Tetracot family
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[[toc|flat]]
The parent of the **tetracot family** is **tetracot**, the 5-limit temperament [[tempering out]] 20000/19683 = |5 -9 4>, the minimal diesis or tetracot comma. The dual of this comma is the wedgie <<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
Map: [<1 1 1|, <0 4 9|]
EDOs: 14c, 27, 34, 75, 109, 470b, 579b
==Seven limit children==
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===
Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie <<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&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&75 temperament. It has <<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.
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, <<4 9 -15 10 ...|| and 11-limit banya, <<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 <355 563 823 997 1230|, with a 52/355 generator.
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 <<4 9 -15 10 -2 ...|| for 13-limit monkey and <<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.
=Monkey=
Commas: 5120/5103, 875/864
[[POTE tuning|POTE generator]]: 175.659
Map: [<1 1 1 5|, <0 4 9 -15|]
EDOs: 7, 34, 41, 321cd
Badness: 0.0734
==11-limit==
Commas: 243/242, 385/384, 100/99
[[POTE tuning|POTE generator]]: 175.570
Map: [<1 1 1 5 2|, <0 4 9 -15 10|]
EDOs: 7, 34, 41, 123c
Badness: 0.0388
==13-limit==
Commas: 100/99, 105/104, 144/143, 243/242
[[POTE tuning|POTE generator]]: 175.622
Map: [<1 1 1 5 2 4|, <0 4 9 -15 10 -2|]
EDOs: 7, 34, 41
Badness: 0.0284
=Bunya=
Commas: 225/224, 15625/15309
[[POTE tuning|POTE generator]]: 175.741
Map: [<1 1 1 -1|, <0 4 9 26|]
EDOs: 41, 116, 157c, 198c
Badness: 0.0629
==11-limit==
Commas: 100/99, 225/224, 1344/1331
[[POTE tuning|POTE generator]]: 175.777
Map: [<1 1 1 -1 2|, <0 4 9 26 10|]
EDOs: 41, 116e, 157ce
Badness: 0.0313
==13-limit==
Commas: 100/99, 144/143, 225/224, 243/242
[[POTE tuning|POTE generator]]: 175.886
Map: [<1 1 1 -1 2 4|, <0 4 9 26 10 -2|]
EDOs: 34d, 41, 75e, 116ef
Badness: 0.0249
=Modus=
Commas: 64/63, 4375/4374
POTE generator: ~10/9 = 177.203
Map: [<1 1 1 4|, <0 4 9 -8|]
EDOs: 7, 27, 61d, 88bcd
Badness: 0.0682
==11-limit==
Commas: 64/63, 100/99, 243/242
POTE generator: ~10/9 = 177.053
Map: [<1 1 1 4 2|, <0 4 9 -8 10|]
EDOs: 7, 20ce, 27e, 34d, 61de
Badness: 0.0351
==13-limit==
Commas: 64/63, 78/77, 100/99, 144/143
POTE generator: ~10/9 = 176.953
Map: [<1 1 1 4 2 4|, <0 4 9 -8 10 -2|]
EDOs: 7, 27e, 34d, 61de
Badness: 0.0238
=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.
Commas: 55/54, 64/63, 363/350
POTE generator: ~10/9 = 177.200
Map: [<1 1 1 4 3|, <0 4 9 -8 3|]
EDOs: 7, 20c, 27, 61de, 88bcde
Badness: 0.0631
==13-limit==
Commas: 55/54, 64/63, 66/65, 143/140
POTE generator: ~10/9 = 177.197
Map: [<1 1 1 4 3 4|, <0 4 9 -8 3 -2|]
EDOs: 7, 20c, 27, 61de, 88bcde
Badness: 0.039
=Wollemia=
Commas: 126/125, 2240/2187
POTE generator: ~10/9 = 177.357
Map: [<1 1 1 0|, <0 4 9 19|]
Wedgie: <<4 9 19 5 19 19||
EDOs: 27, 61, 88bc, 115bc
Badness: 0.0705
==11-limit==
Commas: 56/55, 100/99, 243/242
POTE generator: ~10/9 = 177.413
Map: [<1 1 1 0 2|, <0 4 9 19 10|]
EDOs: 27e, 34, 61e
Badness: 0.0376
==13-limit==
Commas: 56/55, 91/90, 100/99, 352/351
POTE generator: ~10/9 = 177.231
Map: [<1 1 1 0 2 4|, <0 4 9 19 10 -2|]
EDOs: 27e, 34, 61e
Badness: 0.0312
=Octacot=
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 <<8 18 11 10 -5 -25|| and may also be described as 41&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 <<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.
