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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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 }}).  
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2012-04-10 16:57:58 UTC</tt>.<br>
: The original revision id was <tt>319222690</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
The parent of the **tetracot family** is **tetracot**, the 5-limit temperament [[tempering out]] 20000/19683 = |5 -9 4&gt;, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &lt;&lt;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.


[[POTE tuning|POTE generator]]: 176.160
== Tetracot ==
{{Main| Tetracot }}


Map: [&lt;1 1 1|, &lt;0 4 9|]
The [[generator]] of tetracot is [[~]][[10/9]], and that four of these give [[~]][[3/2]]. In fact, (10/9)<sup>4</sup> = (20000/19683)⋅(3/2). We also have (10/9)<sup>9</sup> = (20000/19683)<sup>2</sup>⋅(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)<sup>1/9</sup> for a generator. The 13-note [[mos]] gives enough space for eight triads, with the 20-note mos supplying many more.
EDOs: 14c, 27, 34, 75, 109, 470b, 579b


==Seven limit children==
The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).
The second comma of the [[Normal lists|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 (the names come from members of the Araucaria family of conifers, which have four cotyledons, though sometimes these are fused.) Adding 245/243 gives octacot, which splits the generator in half.


===Monkey and Bunya===
[[Subgroup]]: 2.3.5
Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie &lt;&lt;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&amp;41 temperament, if the vals in question are taken to be [[Patent val|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&amp;75 temperament. It has &lt;&lt;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.
[[Comma list]]: 20000/19683


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, &lt;&lt;4 9 -15 10 ...|| and 11-limit banya, &lt;&lt;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 &lt;355 563 823 997 1230|, with a 52/355 generator.
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}


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 &lt;&lt;4 9 -15 10 -2 ...|| for 13-limit monkey and &lt;&lt;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.
[[Optimal tuning]]s:
* [[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 }}


=Monkey=
[[Minimax tuning]]:
Commas: 5120/5103, 875/864
* [[5-odd-limit]]: ~10/9 = {{monzo| -1/9 0 1/9 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[POTE tuning|POTE generator]]: 175.659
{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 75, 109 }}


Map: [&lt;1 1 1 5|, &lt;0 4 9 -15|]
[[Badness]] (Sintel): 1.14
EDOs: 7, 34, 41, 321cd
Badness: 0.0734


==11-limit==  
=== Overview to extensions ===
Commas: 243/242, 385/384, 100/99
==== Subgroup extensions ====
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.


[[POTE tuning|POTE generator]]: 175.570
Since [[16/13]] is shy of (10/9)<sup>2</sup> 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]]}.  


Map: [&lt;1 1 1 5 2|, &lt;0 4 9 -15 10|]
==== Full 7-limit extensions ====
EDOs: 7, 34, 41, 123c
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.  
Badness: 0.0388


==13-limit==
[[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.
Commas: 100/99, 105/104, 144/143, 243/242


[[POTE tuning|POTE generator]]: 175.622
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11


Map: [&lt;1 1 1 5 2 4|, &lt;0 4 9 -15 10 -2|]
Comma list: 100/99, 243/242
EDOs: 7, 34, 41
Badness: 0.0284


=Bunya=
Subgroup-val mapping: {{mapping| 1 1 1 2 | 0 4 9 10 }}
Commas: 225/224, 15625/15309


[[POTE tuning|POTE generator]]: 175.741
Optimal tunings:  
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}


Map: [&lt;1 1 1 -1|, &lt;0 4 9 26|]
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}
EDOs: 41, 116, 157c, 198c
Badness: 0.0629


==11-limit==
Badness (Sintel): 0.459
Commas: 100/99, 225/224, 1344/1331


[[POTE tuning|POTE generator]]: 175.777
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13


Map: [&lt;1 1 1 -1 2|, &lt;0 4 9 26 10|]
Comma list: 100/99, 144/143, 243/242
EDOs: 41, 116e, 157ce
Badness: 0.0313


==13-limit==
Subgroup-val mapping: {{mapping| 1 1 1 2 4 | 0 4 9 10 -2 }}
Commas: 100/99, 144/143, 225/224, 243/242


[[POTE tuning|POTE generator]]: 175.886
Optimal tunings:  
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}


Map: [&lt;1 1 1 -1 2 4|, &lt;0 4 9 26 10 -2|]
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}
EDOs: 34d, 41, 75e, 116ef
Badness: 0.0249


