Tetracot family
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]]
Vals: 7, 20c, 27, 34, 75, 109, 470b, 579b
Badness: 0.048518
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 bunya, ⟨⟨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 bunya 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
Badness: 0.073437
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
Badness: 0.038836
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 144/143, 243/242
Mapping: [⟨1 1 1 5 2 4], ⟨0 4 9 -15 10 -2]]
POTE generator: ~10/9 = 175.622
Badness: 0.028410
Bunya
Subgroup: 2.3.5.7
Comma list: 225/224, 15625/15309
Mapping: [⟨1 1 1 -1], ⟨0 4 9 26]]
Wedgie: ⟨⟨4 9 26 5 30 35]]
POTE generator: ~10/9 = 175.741
Vals: 34d, 41, 116, 157c, 198c
Badness: 0.062897
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 1344/1331
Mapping: [⟨1 1 1 -1 2], ⟨0 4 9 26 10]]
POTE generator: ~10/9 = 175.777
Badness: 0.031332
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 144/143, 225/224, 243/242
Mapping: [⟨1 1 1 -1 2 4], ⟨0 4 9 26 10 -2]]
POTE generator: ~10/9 = 175.886
Badness: 0.024886
Modus
Subgroup: 2.3.5.7
Comma list: 64/63, 4375/4374
Mapping: [⟨1 1 1 4], ⟨0 4 9 -8]]
POTE generator: ~10/9 = 177.203
Badness: 0.068184
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 243/242
Mapping: [⟨1 1 1 4 2], ⟨0 4 9 -8 10]]
POTE generator: ~10/9 = 177.053
Badness: 0.035149
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 100/99, 144/143
Mapping: [⟨1 1 1 4 2 4], ⟨0 4 9 -8 10 -2]]
POTE generator: ~10/9 = 176.953
Badness: 0.023806
Musical examples
- Tetracot Perc-Sitar by Dustin Schallert
- Tetracot Jam by Dustin Schallert
- Tetracot Pump by Dustin Schallert all in 27edo
Ponens
The error of 11 is about the same as that of Modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 363/350
Mapping: [⟨1 1 1 4 3], ⟨0 4 9 -8 3]]
POTE generator: ~10/9 = 177.200
Vals: 7, 20c, 27, 61dee, 88bcdee
Badness: 0.063077
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 64/63, 66/65, 143/140
Mapping: [⟨1 1 1 4 3 4], ⟨0 4 9 -8 3 -2]]
POTE generator: ~10/9 = 177.197
Vals: 7, 20c, 27, 61dee, 88bcdee
Badness: 0.039043
Wollemia
Subgroup: 2.3.5.7
Comma list: 126/125, 2240/2187
Mapping: [⟨1 1 1 0], ⟨0 4 9 19]]
Wedgie: ⟨⟨4 9 19 5 19 19]]
POTE generator: ~10/9 = 177.357
Badness: 0.070522
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 100/99, 243/242
Mapping: [⟨1 1 1 0 2], ⟨0 4 9 19 10]]
POTE generator: ~10/9 = 177.413
Badness: 0.037551
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 91/90, 100/99, 352/351
Mapping: [⟨1 1 1 0 2 4], ⟨0 4 9 19 10 -2]]
POTE generator: ~10/9 = 177.231
Badness: 0.031219
Octacot
- See also: Chords of octacot
Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It may also be described as 41&68. 68EDO or 109EDO can be used as tunings, as can (5/2)(1/18), which gives just major thirds. Another tuning is 150EDO, which has a generator, 11/150, of exactly 88 cents. This relates octacot to the 88cET non-octave temperament, which like Carlos Alpha arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them.
Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving ⟨⟨8 18 11 20 -4 …]] as the octave part of the wedgie. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.
Subgroup: 2.3.5.7
Comma list: 245/243, 2401/2400
Mapping: [⟨1 1 1 2], ⟨0 8 18 11]]
Wedgie: ⟨⟨8 18 11 10 -5 -25]]
POTE generator: ~21/20 = 88.076
Badness: 0.033845
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 243/242, 245/242
Mapping: [⟨1 1 1 2 2], ⟨0 8 18 11 20]]
POTE generator: ~21/20 = 87.975
Vals: 27e, 41, 109e, 150e, 191e
Badness: 0.024078
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 144/143, 196/195, 243/242
Mapping: [⟨1 1 1 2 2 4], ⟨0 8 18 11 20 -4]]
POTE generator: ~21/20 = 88.106
Badness: 0.023276
Octocat
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 91/90, 100/99, 245/242
Mapping: [⟨1 1 1 2 2 2], ⟨0 8 18 11 20 23]]
POTE generator: ~21/20 = 88.179
Badness: 0.027601
Octopod
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 243/242, 245/242
Mapping: [⟨1 1 1 2 2 1], ⟨0 8 18 11 20 37]]
POTE generator: ~21/20 = 87.697
Badness: 0.028326
Dificot
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 243/242, 245/242, 343/338
Mapping: [⟨1 9 19 13 22 19], ⟨0 -16 -36 -22 -40 -33]]
POTE generator: ~13/9 = 643.989
Badness: 0.051876
Dodecacot
Subgroup: 2.3.5.7
Comma list: 3125/3087, 10976/10935
Mapping: [⟨1 1 1 1], ⟨0 12 27 37]]
Wedgie: ⟨⟨12 27 37 15 25 10]]
POTE generator: ~28/27 = 58.675
Vals: 41, 143d, 184, 225, 409bcd
Badness: 0.119761