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 & 68), 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:

• Blackwood (+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

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 ¢