Tetracot family: Difference between revisions

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

The second comma of the [[normal forms|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 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 {{nowrap| 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 {{nowrap| 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 14^1/26 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 and 11-limit bunya. 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 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 13-limit monkey and 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.

=== 2.3.5.13 subgroup ===

 

 

Subgroup-val mapping: {{mapping| 1 1 1 4 | 0 4 9 -2 }}

* WE: ~2 = 1198.8502{{c}}, ~10/9 = 176.2195{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.2975{{c}}

Modus was named by [[Mike Battaglia]] in 2012 for its fantastic modmos structures<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_102416.html#102467 Yahoo! Tuning Group | ''Guaranteed meantone successor'']</ref>.

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 {{nowrap| 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.

=== 17-limit ===

=== 19-limit ===

==== 2.3.5.7.19 subgroup ====