Xenharmonic Wiki

Tetracot family: Difference between revisions

VisualWikitext
Wikispaces>genewardsmith
**Imported revision 151425695 - Original comment: **
 
m Recategorize
 
(116 intermediate revisions by 21 users not shown)
Line 1: Line 1:
<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>2010-07-03 01:21:19 UTC</tt>.<br>
: The original revision id was <tt>151425695</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">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.


==Seven limit children==
== Tetracot ==
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.</pre></div>
{{Main| Tetracot }}
<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;The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = |5 -9 4&amp;gt;, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &amp;lt;&amp;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 &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; 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.&lt;br /&gt;
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.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
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 &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 20000/19683
 
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}
 
[[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 }}
 
[[Minimax tuning]]:
* [[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
 
=== Overview to extensions ===
==== 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.  
 
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]]}.
 
==== 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.  
 
[[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 }}
 
Optimal tunings:
* 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 }}
 
Optimal tunings:
* 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 ==
{{Main| 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
 
Mapping: {{mapping| 1 1 1 5 2 | 0 4 9 -15 10 }}
 
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 }}
 
Optimal tunings:
* 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 }}
 
Optimal tunings:
* 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:
* 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
 
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}}
: [[error map]]: {{val| -3.212 +1.750 +1.038 +4.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1188{{c}}
: error map: {{val| 0.000 +6.520 +7.755 +14.224 }}
 
{{Optimal ET sequence|legend=1| 7, 20c, 27, 61d, 88bcd, 149bccddd }}
 
[[Badness]] (Sintel): 1.73
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 64/63, 100/99, 243/242
 
Mapping: {{mapping| 1 1 1 4 2 | 0 4 9 -8 10 }}
 
Optimal tunings:
* WE: ~2 = 1196.4227{{c}}, ~10/9 = 176.5252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9286{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}
 
Badness (Sintel): 1.16
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 64/63, 78/77, 100/99, 144/143
 
Mapping: {{mapping| 1 1 1 4 2 4 | 0 4 9 -8 10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1196.8686{{c}}, ~10/9 = 176.4915{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8735{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}
 
Badness (Sintel): 0.984
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 64/63, 78/77, 100/99, 120/119, 144/143
 
Mapping: {{mapping| 1 1 1 4 2 4 1 | 0 4 9 -8 10 -2 21 }}
 
Optimal tunings:
* WE: ~2 = 1196.8783{{c}}, ~10/9 = 176.5241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8969{{c}}
 
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34d }}
 
Badness (Sintel): 1.10
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 64/63, 78/77, 96/95, 100/99, 120/119, 144/143
 
Mapping: {{mapping| 1 1 1 4 2 4 1 5 | 0 4 9 -8 10 -2 21 -5 }}
 
Optimal tunings:
* WE: ~2 = 1196.6939{{c}}, ~10/9 = 176.5426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9645{{c}}
 
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh, 61degh }}
 
Badness (Sintel): 1.09
 
=== Ponens ===
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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 64/63, 363/350
 
Mapping: {{mapping| 1 1 1 4 3 | 0 4 9 -8 3 }}
 
Optimal tunings:
* WE: ~2 = 1198.5026{{c}}, ~10/9 = 176.9786{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1589{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20c, 27 }}
 
Badness (Sintel): 2.09
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 64/63, 66/65, 143/140
 
Mapping: {{mapping| 1 1 1 4 3 4 | 0 4 9 -8 3 -2 }}
 
Optimal tunings:
* WE: ~2 = 1198.5149{{c}}, ~10/9 = 176.9778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1681{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20c, 27 }}
 
Badness (Sintel): 1.61
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 52/51, 55/54, 64/63, 66/65, 143/140
 
Mapping: {{mapping| 1 1 1 4 3 4 5 | 0 4 9 -8 3 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1197.4542{{c}}, ~10/9 = 177.1828{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5355{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20c }}
 
Badness (Sintel): 1.79
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 52/51, 55/54, 64/63, 66/65, 77/76, 143/140
 
Mapping: {{mapping| 1 1 1 4 3 4 5 5 | 0 4 9 -8 3 -2 -6 -5 }}
 
Optimal tunings:
* WE: ~2 = 1197.3233{{c}}, ~10/9 = 177.2025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5878{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20c }}
 
Badness (Sintel): 1.70
 
== Wollemia ==
{{Main| Wollemia }}
 
Wollemia tempers out [[126/125]] as well as [[2240/2187]], and may be described as the {{nowrap| 27 & 34 }} temperament. [[27edo]] may be recommended as a tuning, in which case it is identical to modus with the same tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 2240/2187
 