Commas: 245/243, 2401/2400
[[POTE tuning|POTE generator]]: 88.076
Map: [<1 1 1 2|, <0 8 18 11|]
EDOs: 14c, 27, 41, 68, 109
==11-limit==
Commas: 100/99, 243/242, 245/242
[[POTE tuning|POTE generator]]: 87.975
Map: [<1 1 1 2 2|, <0 8 18 11 20|]
EDOs: 27e, 41, 109e, 150e, 191e
See also: [[Chords of octacot]]
==13-limit==
Commas: 100/99, 144/143, 196/195, 243/242
[[POTE tuning|POTE generator]]: ~22/21 = 88.106
Map: [<1 1 1 2 2 4|, <0 8 18 11 20 -4|]
EDOs: 27e, 41, 68e, 109ef
Badness: 0.0233
==Octocat==
Commas: 78/77, 91/90, 100/99, 245/242
POTE generator: ~22/21 = 88.179
Map: [<1 1 1 2 2 2|, <0 8 18 11 20 23|]
EDOs: 27e, 41f, 68ef
Badness: 0.0276
==Octopod==
Commas: 100/99 105/104 243/242 245/242
POTE generator: ~22/21 = 87.697
Map: [<1 1 1 2 2 1|, <0 8 18 11 20 37|]
EDOs: 41, 137cd, 178cd
Badness: 0.0283
=Dificot=
Commas: 100/99, 243/242, 245/242, 343/338
POTE generator: ~13/9 = 643.989
Map: [<1 9 19 13 22 19|, <0 -16 -36 -22 -40 -33|]
EDOs: 41
Badness: 0.0519
=Duodecicot=
Commas: 3087/3125, 10976/10935
POTE generator: ~28/27 = 58.675
Map: [<1 1 1 1|, <0 12 27 37|]
Wedgie: <<12 27 37 15 25 10||
EDOs: 41, 184, 225, 409bcd
Badness: 0.1198
==Musical Examples==
* [[http://soundcloud.com/dustin-schallert/tetracot-perc-sitar|Tetracot Perc-Sitar]] by Dustin Schallert
* [[http://soundcloud.com/dustin-schallert/tetracot-jam|Tetracot Jam]] by Dustin SchallertOriginal HTML content:
<html><head><title>Tetracot family</title></head><body><!-- ws:start:WikiTextTocRule:48:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --> | <a href="#Monkey">Monkey</a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --> | <a href="#Bunya">Bunya</a><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --> | <a href="#Modus">Modus</a><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --> | <a href="#Ponens">Ponens</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Wollemia">Wollemia</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --> | <a href="#Octacot">Octacot</a><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | <a href="#Dificot">Dificot</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --> | <a href="#Duodecicot">Duodecicot</a><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: -->
<!-- ws:end:WikiTextTocRule:73 -->The parent of the <strong>tetracot family</strong> is <strong>tetracot</strong>, the 5-limit temperament <a class="wiki_link" href="/tempering%20out">tempering out</a> 20000/19683 = |5 -9 4>, the minimal diesis or tetracot comma. The dual of this comma is the wedgie <<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 <a class="wiki_link" href="/34edo">34edo</a> 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.<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 176.160<br />
<br />
Map: [<1 1 1|, <0 4 9|]<br />
EDOs: 14c, 27, 34, 75, 109, 470b, 579b<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2>
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Seven limit children-Monkey and Bunya"></a><!-- ws:end:WikiTextHeadingRule:2 -->Monkey and Bunya</h3>
Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie <<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&41 temperament, if the vals in question are taken to be <a class="wiki_link" href="/Patent%20val">patent vals</a>, meaning that n*log2(prime) rounded to the nearest integer gives the mapping. <a class="wiki_link" href="/41edo">41edo</a> is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.<br />
<br />
Bunya, the bunya-bunya tree temperament, adds 225/224 to the list of commas and may be described as the 41&75 temperament. It has <<4 9 26 5 30 35|| as a wedgie, and <a class="wiki_link" href="/41edo">41edo</a> 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.<br />
<br />
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, <<4 9 -15 10 ...|| and 11-limit banya, <<4 9 26 10...||. Again, <a class="wiki_link" href="/41edo">41edo</a> 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 <355 563 823 997 1230|, with a 52/355 generator.<br />
<br />
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 <<4 9 -15 10 -2 ...|| for 13-limit monkey and <<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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Monkey"></a><!-- ws:end:WikiTextHeadingRule:4 -->Monkey</h1>
Commas: 5120/5103, 875/864<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.659<br />
<br />
Map: [<1 1 1 5|, <0 4 9 -15|]<br />
