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Tetracot family

Revision as of 04:12, 10 October 2023 by Xenllium (talk | contribs)

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.028

Minimax tuning:

eigenmonzo (unchanged-interval) basis: 2.5

Optimal ET sequence: 7, 20c, 27, 34, 41, 75, 334bc, 409bc, 484bc, 559bbc

Badness: 0.048518

Overview to 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. 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 34d & 41 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 ~7's 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 subgroup 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.

2.3.5.11 subgroup

Main article: Tetracot
See also: No-sevens subgroup temperaments § Tetracot

As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which tempering out 100/99, 144/143 and 243/242.

Subgroup: 2.3.5.11

Comma list: 100/99, 243/242

Sval mapping: [⟨1 1 1 2], ⟨0 4 9 10]]

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.985

Optimal ET sequence: 7, 27e, 34, 41, 75e

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

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

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 176.196

Optimal ET sequence: 7, 27e, 34, 41, 75e

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 ]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.676

Optimal ET sequence: 7, 27d, 34, 41, 239cd, 280ccdd, 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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.598

Optimal ET 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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.618

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

Badness: 0.028410

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 154/153, 170/169

Mapping: [⟨1 1 1 5 2 4 6], ⟨0 4 9 -15 10 -2 -13]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.754

Optimal ET sequence: 7, …, 27de, 34, 41, 116defg, 157cdeffgg

Badness: 0.025936

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 144/143, 154/153, 170/169, 171/169

Mapping: [⟨1 1 1 5 2 4 6 6], ⟨0 4 9 -15 10 -2 -13 -12]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.697

Optimal ET sequence: 7, 27deh, 34, 41

Badness: 0.022158

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 ]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.785

Optimal ET sequence: 34d, 41, 116, 157c, 355cc

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.738

Optimal ET sequence: 34d, 41, 116e, 157ce, 198cee, 239cee

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.748

Optimal ET sequence: 34d, 41, 116ef, 157ceff, 198ceeff

Badness: 0.024886

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 144/143, 170/169, 225/224

Mapping: [⟨1 1 1 -1 2 4 6], ⟨0 4 9 26 10 -2 -13]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.811

Optimal ET sequence: 34d, 41, 75e, 116efg, 157ceffgg

Badness: 0.023404

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 144/143, 170/169, 190/189, 225/224

Mapping: [⟨1 1 1 -1 2 4 6 0], ⟨0 4 9 26 10 -2 -13 29]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.802

Optimal ET sequence: 34dh, 41, 116efg, 157ceffgg

Badness: 0.019460

Modus

See also: Tetracot

Modus was named by Mike Battaglia in 2012 for its fantastic modmos structures[1].

Subgroup: 2.3.5.7

Comma list: 64/63, 4375/4374

Mapping: [⟨1 1 1 4], ⟨0 4 9 -8]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.818

Optimal ET sequence: 7, 20c, 27, 34d, 61d, 95dd

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.446

Optimal ET sequence: 7, 20ce, 27e, 34d

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.471

Optimal ET sequence: 7, 20ce, 27e, 34d

Badness: 0.023806

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 64/63, 78/77, 100/99, 120/119, 144/143

Mapping: [⟨1 1 1 4 2 4 1], ⟨0 4 9 -8 10 -2 21]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.453

Optimal ET sequence: 7g, …, 27eg, 34d

Badness: 0.021501

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 64/63, 78/77, 96/95, 100/99, 120/119, 144/143

Mapping: [⟨1 1 1 4 2 4 1 5], ⟨0 4 9 -8 10 -2 21 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.538

Optimal ET sequence: 7g, …, 27eg, 34dh

Badness: 0.017941

Music

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.990

Optimal ET sequence: 7, 20c, 27, 34de, 61dee

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.017

Optimal ET sequence: 7, 20c, 27, 34de, 61dee

Badness: 0.039043

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 52/51, 55/54, 64/63, 66/65, 143/140

Mapping: [⟨1 1 1 4 3 4 5], ⟨0 4 9 -8 3 -2 -6]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.378

Optimal ET sequence: 7, 20c, 27g

Badness: 0.035200

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 52/51, 55/54, 64/63, 66/65, 77/76, 143/140

Mapping: [⟨1 1 1 4 3 4 5 5], ⟨0 4 9 -8 3 -2 -6 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.505

Optimal ET sequence: 7, 20c, 27g

Badness: 0.028026

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 ]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.900

Optimal ET sequence: 7d, 20cd, 27, 34, 61, 95d, 156bcd

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.704

Optimal ET sequence: 7d, 20cde, 27e, 34, 95dee

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]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.716

Optimal ET sequence: 7d, 20cde, 27e, 34, 95dee

Badness: 0.031219

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 56/55, 91/90, 100/99, 136/135, 154/153

Mapping: [⟨1 1 1 0 2 4 1], ⟨0 4 9 19 10 -2 21]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.641

Optimal ET sequence: 27eg, 34

Badness: 0.024471

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 56/55, 76/75, 91/90, 100/99, 136/135, 154/153

Mapping: [⟨1 1 1 0 2 4 1 1], ⟨0 4 9 19 10 -2 21 22]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.749

Optimal ET sequence: 27eg, 34, 95deegh

Badness: 0.021108

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-2 temperament with octaves rather than rank-1 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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.076

Optimal ET sequence: 14c, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 87.975

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.106

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 88.102

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 88.111

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.179

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 87.697

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 643.989

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.035

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.075

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.104

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~19/18 = 88.113

Optimal ET 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]]

mapping generators: ~2, ~28/27

Wedgie: ⟨⟨ 12 27 37 15 25 10 ]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 58.675

Optimal ET sequence: 41, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 58.665

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.639

Optimal ET sequence: 20cdef, 41

Badness: 0.043645

  1. ↑ Yahoo! Tuning Group | Guaranteed meantone successor
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