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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:genewardsmith|genewardsmith]] and made on <tt>2012-05-18 19:22:32 UTC</tt>.<br>
: The original revision id was <tt>337197878</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.
== Tetracot ==
{{Main| Tetracot }}


[[POTE tuning|POTE generator]]: 176.160
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.


Map: [&lt;1 1 1|, &lt;0 4 9|]
The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).
EDOs: 14c, 27, 34, 75, 109, 470b, 579b


==Seven limit children==
[[Subgroup]]: 2.3.5
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===
[[Comma list]]: 20000/19683
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.
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}


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.
[[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 }}


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.
[[Minimax tuning]]:
* [[5-odd-limit]]: ~10/9 = {{monzo| -1/9 0 1/9 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


=Monkey=
{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 75, 109 }}
Commas: 5120/5103, 875/864


[[POTE tuning|POTE generator]]: 175.659
[[Badness]] (Sintel): 1.14


Map: [&lt;1 1 1 5|, &lt;0 4 9 -15|]
=== Overview to extensions ===
EDOs: 7, 34, 41, 321cd
==== Subgroup extensions ====
Badness: 0.0734
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.  


==11-limit==
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]]}.
Commas: 243/242, 385/384, 100/99


[[POTE tuning|POTE generator]]: 175.570
==== Full 7-limit extensions ====
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.  


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


==13-limit==  
=== 2.3.5.11 subgroup ===
Commas: 100/99, 105/104, 144/143, 243/242
Subgroup: 2.3.5.11


[[POTE tuning|POTE generator]]: 175.622
Comma list: 100/99, 243/242


Map: [&lt;1 1 1 5 2 4|, &lt;0 4 9 -15 10 -2|]
Subgroup-val mapping: {{mapping| 1 1 1 2 | 0 4 9 10 }}
EDOs: 7, 34, 41
Badness: 0.0284


=Bunya=  
Optimal tunings:
Commas: 225/224, 15625/15309
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}


[[POTE tuning|POTE generator]]: 175.741
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}


Map: [&lt;1 1 1 -1|, &lt;0 4 9 26|]
Badness (Sintel): 0.459
EDOs: 41, 116, 157c, 198c
Badness: 0.0629


==11-limit==  
==== 2.3.5.11.13 subgroup ====
Commas: 100/99, 225/224, 1344/1331
Subgroup: 2.3.5.11.13


[[POTE tuning|POTE generator]]: 175.777
Comma list: 100/99, 144/143, 243/242


Map: [&lt;1 1 1 -1 2|, &lt;0 4 9 26 10|]
Subgroup-val mapping: {{mapping| 1 1 1 2 4 | 0 4 9 10 -2 }}
EDOs: 41, 116e, 157ce
Badness: 0.0313


==13-limit==
Optimal tunings:
Commas: 100/99, 144/143, 225/224, 243/242
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}


[[POTE tuning|POTE generator]]: 175.886
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}


Map: [&lt;1 1 1 -1 2 4|, &lt;0 4 9 26 10 -2|]
Badness (Sintel): 0.489
EDOs: 34d, 41, 75e, 116ef
Badness: 0.0249


=Modus=  
== Monkey ==
Commas: 64/63, 4375/4374
{{Main| Monkey }}


POTE generator: ~10/9 = 177.203
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.


Map: [&lt;1 1 1 4|, &lt;0 4 9 -8|]
[[Subgroup]]: 2.3.5.7
EDOs: 7, 27, 61d, 88bcd
Badness: 0.0682


==11-limit==
[[Comma list]]: 875/864, 5120/5103
Commas: 64/63, 100/99, 243/242


POTE generator: ~10/9 = 177.053
{{Mapping|legend=1| 1 1 1 5 | 0 4 9 -15 }}


Map: [&lt;1 1 1 4 2|, &lt;0 4 9 -8 10|]
[[Optimal tuning]]s:  
EDOs: 7, 20ce, 27e, 34d, 61de
* [[WE]]: ~2 = 1200.7982{{c}}, ~10/9 = 175.7757{{c}}
Badness: 0.0351
: [[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 7, 34, 41 }}
Commas: 64/63, 78/77, 100/99, 144/143


POTE generator: ~10/9 = 176.953
[[Badness]] (Sintel): 1.86


Map: [&lt;1 1 1 4 2 4|, &lt;0 4 9 -8 10 -2|]
=== 11-limit ===
EDOs: 7, 27e, 34d, 61de
Subgroup: 2.3.5.7.11
Badness: 0.0238


=Ponens=
Comma list: 100/99, 243/242, 385/384
The error of 11 is about the same as that of Modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.


Commas: 55/54, 64/63, 363/350
Mapping: {{mapping| 1 1 1 5 2 | 0 4 9 -15 10 }}


POTE generator: ~10/9 = 177.200
Optimal tunings:  
* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}


Map: [&lt;1 1 1 4 3|, &lt;0 4 9 -8 3|]
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
EDOs: 7, 20c, 27, 61de, 88bcde
Badness: 0.0631


==13-limit==
Badness (Sintel): 1.28
Commas: 55/54, 64/63, 66/65, 143/140


POTE generator: ~10/9 = 177.197
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 1 4 3 4|, &lt;0 4 9 -8 3 -2|]
Comma list: 100/99, 105/104, 144/143, 243/242
EDOs: 7, 20c, 27, 61de, 88bcde
Badness: 0.039