EDOs: 7, 34, 41, 321cd<br />
Badness: 0.0734<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Monkey-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2>
Commas: 243/242, 385/384, 100/99<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.570<br />
<br />
Map: [<1 1 1 5 2|, <0 4 9 -15 10|]<br />
EDOs: 7, 34, 41, 123c<br />
Badness: 0.0388<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Monkey-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2>
Commas: 100/99, 105/104, 144/143, 243/242<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.622<br />
<br />
Map: [<1 1 1 5 2 4|, <0 4 9 -15 10 -2|]<br />
EDOs: 7, 34, 41<br />
Badness: 0.0284<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Bunya"></a><!-- ws:end:WikiTextHeadingRule:10 -->Bunya</h1>
Commas: 225/224, 15625/15309<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.741<br />
<br />
Map: [<1 1 1 -1|, <0 4 9 26|]<br />
EDOs: 41, 116, 157c, 198c<br />
Badness: 0.0629<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Bunya-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2>
Commas: 100/99, 225/224, 1344/1331<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.777<br />
<br />
Map: [<1 1 1 -1 2|, <0 4 9 26 10|]<br />
EDOs: 41, 116e, 157ce<br />
Badness: 0.0313<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Bunya-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2>
Commas: 100/99, 144/143, 225/224, 243/242<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.886<br />
<br />
Map: [<1 1 1 -1 2 4|, <0 4 9 26 10 -2|]<br />
EDOs: 34d, 41, 75e, 116ef<br />
Badness: 0.0249<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Modus"></a><!-- ws:end:WikiTextHeadingRule:16 -->Modus</h1>
Commas: 64/63, 4375/4374<br />
<br />
POTE generator: ~10/9 = 177.203<br />
<br />
Map: [<1 1 1 4|, <0 4 9 -8|]<br />
EDOs: 7, 27, 61d, 88bcd<br />
Badness: 0.0682<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Modus-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2>
Commas: 64/63, 100/99, 243/242<br />
<br />
POTE generator: ~10/9 = 177.053<br />
<br />
Map: [<1 1 1 4 2|, <0 4 9 -8 10|]<br />
EDOs: 7, 20ce, 27e, 34d, 61de<br />
Badness: 0.0351<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="Modus-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2>
Commas: 64/63, 78/77, 100/99, 144/143<br />
<br />
POTE generator: ~10/9 = 176.953<br />
<br />
Map: [<1 1 1 4 2 4|, <0 4 9 -8 10 -2|]<br />
EDOs: 7, 27e, 34d, 61de<br />
Badness: 0.0238<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Ponens"></a><!-- ws:end:WikiTextHeadingRule:22 -->Ponens</h1>
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.<br />
<br />
Commas: 55/54, 64/63, 363/350<br />
<br />
POTE generator: ~10/9 = 177.200<br />
<br />
Map: [<1 1 1 4 3|, <0 4 9 -8 3|]<br />
EDOs: 7, 20c, 27, 61de, 88bcde<br />
Badness: 0.0631<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="Ponens-13-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->13-limit</h2>
Commas: 55/54, 64/63, 66/65, 143/140<br />
<br />
POTE generator: ~10/9 = 177.197<br />
<br />
Map: [<1 1 1 4 3 4|, <0 4 9 -8 3 -2|]<br />
EDOs: 7, 20c, 27, 61de, 88bcde<br />
Badness: 0.039<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:<h1> --><h1 id="toc13"><a name="Wollemia"></a><!-- ws:end:WikiTextHeadingRule:26 -->Wollemia</h1>
Commas: 126/125, 2240/2187<br />
<br />
POTE generator: ~10/9 = 177.357<br />
<br />
Map: [<1 1 1 0|, <0 4 9 19|]<br />
Wedgie: <<4 9 19 5 19 19||<br />
EDOs: 27, 61, 88bc, 115bc<br />
Badness: 0.0705<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc14"><a name="Wollemia-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h2>
Commas: 56/55, 100/99, 243/242<br />
<br />
POTE generator: ~10/9 = 177.413<br />
<br />
Map: [<1 1 1 0 2|, <0 4 9 19 10|]<br />
EDOs: 27e, 34, 61e<br />
Badness: 0.0376<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="Wollemia-13-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->13-limit</h2>
Commas: 56/55, 91/90, 100/99, 352/351<br />
<br />
POTE generator: ~10/9 = 177.231<br />
<br />
Map: [<1 1 1 0 2 4|, <0 4 9 19 10 -2|]<br />
EDOs: 27e, 34, 61e<br />
Badness: 0.0312<br />
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<!-- ws:start:WikiTextHeadingRule:32:<h1> --><h1 id="toc16"><a name="Octacot"></a><!-- ws:end:WikiTextHeadingRule:32 -->Octacot</h1>