{{Mapping|legend=1| 1 1 1 0 | 0 4 9 19 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.6555{{c}}, ~10/9 = 177.0104{{c}}
: [[error map]]: {{val| -2.345 +3.742 +4.435 -5.628 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1667{{c}}
: error map: {{val| 0.000 +6.712 +8.186 -2.659 }}
 
{{Optimal ET sequence|legend=1| 7d, 20cd, 27, 61, 88bc, 115bc }}
 
[[Badness]] (Sintel): 1.78
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 100/99, 243/242
 
Mapping: {{mapping| 1 1 1 0 2 | 0 4 9 19 10 }}
 
Optimal tunings:
* WE: ~2 = 1196.6462{{c}}, ~10/9 = 176.9174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1370{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}
 
Badness (Sintel): 1.24
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 91/90, 100/99, 243/242
 
Mapping: {{mapping| 1 1 1 0 2 4 | 0 4 9 19 10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1197.4576{{c}}, ~10/9 = 176.8557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0949{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}
 
Badness (Sintel): 1.29
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 56/55, 91/90, 100/99, 136/135, 154/153
 
Mapping: {{mapping| 1 1 1 0 2 4 1 | 0 4 9 19 10 -2 21 }}
 
Optimal tunings:
* WE: ~2 = 1197.4770{{c}}, ~10/9 = 176.7733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0123{{c}}
 
{{Optimal ET sequence|legend=0| 7dg, 27eg }}
 
Badness (Sintel): 1.25
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 56/55, 76/75, 91/90, 100/99, 136/135, 154/153
 
Mapping: {{mapping| 1 1 1 0 2 4 1 1 | 0 4 9 19 10 -2 21 22 }}
 
Optimal tunings:
* WE: ~2 = 1197.4380{{c}}, ~10/9 = 176.8774{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1216{{c}}
 
{{Optimal ET sequence|legend=0| 7dgh, 27eg }}
 
Badness (Sintel): 1.28
 
== Octacot ==
{{See also| Chords of octacot }}
 
Octacot splits the difference between the [[#Monkey|monkey]] and [[#Bunya|bunya]] mappings for 7 by cutting the generator in half. 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)<sup>1/18</sup>, 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.
 
Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 2401/2400
 
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6782{{c}}, ~21/20 = 88.0528{{c}}
: [[error map]]: {{val| -0.322 +2.145 -1.686 -0.889 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 88.0525{{c}}
: error map: {{val| 0.000 +2.465 -1.369 -0.248 }}
 
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
 
[[Badness]] (Sintel): 0.857
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 243/242, 245/242
 
Mapping: {{mapping| 1 1 1 2 2 | 0 8 18 11 20 }}
 
Optimal tunings:
* WE: ~2 = 1199.6025{{c}}, ~21/20 = 87.9460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.9453{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 109e }}
 
Badness (Sintel): 0.796
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 144/143, 196/195, 243/242
 
Mapping: {{mapping| 1 1 1 2 2 4 | 0 8 18 11 20 -4 }}
 
Optimal tunings:
* WE: ~2 = 1198.8609{{c}}, ~21/20 = 87.0219{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0557{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 68e, 109ef }}
 
Badness (Sintel): 0.962
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 120/119, 144/143, 154/153, 189/187
 
Mapping: {{mapping| 1 1 1 2 2 4 3 | 0 8 18 11 20 -4 15 }}
 
Optimal tunings:
* WE: ~2 = 1198.4494{{c}}, ~21/20 = 87.9878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0324{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}
 
Badness (Sintel): 1.07
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 100/99, 120/119, 133/132, 144/143, 154/153, 189/187
 
Mapping: {{mapping| 1 1 1 2 2 4 3 3 | 0 8 18 11 20 -4 15 17 }}
 
Optimal tunings:
* WE: ~2 = 1198.5995{{c}}, ~20/19 = 88.0081{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0471{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}
 
Badness (Sintel): 1.01
 
==== Octocat ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 91/90, 100/99, 245/242
 
Mapping: {{mapping| 1 1 1 2 2 2 | 0 8 18 11 20 23 }}
 
Optimal tunings:
* WE: ~2 = 1199.4441{{c}}, ~21/20 = 88.1380{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1375{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27e, 41f }}
 
Badness (Sintel): 1.14
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 52/51, 78/77, 91/90, 100/99, 189/187
 
Mapping: {{mapping| 1 1 1 2 2 2 3 | 0 8 18 11 20 23 15 }}
 
Optimal tunings:
* WE: ~2 = 1198.4257{{c}}, ~21/20 = 88.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1642{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27eg }}
 
