Buzzardsmic clan
- 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 & 5), which tempers out 245/243. All are considered below.
Weak extensions include submajor (10 & 43), which tempers out 225/224 and splits 32/21 (the superfifth) in two; and thuja (15 & 43), which tempers out 126/125 and splits 21/8 into three.
Full 7-limit temperaments discussed elsewhere are:
- Blacksmith (+28/27) → Limmic temperaments
- Quadrasruta (+2048/2025) → Diaschismic family
- Hemikleismic (+4000/3969) → Kleismic family
- Cohemimabila (+3136/3125) → Mabila family
The rest are considered below.
2.3.7 subgroup
Buzzard
Subgroup: 2.3.7
Comma list: 65536/64827
Mapping: [⟨1 0 4], ⟨0 4 -3]]
Optimal ET sequence: 5, 33, 38, 43, 48, 53, 58
- Smith: 0.0240
- Dirichlet: 0.824
Strong extensions
Septimal buzzard
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, 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 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]]
- 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]]
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]]
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
Subfourth
Subgroup: 2.3.5.7
Comma list: 10976/10935, 65536/64827
Mapping: [⟨1 0 17 4], ⟨0 4 -37 -3]]
- mapping generators: ~2, ~21/16
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.991
Optimal ET sequence: 58, 121, 179, 300bd, 479bcd
Badness: 0.140722
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 tuning (POTE): ~2 = 1\1, ~21/16 = 475.995
Optimal ET sequence: 58, 121, 179e, 300bde
Badness: 0.045323
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 tuning (POTE): ~2 = 1\1, ~21/16 = 475.996
Optimal ET sequence: 58, 121, 179ef, 300bdef
Badness: 0.023800
Lemongrass
Subgroup: 2.3.5.7
Comma list: 245/243, 65536/64827
Mapping: [⟨1 0 17 4], ⟨0 4 26 -3]]
- mapping generators: ~2, ~21/16
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 476.4448
Optimal ET sequence: 5, ..., 63, 68
Badness (Sintel): 2.902
Weak extensions
Submajor
7-limit
Subgroup: 2.3.5.7
Comma list: 225/224, 51200/50421
Mapping: [⟨1 4 -1 1], ⟨0 -8 11 6]]
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.255
Optimal ET sequence: 10, 33, 43, 53
Badness: 0.060533
2.3.5.7.13 subgroup
This temperament naturally comes about from a structure in EDOs like 43 and 53 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.
Subgroup: 2.3.5.7.13
Comma list: 169/168, 225/224, 640/637
Mapping: [⟨1 4 -1 1 4], ⟨0 -8 11 6 -1]]
Optimal tuning (CTE): ~2 = 1\1, ~16/13 = 362.242
Badness (Sintel): 0.847
11-limit
Submajor diverges into two extensions to prime 11: this one favoring sharp fifths, and interpental, favoring flat fifths; the two mappings meet at 53edo.
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 6655/6561
Mapping: [⟨1 4 -1 1 11], ⟨0 -8 11 6 -25]]
Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101
Optimal ET sequence: 10, 43e, 53, 116, 169de, 285cde
Badness: 0.050582
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 275/273, 385/384
Mapping: [⟨1 4 -1 1 11 4], ⟨0 -8 11 6 -25 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105
Optimal ET sequence: 10, 43e, 53, 116, 169de, 285cdef
Badness: 0.027689
Interpental
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 51200/50421
Mapping: [⟨1 4 -1 1 -5], ⟨0 -8 11 6 28]]
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418
Optimal ET sequence: 43, 53, 96, 149d
Badness: 0.051806
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 169/168, 176/175, 640/637
Mapping: [⟨1 4 -1 1 -5 4], ⟨0 -8 11 6 28 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402
Optimal ET sequence: 43, 53, 96, 149d
Badness: 0.029680
Thuja
- For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Thuja.
Subgroup: 2.3.5.7
Comma list: 126/125, 65536/64827
Mapping: [⟨1 -4 0 7], ⟨0 12 5 -9]]
Optimal tuning (POTE): ~2 = 1\1, ~175/128 = 558.605
Optimal ET sequence: 15, 43, 58
Badness: 0.088441
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 tuning (POTE): ~2 = 1\1, ~11/8 = 558.620
Optimal ET sequence: 15, 43, 58
Badness: 0.033078
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 tuning (POTE): ~2 = 1\1, ~11/8 = 558.589
Optimal ET sequence: 15, 43, 58
Badness: 0.022838
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 tuning (POTE): ~2 = 1\1, ~11/8 = 558.509
Optimal ET sequence: 15, 43, 58
Badness: 0.022293
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 tuning (POTE): ~2 = 1\1, ~11/8 = 558.504
Optimal ET sequence: 15, 43, 58h
Badness: 0.018938
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 tuning (POTE): ~2 = 1\1, ~11/8 = 558.522
Optimal ET sequence: 15, 43, 58hi
Badness: 0.016581
29-limit
The raison d'etre of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.
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 tuning (POTE): ~2 = 1\1, ~11/8 = 558.520
Optimal ET sequence: 15, 43, 58hi
Badness: 0.013762
Anthoine
Anthoine is generated by 5/4 and tempers out 3125/3087 in addition to the buzzardsma, so that 32/21 is found at 5 generators up. It is most notable as the 25 & 28 temperament and as the chain of 5/4s present in 53edo.
7-limit
Subgroup: 2.3.5.7
Comma list: 3125/3087, 65536/64827
Mapping: [⟨1 8 2 -2], ⟨0 -20 1 15]]
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 384.856
Optimal ET sequence: 25, 53, 184, 237d, 290d, 343dd
Badness (Sintel): 4.571