Tetracot family

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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

Main article: Tetracot

Subgroup: 2.3.5

Comma list: 20000/19683

Mapping: [1 1 1], 0 4 9]]

POTE generator: ~10/9 = 176.160

Minimax tuning:

  • 5-odd-limit: ~10/9 = [-1/9 0 1/9
Eigenmonzos (unchanged intervals): 2, 5

Optimal GPV sequence7, 20c, 27, 34, 75, 109, 470b, 579b

Badness: 0.048518

Scales: Tetracot7, Tetracot13, Tetracot20

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 141/26 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 141/26 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, 41edo 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.

Subgroup temperament

See also: No-sevens subgroup temperaments #Tetracot

The tetracot temperament works well for the 2.3.5.11.13 subgroup, in which tempering out 100/99, 144/143 and 243/242. In this temperament, 3/2 is divided into four equal parts, one of which represents both 10/9 and 11/10.

Subgroup: 2.3.5.11

Comma list: 100/99, 243/242

Gencom: [2 10/9; 100/99 243/242]

Sval mapping: [1 1 1 2], 0 4 9 10]]

POL2 generator: ~10/9 = 175.985

Optimal GPV sequence7, 27e, 34, 41, 75e

Scales: Tetracot7, Tetracot13, Tetracot20

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 100/99, 144/143, 243/242

Gencom: [2 10/9; 100/99 144/143 243/242]

Sval mapping: [1 1 1 2 4], 0 4 9 10 -2]]

POL2 generator: ~10/9 = 176.196

Optimal GPV sequence7, 27e, 34, 41, 75e

Scales: Tetracot7, Tetracot13, Tetracot20

Monkey

See also: Tetracot

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

Optimal GPV sequence7, 27d, 34, 41, 321ccdd

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

Optimal GPV sequence: 7, 27de, 34, 41

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

Optimal GPV sequence: 7, 27de, 34, 41

Badness: 0.028410

Bunya

See also: Tetracot

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

Optimal GPV sequence34d, 41, 116, 157c, 198c

Badness: 0.062897

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 243/242

Mapping: [1 1 1 -1 2], 0 4 9 26 10]]

POTE generator: ~10/9 = 175.777

Optimal GPV sequence: 34d, 41, 116e, 157ce

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

Optimal GPV sequence: 34d, 41, 75e, 116ef

Badness: 0.024886

Modus

See also: Tetracot

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

Optimal GPV sequence7, 20c, 27, 61d, 88bcd

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

Optimal GPV sequence: 7, 20ce, 27e, 34d, 61de

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

Optimal GPV sequence: 7, 20ce, 27e, 34d, 61de

Badness: 0.023806

Musical examples

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

Optimal GPV sequence: 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

Optimal GPV sequence: 7, 20c, 27, 61dee, 88bcdee

Badness: 0.039043

Wollemia

See also: Tetracot

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

Optimal GPV sequence27, 61, 88bc, 115bc

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

Optimal GPV sequence: 27e, 34, 61e

Badness: 0.037551

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99, 243/242

Mapping: [1 1 1 0 2 4], 0 4 9 19 10 -2]]

POTE generator: ~10/9 = 177.231

Optimal GPV sequence: 27e, 34, 61e

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

Optimal GPV sequence14c, 27, 41, 68, 109

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

Optimal GPV sequence: 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

Optimal GPV sequence: 27e, 41, 68e, 109ef

Badness: 0.023276

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 144/143, 154/153, 189/187

Mapping: [1 1 1 2 2 4 3], 0 8 18 11 20 -4 15]]

POTE generator: ~18/17 = 88.102

Optimal GPV sequence: 14c, 27eg, 41, 68egg, 109efgg

Badness: 0.021088

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 133/132, 144/143, 154/153, 189/187

Mapping: [1 1 1 2 2 4 3 3], 0 8 18 11 20 -4 15 17]]

POTE generator: ~18/17 = 88.111

Optimal GPV sequence: 14c, 27eg, 41, 68egg, 109efgg

Badness: 0.016652

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

Optimal GPV sequence: 27e, 41f, 68ef

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

Optimal GPV sequence: 41, 137cd, 178cd

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

Optimal GPV sequence: 13cdeef, 28ccdef, 41

Badness: 0.051876

October

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 1375/1372

Mapping: [1 1 1 2 5], 0 8 18 11 -21]]

POTE generator: ~21/20 = 88.035

Optimal GPV sequence: 27, 41, 68, 109, 150, 259

Badness: 0.039643

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 275/273, 385/384

Mapping: [1 1 1 2 5 4], 0 8 18 11 -21 -4]]

POTE generator: ~21/20 = 88.075

Optimal GPV sequence: 27, 41, 68, 109f

Badness: 0.031136

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 170/169, 196/195, 245/243, 256/255

Mapping: [1 1 1 2 5 4 6], 0 8 18 11 -21 -4 -26]]

POTE generator: ~21/20 = 88.104

Optimal GPV sequence: 27, 41, 68, 109f

Badness: 0.026833

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 154/153, 170/169, 190/189, 196/195, 209/208, 245/243

Mapping: [1 1 1 2 5 4 6 3], 0 8 18 11 -21 -4 -26 17]]

POTE generator: ~19/18 = 88.113

Optimal GPV sequence: 27, 41, 68, 109f, 177ffg

Badness: 0.020511

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

Optimal GPV sequence41, 143d, 184, 225, 409bcd

Badness: 0.119761

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 243/242, 1375/1372

Mapping: [1 1 1 1 2], 0 12 27 37 30]]

POTE generator: ~28/27 = 58.665

Optimal GPV sequence: 20cde, 41

Badness: 0.059528

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 243/242, 275/273

Mapping: [1 1 1 1 2 2], 0 12 27 37 30 35]]

POTE generator: ~27/26 = 58.639

Optimal GPV sequence: 20cdef, 41

Badness: 0.043645