), a small [[5-limit]] comma of 4.2 [[cent]]s.  
{{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 ==
== Vulture ==
The generator of the vulture temperament is a grave fourth of [[320/243]], that is, a [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Four of these make a [[3/1|perfect twelfth]]. Its [[ploidacot]] is alpha-tetracot.  
The generator of the vulture temperament is a grave fourth of [[320/243]], that is, a [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Four of these make a [[3/1|perfect twelfth]]. Its [[ploidacot]] is alpha-tetracot. It is a member of the [[syntonic–diatonic equivalence continuum]] with {{nowrap|''n'' {{=}} 4}}, so it equates a [[256/243|Pythagorean limma]] with a stack of four syntonic commas. It is also in the [[schismic–Mercator equivalence continuum]] with {{nowrap|''n'' {{=}} 4}}, so unless [[53edo]] is used as a tuning, the [[schisma]] is always observed.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 28: Line 29:
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}


[[Badness]] (Smith): 0.041431
[[Badness]]:
* Smith: 0.041431
* Dirichlet: 0.972


== Septimal vulture ==
== Septimal vulture ==
Septimal vulture can be described as the {{nowrap| 53 & 270 }} microtemperament, tempering out the [[ragisma]], 4375/4374 and the [[garischisma]], 33554432/33480783 ({{monzo| 25 -14 0 -1 }}) aside from the vulture comma. [[270edo]] is a good tuning for this temperament, with generator 107\270. Due to being a microtemperament, to find the mapping of 7, you need 56 generators, so that the smallest mos scale that finds 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 of 7 mapping at the cost of higher error, you could try [[#Buzzard]].)
Septimal vulture can be described as the {{nowrap| 53 & 270 }} microtemperament, tempering out the [[ragisma]], 4375/4374 and the [[garischisma]], 33554432/33480783 ({{monzo| 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 {{nowrap| (-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|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
[[Subgroup]]: 2.3.5.7
Line 38: Line 43:


{{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }}
{{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }}
{{Multival|legend=1| 4 21 -56 24 -100 -189 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 81: Line 84:
Badness (Smith): 0.018758
Badness (Smith): 0.018758


==== 17-limit ====
==== 2.3.5.7.11.13.19 subgroup ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 676/675, 936/935, 1001/1000, 1225/1224, 4096/4095
 
Mapping: {{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|legend=0| 53, 217, 270, 487, 757g }}
 
Badness (Smith): 0.020103
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 676/675, 936/935, 1001/1000, 1216/1215, 1225/1224, 1540/1539
Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728


Mapping: {{mapping| 1 0 -6 25 -33 -7 35 -12 | 0 4 21 -56 92 27 -78 41 }}
Mapping: {{mapping| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~25/19 = 475.5606
* CTE: ~2 = 1200.0000, ~25/19 = 475.5561
* POTE: ~2 = 1200.0000, , ~25/19 = 475.5615
* CWE: ~2 = 1200.0000, , ~25/19 = 475.5569


{{Optimal ET sequence|legend=0| 53, 217, 270, 487, 757g }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}


Badness (Smith): 0.013850
Badness (Smith): 0.00704


=== Semivulture ===
=== Semivulture ===
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Badness (Smith): 0.035458
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 [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]}, with the structure of its 7-limit implied by these 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 ==
== Condor ==
Line 269: Line 137:


{{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }}
{{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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791
Line 325: Line 191:


: mapping generators: ~177147/125440, ~28/27
: 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
[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229
Line 366: Line 230:


{{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }}
{{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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120
Line 402: Line 264:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Vulture family| ]] <!-- main article -->
[[Category:Vulture family| ]] <!-- main article -->
[[Category:Vulture| ]] <!-- key article -->
[[Category:Vulture| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:36, 24 June 2025

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 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 = 1200.000, ~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]]

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 = 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

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]]

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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