=Modus=
Badness (Sintel): 0.489
Commas: 64/63, 4375/4374


POTE generator: ~10/9 = 177.203
== Monkey ==
{{Main| Monkey }}


Map: [&lt;1 1 1 4|, &lt;0 4 9 -8|]
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.
EDOs: 7, 27, 61d, 88bcd
Badness: 0.0682


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 64/63, 100/99, 243/242


POTE generator: ~10/9 = 177.053
[[Comma list]]: 875/864, 5120/5103


Map: [&lt;1 1 1 4 2|, &lt;0 4 9 -8 10|]
{{Mapping|legend=1| 1 1 1 5 | 0 4 9 -15 }}
EDOs: 7, 20ce, 27e, 34d, 61de
Badness: 0.0351


==13-limit==  
[[Optimal tuning]]s:
Commas: 64/63, 78/77, 100/99, 144/143
* [[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 }}


POTE generator: ~10/9 = 176.953
{{Optimal ET sequence|legend=1| 7, 34, 41 }}


Map: [&lt;1 1 1 4 2 4|, &lt;0 4 9 -8 10 -2|]
[[Badness]] (Sintel): 1.86
EDOs: 7, 27e, 34d, 61de
Badness: 0.0238


=Ponens=  
=== 11-limit ===
Commas:
Subgroup: 2.3.5.7.11


POTE generator: ~176.953 (?)
Comma list: 100/99, 243/242, 385/384


Map: [&lt;1 1 1 4 3 4|, &lt;0 4 9 -8 3 -2|]
Mapping: {{mapping| 1 1 1 5 2 | 0 4 9 -15 10 }}
*The 11 is about as bad as that of Modus, but flat instead of sharp, and much more abundant.


=Wollemia=  
Optimal tunings:
Commas: 126/125, 2240/2187
* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}


POTE generator: ~10/9 = 177.357
{{Optimal ET sequence|legend=0| 7, 34, 41 }}


Map: [&lt;1 1 1 0|, &lt;0 4 9 19|]
Badness (Sintel): 1.28
Wedgie: &lt;&lt;4 9 19 5 19 19||
EDOs: 27, 61, 88bc, 115bc
Badness: 0.0705


=Octacot=  
=== 13-limit ===
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 &lt;&lt;8 18 11 10 -5 -25|| and may also be described as 41&amp;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.
Subgroup: 2.3.5.7.11.13


Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving &lt;&lt;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.
Comma list: 100/99, 105/104, 144/143, 243/242


Commas: 245/243, 2401/2400
Mapping: {{mapping| 1 1 1 5 2 4 | 0 4 9 -15 10 -2 }}


[[POTE tuning|POTE generator]]: 88.076
Optimal tunings:  
* WE: ~2 = 1199.9206{{c}}, ~10/9 = 175.6108{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6217{{c}}


Map: [&lt;1 1 1 2|, &lt;0 8 18 11|]
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
EDOs: 14c, 27, 41, 68, 109


==11-limit==
Badness (Sintel): 1.17
Commas: 100/99, 243/242, 245/242


[[POTE tuning|POTE generator]]: 87.975
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 1 2 2|, &lt;0 8 18 11 20|]
Comma list: 100/99, 105/104, 144/143, 154/153, 170/169
EDOs: 27e, 41, 109e, 150e, 191e


See also: [[Chords of octacot]]
Mapping: {{mapping| 1 1 1 5 2 4 6 | 0 4 9 -15 10 -2 -13 }}


==13-limit==
Optimal tunings:
Commas: 100/99, 144/143, 196/195, 243/242
* WE: ~2 = 1199.5029{{c}}, ~10/9 = 175.6832{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7558{{c}}


[[POTE tuning|POTE generator]]: ~22/21 = 88.106
{{Optimal ET sequence|legend=0| 7, 34, 41 }}


Map: [&lt;1 1 1 2 2 4|, &lt;0 8 18 11 20 -4|]
Badness (Sintel): 1.32
EDOs: 27e, 41, 68e, 109ef
 
Badness: 0.0233</pre></div>
=== 19-limit ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11.13.17.19
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&
 
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:
* WE: ~2 = 1199.7318{{c}}, ~10/9 = 175.6498{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6901{{c}}
 
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
 
Badness (Sintel): 1.35
 
== Bunya ==
{{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<sup>1/26</sup>, 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
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