Buzzardsmic clan

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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 2.3.7-subgroup comma for the buzzardsmic clan is the buzzardsma, 65536/64827, with monzo [16 -3 0 -4⟩, which implies that the tritave, 3/1, is divided into four intervals each representing a 21/16 subfourth. Tempering out this comma implies a sharpened 7th harmonic, and especially a sharpened ~21/16 generator, which approaches the 480 ¢ fourth of 5edo.

Extensions of buzzard to incorporate prime 5 along its chain of generators (and therefore the full 7-limit) include septimal buzzard (53 & 58), which tempers out 1728/1715 (and 5120/5103); subfourth (58 & 63), which tempers out 10976/10935; and lemongrass (63 & 68), which tempers out 245/243. All are considered below.

Weak extensions include demibuzzard (10 & 53), which tempers out 225/224 and splits 32/21 (the superfifth) in two; thuja (15 & 43), which tempers out 126/125 and splits 21/8 into three; subsedia (10 & 111), which tempers out 16875/16807 and splits 21/16 in four; and anthoine (25 & 53), which tempers out 3125/3087 and splits 21/2 in five.

Full 7-limit temperaments discussed elsewhere are:

The rest are considered below.

2.3.7 subgroup

Buzzard

Main article: Buzzard

Subgroup: 2.3.7

Comma list: 65536/64827

Mapping: [⟨1 0 4], ⟨0 4 -3]]

Optimal tunings:

  • WE: ~2 = 1199.2548 ¢, ~21/16 = 475.5761 ¢
error map: ⟨-0.745 +0.350 +1.465]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.8328 ¢
error map: ⟨0.000 +1.376 +3.676]

Optimal ET sequence: 5, 33, 38, 43, 48, 53, 58

Badness (Sintel): 0.824

Strong extensions

Septimal buzzard

Main article: Buzzard
See also: Vulture family

Septimal buzzard is not only a naturally motivated extension to 2.3.7 buzzard, but 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, though buzzard is powerful as 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 (111 = 53 + 58) is a great tuning for it. Mos scales of 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 vanishing 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]]

Optimal tunings:

  • WE: ~2 = 1199.3061 ¢, ~21/16 = 475.3611 ¢
error map: ⟨-0.694 -0.511 +0.432 +2.315]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6144 ¢
error map: ⟨0.000 +0.503 +1.589 +4.331]

Optimal ET sequence: 5, 48, 53, 111, 164d, 275d

Badness (Sintel): 1.21

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]]

Optimal tunings:

  • WE: ~2 = 1199.2516 ¢, ~21/16 = 475.4037 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6806 ¢

Optimal ET sequence: 53, 58, 111, 280cd

Badness (Sintel): 1.14

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]]

Optimal tunings:

  • WE: ~2 = 1199.2391 ¢, ~21/16 = 475.3956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6760 ¢

Optimal ET sequence: 53, 58, 111, 280cdf

Badness (Sintel): 0.779

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:

  • WE: ~2 = 1199.2723 ¢, ~21/16 = 475.4039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6837 ¢

Optimal ET sequence: 53, 58, 111

Badness (Sintel): 0.938

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:

  • WE: ~2 = 1199.2457 ¢, ~21/16 = 475.3797 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6690 ¢

Optimal ET sequence: 53, 58h, 111

Badness (Sintel): 0.952

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:

  • WE: ~2 = 1200.2867 ¢, ~21/16 = 475.5498 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.4393 ¢

Optimal ET sequence: 5, 48, 53

Badness (Sintel): 1.99

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:

  • WE: ~2 = 1200.3416 ¢, ~21/16 = 475.5998 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.4696 ¢

Optimal ET sequence: 5, 48f, 53

Badness (Sintel): 1.65

Subfourth

Subfourth tempers out 10976/10935 and may be described as the 58 & 63 temperament, more notable in the higher limits than the lower as it supplies a lot of essentially tempered chords there, including everything from parapyth. Among the good tunings are 121edo and 179edo using the 179ef val in the 13-limit.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65536/64827

Mapping: [⟨1 0 17 4], ⟨0 4 -37 -3]]

Optimal tunings:

