Tetracot family: Difference between revisions

The parent of the '''tetracot family''' is '''tetracot''', the 5-limit temperament [[tempering out]] [[20000/19683]] = {{monzo| 5 -9 4 }}, the minimal diesis or tetracot comma. The dual of this comma is the wedgie {{multival| 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|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.

Comma list: 20000/19683

[[Mapping]]: [{{val| 1 1 1 }}, {{val| 0 4 9 }}]

'''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 val]]s, meaning that ''n''×log₂(prime) rounded to the nearest integer gives the mapping. [[41edo|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, {{multival| 4 9 -15 10 … }} and 11-limit bunya, {{multival| 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 {{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 {{multival| 4 9 -15 10 -2 … }} for 13-limit monkey and {{multival| 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.

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|68EDO]] or [[109edo|109EDO]] can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is [[150edo|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.