Tetracot family: Difference between revisions
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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<sup>1/26</sup> 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. | 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<sup>1/26</sup> 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<sup>1/26</sup> generator supplies, or even sharper yet, as for instance by the val {{val| 355 563 823 997 1230 }}, with a 52/355 generator. | 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|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<sup>1/26</sup> 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)<sup>2</sup> 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. | Since [[16/13]] is shy of (10/9)<sup>2</sup> 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. | ||
==== Subgroup extensions ==== | |||
As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which it tempers out 100/99, 144/143 and 243/242, considered right below. The [[S-expression]]-based comma list of this temperament is {[[243/242|S9/S11]], [[100/99|S10]], [[144/143|S12]]}. | |||
=== 2.3.5.11 subgroup === | === 2.3.5.11 subgroup === | ||
Subgroup: 2.3.5.11 | Subgroup: 2.3.5.11 | ||
| Line 76: | Line 75: | ||
Badness (Sintel): 0.489 | Badness (Sintel): 0.489 | ||
== Monkey == | == Monkey == | ||
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== Other subgroup extensions == | == Other subgroup extensions == | ||
=== Tetracot (2.3.5.13) === | |||
Subgroup: 2.3.5.13 | |||
Comma list: 325/324, 512/507 | |||
Subgroup-val mapping: {{mapping| 1 1 1 4 | 0 4 9 -2 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1198.8502{{c}}, ~10/9 = 176.2195{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.2975{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 20c, 27, 34, 245bff, 279bfff }} | |||
Badness (Sintel): 0.551 | |||
=== Devisemi (2.3.5.19) === | === Devisemi (2.3.5.19) === | ||
[[Subgroup]]: 2.3.5.19 | [[Subgroup]]: 2.3.5.19 | ||
| Line 906: | Line 905: | ||
{{Mapping|legend=3| 1 1 1 0 0 0 0 3 | 0 8 18 0 0 0 0 17 }} | {{Mapping|legend=3| 1 1 1 0 0 0 0 3 | 0 8 18 0 0 0 0 17 }} | ||
: mapping generators: ~2, ~20/19 | : mapping generators: ~2, ~20/19 | ||
| Line 919: | Line 917: | ||
[[Badness]] (Sintel): 1.30 | [[Badness]] (Sintel): 1.30 | ||
=== | === Devisemi (2.3.5.7.19) === | ||
Subgroup: 2.3.5.7.19 | Subgroup: 2.3.5.7.19 | ||