  • WE: ~2 = 1199.1804 ¢, ~21/16 = 475.6659 ¢
error map: ⟨-0.820 +0.709 +0.113 +0.898]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.0019 ¢
error map: ⟨0.000 +2.052 +1.617 +3.168]

Optimal ET sequence: 58, 121, 179, 300bd, 479bcdd

Badness (Sintel): 3.56

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 12005/11979

Mapping: [⟨1 0 17 4 11], ⟨0 4 -37 -3 -19]]

Optimal tunings:

  • WE: ~2 = 1199.0801 ¢, ~21/16 = 475.6303 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.0088 ¢

Optimal ET sequence: 58, 121, 179e, 300bdee, 479bcddeee

Badness (Sintel): 1.50

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: [⟨1 0 17 4 11 16], ⟨0 4 -37 -3 -19 -31]]

Optimal tunings:

  • WE: ~2 = 1199.0747 ¢, ~21/16 = 475.6291 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.0113 ¢

Optimal ET sequence: 58, 121, 179ef, 300bdeef

Badness (Sintel): 0.983

Lemongrass

Named by Lériendil in 2025, lemongrass tempers out 245/243 and may be described as the 63 & 68 temperament. Characterized by a sharper generator than septimal buzzard, lemongrass compresses the septimal comma so much that the syntonic comma is no longer equated with it but with twice of it, or the large septimal diesis. 68edo itself is a great tuning for this, though 63edo and 73edo are also possible.

Subgroup: 2.3.5.7

Comma list: 245/243, 65536/64827

Mapping: [⟨1 0 17 4], ⟨0 4 26 -3]]

Optimal tunings:

  • WE: ~2 = 1199.0957 ¢, ~21/16 = 476.0857 ¢
error map: ⟨-0.904 +2.388 -0.851 -0.700]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.4221 ¢
error map: ⟨0.000 +3.733 +0.660 +1.908]

Optimal ET sequence: 5, …, 63, 68

Badness (Sintel): 2.90

Weak extensions

Demibuzzard

For the 5-limit version, see Schismic–Mercator equivalence continuum #Demibuzzard.

Demibuzzard may be described as the 10 & 53 temperament. It is generated by a submajor third; note that in the data below, the generator is the octave complement, a supraminor sixth, since two of it minus an octave make buzzard's generator of ~21/16. The ploidacot for this temperament is epsilon-octacot.

This temperament naturally comes about from a structure in edos like 43-, 53-, and 63edo where two flattened ~13/8 intervals reach the buzzard generator of ~21/16, two of which produce a semitritave that can here be equated to 26/15 – providing a mapping of 5 significantly less complex than the vulture mapping – and two of those finally reach 3/1.

It diverges into two extensions for prime 11: submajor (53 & 63) favoring sharp fifths, and interpental (43 & 53), favoring flat fifths; the two mappings meet at 53edo. Note that submajor (referring to the submajor third, not the supraminor sixth) used to be the name for the 7-limit temperament.

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 51200/50421

Mapping: [⟨1 -4 10 7], ⟨0 8 -11 -6]]

mapping generators: ~2, ~80/49

Optimal tunings:

  • WE: ~2 = 1199.7399 ¢, ~80/49 = 837.5637 ¢
error map: ⟨-0.260 -0.405 -2.116 +3.971]
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 837.7471 ¢
error map: ⟨0.000 +0.022 -1.532 +4.691]

Optimal ET sequence: 10, 33, 43, 53

Badness (Sintel): 1.53

2.3.5.7.13 subgroup

See also: Greater tendoneutralic

Subgroup: 2.3.5.7.13

Comma list: 169/168, 225/224, 640/637

Mapping: [⟨1 -4 10 7 3], ⟨0 8 -11 -6 1]]

Optimal tunings:

  • WE: ~2 = 1199.9444 ¢, ~13/8 = 837.7178 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 837.7569 ¢

Optimal ET sequence: 10, 33, 43, 53

Badness (Sintel): 0.847

Submajor

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 6655/6561

Mapping: [⟨1 -4 10 7 -14], ⟨0 8 -11 -6 25]]

Optimal tunings:

