Tetracot family: Difference between revisions

The parent of the '''tetracot family''' is [[tetracot]], the [[5-limit]] [[regular temperament|temperament]] [[tempering out]] the [[tetracot comma]] ([[ratio]]: 20000/19683, {{monzo|legend=1| 5 -9 4 }}).  

{{Main| Tetracot }}

The [[generator]] of tetracot is [[~]][[10/9]], and that four of these 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.

[[Subgroup]]: 2.3.5

[[Comma list]]: 20000/19683

{{Mapping|legend=1| 1 1 1 | 0 4 9 }}

* [[WE]]: ~2 = 1199.5586{{c}}, ~10/9 = 176.0950{{c}}

: [[error map]]: {{val| -0.441 +1.984 -1.900 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 176.0965{{c}}

: error map: {{val| 0.000 +2.431 -1.445 }}

* [[5-odd-limit]]: ~10/9 = {{monzo| -1/9 0 1/9 }}

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

{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 75, 109 }}

[[Badness]] (Sintel): 1.14

Since the generator in all reasonable tunings is between 10/9 and [[11/10]], it is natural to extend tetracot to the [[11-limit]] by tempering out (10/9)/(11/10) = [[100/99]]. This gives the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] version of tetracot, dispensing with 7. For this, [[41edo]] can be used as a tuning.  

Since [[16/13]] is shy of (10/9)² by just [[325/324]], it is likewise natural to extend our winning streak by adding this to the list of commas. This gives us [[2.3.5.11.13 subgroup|2.3.5.11.13-subgroup]] tetracot, which tempers out 100/99, [[144/143]] and [[243/242]], with the [[S-expression]]-based comma list {[[243/242|S9/S11]], [[100/99|S10]], [[144/143|S12]]}.

The second comma of the comma list defines which 7-limit family member we are looking at. [[875/864]], the keema, gives monkey. [[225/224]] gives bunya. [[64/63]] gives modus. [[126/125]] gives wollemia. These all use the same generators as tetracot.  

[[245/243]] gives octacot, which splits the generator in halves. [[3125/3087]] gives dodecacot, which splits the generator in thirds. [[50/49]] gives weasel, which splits the period in halves.  

=== 2.3.5.11 subgroup ===

Subgroup: 2.3.5.11

Comma list: 100/99, 243/242

Subgroup-val mapping: {{mapping| 1 1 1 2 | 0 4 9 10 }}

* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}

{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}

Badness (Sintel): 0.459

==== 2.3.5.11.13 subgroup ====

Subgroup: 2.3.5.11.13

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

Subgroup-val mapping: {{mapping| 1 1 1 2 4 | 0 4 9 10 -2 }}

* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}

{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}

Badness (Sintel): 0.489

== Monkey ==

Monkey tempers out the [[keema]]. The keema, 875/864, is the amount by which three [[6/5|just minor thirds]] fall short of [[7/4]], and tells us the ~7/4 of monkey is reached by three such 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|bunya]] with the same tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 5120/5103

{{Mapping|legend=1| 1 1 1 5 | 0 4 9 -15 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.7982{{c}}, ~10/9 = 175.7757{{c}}

: [[error map]]: {{val| +0.798 +1.946 -3.534 -1.470 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.6622{{c}}

: error map: {{val| 0.000 +0.694 -5.354 -3.759 }}

{{Optimal ET sequence|legend=1| 7, 34, 41 }}

[[Badness]] (Sintel): 1.86

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 100/99, 243/242, 385/384

Optimal tunings:  

* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}

{{Optimal ET sequence|legend=0| 7, 34, 41 }}

Badness (Sintel): 1.28

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 243/242

Mapping: {{mapping| 1 1 1 5 2 4 | 0 4 9 -15 10 -2 }}

* WE: ~2 = 1199.9206{{c}}, ~10/9 = 175.6108{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6217{{c}}

{{Optimal ET sequence|legend=0| 7, 34, 41 }}

Badness (Sintel): 1.17

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

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

Mapping: {{mapping| 1 1 1 5 2 4 6 | 0 4 9 -15 10 -2 -13 }}

* WE: ~2 = 1199.5029{{c}}, ~10/9 = 175.6832{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7558{{c}}

{{Optimal ET sequence|legend=0| 7, 34, 41 }}

Badness (Sintel): 1.32

=== 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: {{mapping| 1 1 1 5 2 4 6 6 | 0 4 9 -15 10 -2 -13 -12 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6901{{c}}

{{Optimal ET sequence|legend=0| 7, 34, 41 }}

Badness (Sintel): 1.35

{{Main| Bunya }}

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|monkey]]. However, bunya profits a little from a slightly sharper fifth. An excellent generator is 14^1/26, 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, or even sharper yet: 17\116 with a fifth a cent and a half sharp, or 11\75 with a fifth two cents sharp. [[Octave stretching]], if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 15625/15309

{{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.2991{{c}}, ~10/9 = 175.7844{{c}}

: [[error map]]: {{val| +0.299 +1.482 -3.955 +1.270 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.7567{{c}}

: error map: {{val| 0.000 +1.072 -4.503 +0.849 }}

{{Optimal ET sequence|legend=1| 7d, …, 34d, 41, 116, 157c, 198c }}

[[Badness]] (Sintel): 1.59

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 243/242

Mapping: {{mapping| 1 1 1 -1 2 | 0 4 9 26 10 }}

Optimal tunings:  

* WE: ~2 = 1199.7481{{c}}, ~10/9 = 175.7401{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7637{{c}}

{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}

Badness (Sintel): 1.04

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 243/242

Mapping: {{mapping| 1 1 1 -1 2 4 | 0 4 9 26 10 -2 }}