Badness (Sintel): 1.19
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 52/51, 78/77, 91/90, 100/99, 133/132, 189/187
 
Mapping: {{mapping| 1 1 1 2 2 2 3 3 | 0 8 18 11 20 23 15 17 }}
 
Optimal tunings:
* WE: ~2 = 1198.5748{{c}}, ~20/19 = 88.1631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1637{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27eg }}
 
Badness (Sintel): 1.09
 
==== Octopod ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 243/242, 245/242
 
Mapping: {{mapping| 1 1 1 2 2 1 | 0 8 18 11 20 37 }}
 
Optimal tunings:
* WE: ~2 = 1200.5116{{c}}, ~21/20 = 87.7346{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7257{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27eff, 41 }}
 
Badness (Sintel): 1.17
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 105/104, 120/119, 154/153, 243/242
 
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 8 18 11 20 37 15 }}
 
Optimal tunings:
* WE: ~2 = 1199.6667{{c}}, ~21/20 = 87.7494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7559{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
 
Badness (Sintel): 1.26
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 100/99, 105/104, 120/119, 133/132, 154/153, 209/208
 
Mapping: {{mapping| 1 1 1 2 2 1 3 3 | 0 8 18 11 20 37 15 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~20/19 = 87.7474{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 87.7476{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
 
Badness (Sintel): 1.19
 
==== Dificot ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 243/242, 245/242, 343/338
 
Mapping: {{mapping| 1 -7 -17 -9 -18 -14 | 0 16 36 22 40 33 }}
: mapping generators: ~2, ~13/9
 
Optimal tunings:
* WE: ~2 = 1199.1496{{c}}, ~13/9 = 643.5328{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 643.9567{{c}}
 
{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}
 
Badness (Sintel): 2.14
 
=== October ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 1375/1372
 
Mapping: {{mapping| 1 1 1 2 5 | 0 8 18 11 -21 }}
 
Optimal tunings:
* WE: ~2 = 1199.8843{{c}}, ~21/20 = 88.0261{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0329{{c}}
 
{{Optimal ET sequence|legend=0| 27, 41, 68, 109, 150, 259 }}
 
Badness (Sintel): 1.31
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 245/243, 275/273, 385/384
 
Mapping: {{mapping| 1 1 1 2 5 4 | 0 8 18 11 -21 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.5060{{c}}, ~21/20 = 88.0388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0697{{c}}
 
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
 
Badness (Sintel): 1.29
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 154/153, 170/169, 196/195, 245/243, 256/255
 
Mapping: {{mapping| 1 1 1 2 5 4 6 | 0 8 18 11 -21 -4 -26 }}
 
Optimal tunings:
* WE: ~2 = 1199.3845{{c}}, ~21/20 = 88.0589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1027{{c}}
 
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
 
Badness (Sintel): 1.37
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 154/153, 170/169, 190/189, 196/195, 209/208, 245/243
 
Mapping: {{mapping| 1 1 1 2 5 4 6 3 | 0 8 18 11 -21 -4 -26 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.4449{{c}}, ~20/19 = 88.0723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1107{{c}}
 
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
 
Badness (Sintel): 1.25
 
== Dodecacot ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3125/3087, 10976/10935
 
{{Mapping|legend=1| 1 1 1 1 | 0 12 27 37 }}
: mapping generators: ~2, ~28/27
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6912{{c}}, ~28/27 = 58.6600{{c}}
: [[error map]]: {{val| -0.309 +1.657 -2.802 +1.287 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 58.6624{{c}}
: error map: {{val| 0.000 +1.993 -2.430 +1.681 }}
 
{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}
 
[[Badness]] (Sintel): 3.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 243/242, 1375/1372
 
Mapping: {{mapping| 1 1 1 1 2 | 0 12 27 37 30 }}
 
Optimal tunings:
* WE: ~2 = 1199.3125{{c}}, ~28/27 = 58.6317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.6360{{c}}
 
{{Optimal ET sequence|legend=0| 20cde, 41 }}
 
Badness (Sintel): 1.97
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 196/195, 243/242, 275/273
 
Mapping: {{mapping| 1 1 1 1 2 2 | 0 12 27 37 30 35 }}
 
Optimal tunings:
* WE: ~2 = 1199.0713{{c}}, ~28/27 = 58.5932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.5982{{c}}
 