  • WE: ~2 = 1200.0666 ¢, ~44/27 = 837.9460 ¢
  • CWE: ~2 = 1200.0000 ¢, ~44/27 = 837.9000 ¢

Optimal ET sequence: 10, 43e, 53, 116

Badness (Sintel): 1.67

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 275/273, 385/384

Mapping: [⟨1 -4 10 7 -14 3], ⟨0 8 -11 -6 25 1]]

Optimal tunings:

  • WE: ~2 = 1200.1769 ¢, ~13/8 = 838.0187 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 837.8965 ¢

Optimal ET sequence: 10, 43e, 53, 116

Badness (Sintel): 1.14

Interpental

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 51200/50421

Mapping: [⟨1 -4 10 7 23], ⟨0 8 -11 -6 -28]]

Optimal tunings:

  • WE: ~2 = 1199.9381 ¢, ~80/49 = 838.5389 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 837.5832 ¢

Optimal ET sequence: 43, 53, 96

Badness (Sintel): 1.71

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 640/637

Mapping: [⟨1 -4 10 7 23 3], ⟨0 8 -11 -6 -28 1]]

Optimal tunings:

  • WE: ~2 = 1200.1048 ¢, ~13/8 = 837.6710 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 837.5964 ¢

Optimal ET sequence: 43, 53, 96

Badness (Sintel): 1.23

Thuja

For the 5-limit version, see Miscellaneous 5-limit temperaments #Thuja.

Thuja tempers out 126/125 and may be described as the 15 & 43 temperament. The generator is a somewhat sharp fourth, which may be taken as a ~11/8 in the 11-limit, and three minus an octave make buzzard's generator of ~21/16. The ploidacot for this temperament is epsilon-dodecacot.

Thuja can be extended up to the 29-limit, with a simple and accurate approximation to 29, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.

Subgroup: 2.3.5.7

Comma list: 126/125, 65536/64827

Mapping: [⟨1 -4 0 7], ⟨0 12 5 -9]]

mapping generators: ~2, ~175/128

Optimal tunings:

  • WE: ~2 = 1198.7356 ¢, ~175/128 = 558.0168 ¢
error map: ⟨-1.264 -0.696 +3.770 +0.172]
  • CWE: ~2 = 1200.0000 ¢, ~175/128 = 558.5795 ¢
error map: ⟨0.000 +0.999 +6.584 +3.959]

Optimal ET sequence: 15, 43, 58

Badness (Sintel): 2.24

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 1344/1331

Mapping: [⟨1 -4 0 7 3], ⟨0 12 5 -9 1]]

Optimal tunings:

  • WE: ~2 = 1198.5470 ¢, ~11/8 = 557.9433 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 558.5942 ¢

Optimal ET sequence: 15, 43, 58

Badness (Sintel): 1.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 364/363

Mapping: [⟨1 -4 0 7 3 -7], ⟨0 12 5 -9 1 23]]

Optimal tunings:

  • WE: ~2 = 1198.5083 ¢, ~11/8 = 557.8942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 558.5565 ¢

Optimal ET sequence: 15, 43, 58

Badness (Sintel): 0.944

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 176/175, 221/220, 256/255

Mapping: [⟨1 -4 0 7 3 -7 12], ⟨0 12 5 -9 1 23 -17]]

Optimal tunings:

  • WE: ~2 = 1198.8533 ¢, ~11/8 = 557.9750 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 558.4979 ¢

Optimal ET sequence: 15, 43, 58, 101e, 159cdef

Badness (Sintel): 1.14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220

Mapping: [⟨1 -4 0 7 3 -7 12 1], ⟨0 12 5 -9 1 23 -17 7]]

Optimal tunings:

  • WE: ~2 = 1198.6460 ¢, ~11/8 = 557.8736 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 558.4905 ¢

Optimal ET sequence: 15, 43, 58h, 101eh

Badness (Sintel): 1.15

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: [⟨1 -4 0 7 3 -7 12 1 5], ⟨0 12 5 -9 1 23 -17 7 -1]]

Optimal tunings:

  • WE: ~2 = 1198.4488 ¢, ~11/8 = 557.7999 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 558.5086 ¢

Optimal ET sequence: 15, 43, 58hi

Badness (Sintel): 1.19

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: [⟨1 -4 0 7 3 -7 12 1 5 3], ⟨0 12 5 -9 1 23 -17 7 -1 4]]

Optimal tunings:

  • WE: ~2 = 1198.5114 ¢, ~11/8 = 557.8276 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 558.5079 ¢

Optimal ET sequence: 15, 43, 58hi

Badness (Sintel): 1.15

Antemka

This temperament has the opposite mappings of 5 and 13 to emka.