=Wollemia=
Mapping: {{mapping| 1 1 1 5 2 4 | 0 4 9 -15 10 -2 }}
Commas: 126/125, 2240/2187


POTE generator: ~10/9 = 177.357
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 0|, &lt;0 4 9 19|]
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
Wedgie: &lt;&lt;4 9 19 5 19 19||
EDOs: 27, 61, 88bc, 115bc
Badness: 0.0705


==11-limit==
Badness (Sintel): 1.17
Commas: 56/55, 100/99, 243/242


POTE generator: ~10/9 = 177.413
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 1 0 2|, &lt;0 4 9 19 10|]
Comma list: 100/99, 105/104, 144/143, 154/153, 170/169
EDOs: 27e, 34, 61e
Badness: 0.0376


==13-limit==
Mapping: {{mapping| 1 1 1 5 2 4 6 | 0 4 9 -15 10 -2 -13 }}
Commas: 56/55, 91/90, 100/99, 352/351


POTE generator: ~10/9 = 177.231
Optimal tunings:  
* WE: ~2 = 1199.5029{{c}}, ~10/9 = 175.6832{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7558{{c}}


Map: [&lt;1 1 1 0 2 4|, &lt;0 4 9 19 10 -2|]
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
EDOs: 27e, 34, 61e
Badness: 0.0312


=Octacot=
Badness (Sintel): 1.32
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.


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.
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Commas: 245/243, 2401/2400
Comma list: 100/99, 105/104, 144/143, 154/153, 170/169, 171/169


[[POTE tuning|POTE generator]]: 88.076
Mapping: {{mapping| 1 1 1 5 2 4 6 6 | 0 4 9 -15 10 -2 -13 -12 }}


Map: [&lt;1 1 1 2|, &lt;0 8 18 11|]
Optimal tunings:  
EDOs: 14c, 27, 41, 68, 109
* WE: ~2 = 1199.7318{{c}}, ~10/9 = 175.6498{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6901{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
Commas: 100/99, 243/242, 245/242


[[POTE tuning|POTE generator]]: 87.975
Badness (Sintel): 1.35


Map: [&lt;1 1 1 2 2|, &lt;0 8 18 11 20|]
== Bunya ==
EDOs: 27e, 41, 109e, 150e, 191e
{{Main| Bunya }}


See also: [[Chords of octacot]]
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.


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 100/99, 144/143, 196/195, 243/242


[[POTE tuning|POTE generator]]: ~22/21 = 88.106
[[Comma list]]: 225/224, 15625/15309


Map: [&lt;1 1 1 2 2 4|, &lt;0 8 18 11 20 -4|]
{{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}
EDOs: 27e, 41, 68e, 109ef
Badness: 0.0233


==Octocat==
[[Optimal tuning]]s:
Commas: 78/77, 91/90, 100/99, 245/242
* [[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 }}


POTE generator: ~22/21 = 88.179
{{Optimal ET sequence|legend=1| 7d, …, 34d, 41, 116, 157c, 198c }}


Map: [&lt;1 1 1 2 2 2|, &lt;0 8 18 11 20 23|]
[[Badness]] (Sintel): 1.59
EDOs: 27e, 41f, 68ef
Badness: 0.0276


==Octopod==
=== 11-limit ===
Commas: 100/99 105/104 243/242 245/242
Subgroup: 2.3.5.7.11


POTE generator: ~22/21 = 87.697
Comma list: 100/99, 225/224, 243/242


Map: [&lt;1 1 1 2 2 1|, &lt;0 8 18 11 20 37|]
Mapping: {{mapping| 1 1 1 -1 2 | 0 4 9 26 10 }}
EDOs: 41, 137cd, 178cd
Badness: 0.0283


Optimal tunings:
* WE: ~2 = 1199.7481{{c}}, ~10/9 = 175.7401{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7637{{c}}


</pre></div>
{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tetracot family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:42:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;
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 }}
 
Optimal tunings:
* WE: ~2 = 1199.1044{{c}}, ~10/9 = 175.7545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8526{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 34d, 41, 116ef }}
 
Badness (Sintel): 1.03
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 120/119, 144/143, 170/169, 225/224
 
Mapping: {{mapping| 1 1 1 -1 2 4 6 | 0 4 9 26 10 -2 -13 }}
 
Optimal tunings:
* WE: ~2 = 1198.7905{{c}}, ~10/9 = 175.7757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9302{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 41, 75e }}
 
Badness (Sintel): 1.19
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 100/99, 120/119, 144/143, 170/169, 190/189, 225/224
 
Mapping: {{mapping| 1 1 1 -1 2 4 6 0 | 0 4 9 26 10 -2 -13 29 }}
 
Optimal tunings:
* WE: ~2 = 1198.7904{{c}}, ~10/9 = 175.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9287{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 41, 75e }}
 
Badness (Sintel): 1.18
 
== Modus ==
{{Main| Modus }}
 
Modus tempers out [[64/63]] as well as [[4375/4374]], and may be described as the {{nowrap| 27 & 34d }} temperament. While less accurate than [[#Monkey|monkey]] or [[#Bunya|bunya]], it is nonetheless very useful because it is simpler and because of the harmonic puns it possesses. [[27edo]], [[34edo]] and [[61edo]] can all be used as tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 4375/4374
 
{{Mapping|legend=1| 1 1 1 4 | 0 4 9 -8 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.7884{{c}}, ~10/9 = 176.7292{{c}}
: [[
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