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 <<8 18 11 10 -5 -25|| and may also be described as 41&68. <a class="wiki_link" href="/68edo">68edo</a> or <a class="wiki_link" href="/109edo">109edo</a> can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is <a class="wiki_link" href="/150edo">150edo</a>, which has a generator, 11/150, of exactly 88 cents. This relates octacot to the <a class="wiki_link" href="/88cET">88cET</a> non-octave temperament, which like <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a> arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them.<br />
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Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving <<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.<br />
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Commas: 245/243, 2401/2400<br />
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 88.076<br />
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Map: [<1 1 1 2|, <0 8 18 11|]<br />
EDOs: 14c, 27, 41, 68, 109<br />
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<!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Octacot-11-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->11-limit</h2>
Commas: 100/99, 243/242, 245/242<br />
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 87.975<br />
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Map: [<1 1 1 2 2|, <0 8 18 11 20|]<br />
EDOs: 27e, 41, 109e, 150e, 191e<br />
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See also: <a class="wiki_link" href="/Chords%20of%20octacot">Chords of octacot</a><br />
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<!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Octacot-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2>
Commas: 100/99, 144/143, 196/195, 243/242<br />
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~22/21 = 88.106<br />
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Map: [<1 1 1 2 2 4|, <0 8 18 11 20 -4|]<br />
EDOs: 27e, 41, 68e, 109ef<br />
Badness: 0.0233<br />
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<!-- ws:start:WikiTextHeadingRule:38:<h2> --><h2 id="toc19"><a name="Octacot-Octocat"></a><!-- ws:end:WikiTextHeadingRule:38 -->Octocat</h2>
Commas: 78/77, 91/90, 100/99, 245/242<br />
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POTE generator: ~22/21 = 88.179<br />
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Map: [<1 1 1 2 2 2|, <0 8 18 11 20 23|]<br />
EDOs: 27e, 41f, 68ef<br />
Badness: 0.0276<br />
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<!-- ws:start:WikiTextHeadingRule:40:<h2> --><h2 id="toc20"><a name="Octacot-Octopod"></a><!-- ws:end:WikiTextHeadingRule:40 -->Octopod</h2>
Commas: 100/99 105/104 243/242 245/242<br />
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POTE generator: ~22/21 = 87.697<br />
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Map: [<1 1 1 2 2 1|, <0 8 18 11 20 37|]<br />
EDOs: 41, 137cd, 178cd<br />
Badness: 0.0283<br />
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<!-- ws:start:WikiTextHeadingRule:42:<h1> --><h1 id="toc21"><a name="Dificot"></a><!-- ws:end:WikiTextHeadingRule:42 -->Dificot</h1>
Commas: 100/99, 243/242, 245/242, 343/338<br />
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POTE generator: ~13/9 = 643.989<br />
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Map: [<1 9 19 13 22 19|, <0 -16 -36 -22 -40 -33|]<br />
EDOs: 41<br />
Badness: 0.0519<br />
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<!-- ws:start:WikiTextHeadingRule:44:<h1> --><h1 id="toc22"><a name="Duodecicot"></a><!-- ws:end:WikiTextHeadingRule:44 -->Duodecicot</h1>
Commas: 3087/3125, 10976/10935<br />
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POTE generator: ~28/27 = 58.675<br />
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Map: [<1 1 1 1|, <0 12 27 37|]<br />
Wedgie: <<12 27 37 15 25 10||<br />
EDOs: 41, 184, 225, 409bcd<br />
Badness: 0.1198<br />
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<!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc23"><a name="Duodecicot-Musical Examples"></a><!-- ws:end:WikiTextHeadingRule:46 -->Musical Examples</h2>
<ul><li><a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-perc-sitar" rel="nofollow">Tetracot Perc-Sitar</a> by Dustin Schallert</li><li><a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-jam" rel="nofollow">Tetracot Jam</a> by Dustin Schallert</li></ul></body></html>