Comma list: 105/104, 126/125, 66/65, 1024/1001

Edo join: 15 & 28

Mapping: [⟨1 8 5 -2 4 8], ⟨0 -12 -5 9 -1 -8]]

Optimal tuning (CWE): ~2 = 1200.0000 ¢, ~16/11 = 641.492 ¢

Subsedia

Named by Xenllium in 2022, subsedia tempers out the mirkwai comma and may be described as the 111 & 121 temperament. The generator for subsedia is 0.5 cents flat of 15/14-wide semitone. In this temperament, three generators make ~16/13, five make ~24/17, twelve make ~16/7, sixteen make ~3/1, and 45 make ~22/1.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 65536/64827

Mapping: [⟨1 0 5 4], ⟨0 16 -27 -12]]

mapping generators: ~2, ~15/14

Optimal tunings:

  • WE: ~2 = 1199.2693 ¢, ~15/14 = 118.8923 ¢
error map: ⟨-0.731 +0.322 -0.060 +1.543]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 118.9682 ¢
error map: ⟨0.000 +1.536 +1.545 +3.556]

Optimal ET sequence: 10, 91cd, 101, 111, 121, 232d

Badness (Sintel): 3.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 65536/64827

Mapping: [⟨1 0 5 4 -1], ⟨0 16 -27 -12 45]]

Optimal tunings:

  • WE: ~2 = 1199.2891 ¢, ~15/14 = 118.8978 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 118.9662 ¢

Optimal ET sequence: 10, 101, 111, 121, 232d

Badness (Sintel): 2.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 676/675, 1375/1372

Mapping: [⟨1 0 5 4 -1 4], ⟨0 16 -27 -12 45 -3]]

Optimal tunings:

  • WE: ~2 = 1199.2920 ¢, ~15/14 = 118.8980 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 118.9666 ¢

Optimal ET sequence: 10, 101, 111, 121, 232d

Badness (Sintel): 1.31

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 442/441, 540/539, 715/714

Mapping: [⟨1 0 5 4 -1 4 3], ⟨0 16 -27 -12 45 -3 11]]

Optimal tunings:

  • WE: ~2 = 1199.2648 ¢, ~15/14 = 118.8946 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 118.9655 ¢

Optimal ET sequence: 10, 101, 111, 121, 232dg

Badness (Sintel): 1.00

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714

Mapping: [⟨1 0 5 4 -1 4 3 10], ⟨0 16 -27 -12 45 -3 11 -58]]

Optimal tunings:

  • WE: ~2 = 1199.2847 ¢, ~15/14 = 118.8929 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 118.9644 ¢

Optimal ET sequence: 10, 111, 121, 232dg

Badness (Sintel): 1.09

Anthoine

For the 5-limit version, see Miscellaneous 5-limit temperaments #Anthoine.

Named by Lériendil in 2025, anthoine is generated by 5/4 and tempers out 3125/3087 in addition to the buzzardsma; note that the data below shows the octave complement generator, ~8/5, so that buzzard's generator is found at 5 generators up. It is most notable as the 25 & 28 temperament and as the chain of 5/4's present in 53edo. Its ploidacot is 13-sheared-20-cot.

Subgroup: 2.3.5.7

Comma list: 3125/3087, 65536/64827

Mapping: [⟨1 -12 3 13], ⟨0 20 -1 -15]]

mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1199.6282 ¢, ~8/5 = 814.9050 ¢
error map: ⟨-0.372 +0.605 -2.334 +2.767]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 815.1546 ¢
error map: ⟨0.000 +1.138 -1.468 +3.854]

Optimal ET sequence: 25, 53, 184, 237d

Badness (Sintel): 4.57

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