{{Optimal ET sequence|legend=0| 20cdef, 41 }}
 
Badness (Sintel): 1.80
 
== Weasel ==
{{See also| No-fives subgroup temperaments #Byhearted }}
 
Weasel, named by [[Mike Battaglia]] in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104304.html Yahoo! Tuning Group | ''This temperament should have a name'']</ref> and also known as ''byhearted''<ref group="note">Alias by [[Xenllium]]. </ref>, tempers out [[50/49]] and splits the octave in halves; its ploidacot is diploid tetracot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 19683/19208
 
{{Mapping|legend=1| 2 2 2 3 | 0 4 9 9 }}
: mapping generators: ~7/5, ~10/9
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6934{{c}}, ~10/9 = 175.5626{{c}}
: [[error map]]: {{val| -0.613 -0.318 -6.864 +10.318 }}
* [[CWE]]: ~7/5 = 1200.0000{{c}}, ~10/9 = 175.5632{{c}}
: error map: {{val| 0.000 +0.298 -6.245 +11.243 }}
 
{{Optimal ET sequence|legend=1| 14c, 34d, 48 }}
 
[[Badness]] (Sintel): 2.82
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 99/98, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 | 0 4 9 9 10 }}
 
Optimal tunings:
* WE: ~7/5 = 599.6525{{c}}, ~10/9 = 175.5103{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.5086{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 34d, 48 }}
 
Badness (Sintel): 1.45
 
=== 13-limit ===
The canonical mapping finds 13/8 at +15 generators rather than using the regular tetracot mapping, in order to find [[15/13]] as being half of [[4/3]].
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 78/77, 99/98, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 3 | 0 4 9 9 10 15 }}
 
Optimal tunings:
* WE: ~7/5 = 599.4539{{c}}, ~10/9 = 175.7393{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.7502{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}
 
Badness (Sintel): 1.32
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 78/77, 85/84, 99/98, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 3 7 | 0 4 9 9 10 15 4 }}
 
Optimal tunings:
* WE: ~7/5 = 599.7509{{c}}, ~10/9 = 175.6684{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6839{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}
 
Badness (Sintel): 1.33
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 50/49, 78/77, 85/84, 99/98, 135/133, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 3 7 5 | 0 4 9 9 10 15 4 12 }}
 
Optimal tunings:
* WE: ~7/5 = 599.6682{{c}}, ~10/9 = 175.5994{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6190{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 20cdefhh, 34dh, 48f }}
 
Badness (Sintel): 1.28
 
=== Weasly ===
{{Todo|review|unify precision}}
The alternative extension uses the same mapping of 13 as in tetracot, though many other intervals of 13 take more generators to reach as a result.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 99/98, 144/143, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 8 | 0 4 9 9 10 -2 }}
 
Optimal tunings:
* WE: ~7/5 = 599.285{{c}}, ~10/9 = 175.641{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.728{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 20cde, 34d, 48 }}
 
Badness (Sintel): 1.72
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 85/84, 99/98, 144/143, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 8 7 | 0 4 9 9 10 -2 4 }}
 
Optimal tunings:
* WE: ~7/5 = 599.494{{c}}, ~10/9 = 175.613{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.681{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 20cde, 34d, 48 }}
 
Badness (Sintel): 1.54
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 50/49, 85/84, 99/98, 144/143, 190/189, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 8 7 5 | 0 4 9 9 10 -2 4 12}}
 
Optimal tunings:
* WE: ~7/5 = 599.464{{c}}, ~10/9 = 175.523{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.593{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 34dh, 48 }}
 
Badness (Sintel): 1.48
 
== 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) ===
[[Subgroup]]: 2.3.5.19
 
[[Comma list]]: 361/360, 20000/19683
 
{{Mapping|legend=2| 1 1 1 3 | 0 8 18 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
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6900{{c}}, ~20/19 = 88.0541{{c}}
: [[error map]]: {{val| -0.310 +2.168 -1.649 -1.523 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/19 = 88.0538{{c}}
: error map: {{val| 0.000 +2.475 -1.345 -0.598 }}
 
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
 
[[Badness]] (Sintel): 1.30
 
=== Devisemi (2.3.5.7.19) ===
Subgroup: 2.3.5.7.19
 
Comma list: 190/189, 245/243, 361/360
 
Subgroup-val mapping: {{mapping| 1 1 1 2 3 | 0 8 18 11 17 }}
 
Gencom mapping: {{mapping| 1 1 1 2 0 0 0 3 | 0 8 18 11 0 0 0 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.7591{{c}}, ~20/19 = 88.0570{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0564{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 27, 41, 68, 109 }}
 
Badness (Sintel): 0.508
 
== Notes ==
<references group="note"/>
 
== References ==
<references/>
 
[[Category:Tetracot family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Tetracot_family"