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

Comma: 20000/19683

POTE generator: 176.160

Map: [<1 1 1|, <0 4 9|]

EDOs: 14c, 27, 34, 75, 109, 470b, 579b

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding 875/864, the keema, gives monkey, and 179200/177147 (or equivalently 225/224) gives bunya. Adding 245/243 gives octacot, which splits the generator in half.

Monkey and Bunya

Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie <<4 9 -15 5 -35 -60||. 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, the bunya-bunya tree temperament, adds 225/224 to the list of commas and may be described as the 41&75 temperament. It has <<4 9 26 5 30 35|| as a wedgie, and 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

Commas: 5120/5103, 875/864

POTE generator: 175.659

Map: [<1 1 1 5|, <0 4 9 -15|]

EDOs: 7, 34, 41, 321cd

Badness: 0.0734

11-limit

Commas: 243/242, 385/384, 100/99

POTE generator: 175.570

Map: [<1 1 1 5 2|, <0 4 9 -15 10|]

EDOs: 7, 34, 41, 123c

Badness: 0.0388

13-limit

Commas: 100/99, 105/104, 144/143, 243/242

POTE generator: 175.622

Map: [<1 1 1 5 2 4|, <0 4 9 -15 10 -2|]

EDOs: 7, 34, 41

Badness: 0.0284

Bunya

Commas: 225/224, 15625/15309

POTE generator: 175.741

Map: [<1 1 1 -1|, <0 4 9 26|]

EDOs: 41, 116, 157c, 198c

Badness: 0.0629

11-limit

Commas: 100/99, 225/224, 1344/1331

POTE generator: 175.777

Map: [<1 1 1 -1 2|, <0 4 9 26 10|]

EDOs: 41, 116e, 157ce

Badness: 0.0313

13-limit

Commas: 100/99, 144/143, 225/224, 243/242

POTE generator: 175.886

Map: [<1 1 1 -1 2 4|, <0 4 9 26 10 -2|]

EDOs: 34d, 41, 75e, 116ef

Badness: 0.0249

Modus

Commas: 64/63, 4375/4374

POTE generator: ~10/9 = 177.203

Map: [<1 1 1 4|, <0 4 9 -8|]

EDOs: 7, 27, 61d, 88bcd

Badness: 0.0682

11-limit

Commas: 64/63, 100/99, 243/242

POTE generator: ~10/9 = 177.053

Map: [<1 1 1 4 2|, <0 4 9 -8 10|]

EDOs: 7, 20ce, 27e, 34d, 61de

Badness: 0.0351

13-limit

Commas: 64/63, 78/77, 100/99, 144/143

POTE generator: ~10/9 = 176.953

Map: [<1 1 1 4 2 4|, <0 4 9 -8 10 -2|]

EDOs: 7, 27e, 34d, 61de

Badness: 0.0238

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.

Commas: 55/54, 64/63, 363/350

POTE generator: ~10/9 = 177.200

Map: [<1 1 1 4 3|, <0 4 9 -8 3|]

EDOs: 7, 20c, 27, 61de, 88bcde

Badness: 0.0631

13-limit

Commas: 55/54, 64/63, 66/65, 143/140

POTE generator: ~10/9 = 177.197

Map: [<1 1 1 4 3 4|, <0 4 9 -8 3 -2|]

EDOs: 7, 20c, 27, 61de, 88bcde

Badness: 0.039

Wollemia

Commas: 126/125, 2240/2187

POTE generator: ~10/9 = 177.357

Map: [<1 1 1 0|, <0 4 9 19|]

Wedgie: <<4 9 19 5 19 19||

EDOs: 27, 61, 88bc, 115bc

Badness: 0.0705

11-limit

Commas: 56/55, 100/99, 243/242

POTE generator: ~10/9 = 177.413

Map: [<1 1 1 0 2|, <0 4 9 19 10|]

EDOs: 27e, 34, 61e

Badness: 0.0376

13-limit

Commas: 56/55, 91/90, 100/99, 352/351

POTE generator: ~10/9 = 177.231

Map: [<1 1 1 0 2 4|, <0 4 9 19 10 -2|]

EDOs: 27e, 34, 61e

Badness: 0.0312

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 has wedgie <<8 18 11 10 -5 -25|| and 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.

Commas: 245/243, 2401/2400

POTE generator: 88.076

Map: [<1 1 1 2|, <0 8 18 11|]

EDOs: 14c, 27, 41, 68, 109

Badness: 0.0338

11-limit

Commas: 100/99, 243/242, 245/242

POTE generator: 87.975

Map: [<1 1 1 2 2|, <0 8 18 11 20|]

EDOs: 27e, 41, 109e, 150e, 191e

Badness: 0.0241

See also: Chords of octacot

13-limit

Commas: 100/99, 144/143, 196/195, 243/242

POTE generator: ~22/21 = 88.106

Map: [<1 1 1 2 2 4|, <0 8 18 11 20 -4|]

EDOs: 27e, 41, 68e, 109ef

Badness: 0.0233

Octocat

Commas: 78/77, 91/90, 100/99, 245/242

POTE generator: ~22/21 = 88.179

Map: [<1 1 1 2 2 2|, <0 8 18 11 20 23|]

EDOs: 27e, 41f, 68ef

Badness: 0.0276

Octopod

Commas: 100/99 105/104 243/242 245/242

POTE generator: ~22/21 = 87.697

Map: [<1 1 1 2 2 1|, <0 8 18 11 20 37|]

EDOs: 41, 137cd, 178cd

Badness: 0.0283

Dificot

Commas: 100/99, 243/242, 245/242, 343/338

POTE generator: ~13/9 = 643.989

Map: [<1 9 19 13 22 19|, <0 -16 -36 -22 -40 -33|]

EDOs: 41

Badness: 0.0519

Dodecacot

Commas: 3087/3125, 10976/10935

POTE generator: ~28/27 = 58.675

Map: [<1 1 1 1|, <0 12 27 37|]

Wedgie: <<12 27 37 15 25 10||

EDOs: 41, 184, 225, 409bcd

Badness: 0.1198