The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = [5 -9 4⟩, the minimal diesis or tetracot comma. The dual of this comma is the wedgie ⟨⟨ 4 9 5 ]], which tells us 10/9 is a generator, and that four of them give 3/2. In fact, (10/9)⁴ = 20000/19683 × 3/2. We also have (10/9)⁹ = (20000/19683)² × 5/2. From this it is evident we should flatten the generator a bit, and 34edo does this and makes for a recommendable tuning. Another possibility is to use (5/2)^(1/9) for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.

The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).

Tetracot

Subgroup: 2.3.5

Comma list: 20000/19683

POTE generator: ~10/9 = 176.160

Mapping: [⟨1 1 1], ⟨0 4 9]]

Template:Val list

Extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

875/864, the keema, gives monkey;
179200/177147 (or equivalently 225/224) gives bunya;
245/243 gives octacot, which splits the generator in half.

Monkey and bunya

Monkey tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the 7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34&41 temperament, if the vals in question are taken to be patent vals, meaning that n×log₂(prime) rounded to the nearest integer gives the mapping. 41edo is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.

Bunya adds 225/224 to the list of commas and may be described as the 41&75 temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 14^(1/26) as a generator, giving just 7s and an improved value for 5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.

Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives 11-limit monkey, ⟨⟨ 4 9 -15 10 … ]] and 11-limit banya, ⟨⟨ 4 9 26 10 … ]]. Again, 41edo can be used as a tuning, making the two identical, which is also the case if we turn to the 2.3.5.11 temperament, dispensing with 7. However 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 14^(1/26) generator supplies, or even sharper yet, as for instance by the val ⟨355 563 823 997 1230], with a 52/355 generator.

Since 16/13 is shy of (10/9)² by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us ⟨⟨ 4 9 -15 10 -2 … ]] for 13-limit monkey and ⟨⟨ 4 9 26 10 -2 … ]] for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp.

Monkey

Subgroup: 2.3.5.7

Comma list: 875/864, 5120/5103

Mapping: [⟨1 1 1 5], ⟨0 4 9 -15]]

Wedgie: ⟨⟨ 4 9 -15 5 -35 -60 ]]

POTE generator: ~10/9 = 175.659

Template:Val list

Badness: 0.0734

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 243/242, 385/384

Mapping: [⟨1 1 1 5 2], ⟨0 4 9 -15 10]]

POTE generator: ~10/9 = 175.570

Tetracot family

Tetracot

Extensions

Monkey and bunya

Monkey

11-limit

13-limit

Bunya

11-limit

13-limit

Modus

11-limit

13-limit

Musical examples

Ponens

13-limit

Wollemia

11-limit

13-limit

Octacot

11-limit

13-limit

Octocat

Octopod

Dificot

Dodecacot

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