Tetracot family: Difference between revisions

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.

=Tetracot=  

[[POTE tuning|POTE generator]]: 176.160

==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. 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.

 

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.

=Monkey=  

[[POTE tuning|POTE generator]]: 175.659

==11-limit==  

[[POTE tuning|POTE generator]]: 175.570

==13-limit==  

[[POTE tuning|POTE generator]]: 175.622

=Bunya=  

[[POTE tuning|POTE generator]]: 175.741

==11-limit==  

[[POTE tuning|POTE generator]]: 175.777

==13-limit==  

[[POTE tuning|POTE generator]]: 175.886

=Modus=  

==11-limit==  

==13-limit==  

===Musical Examples===  

[[http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3|Tetracot Perc-Sitar]] by [[http://soundcloud.com/dustin-schallert/tetracot-perc-sitar|Dustin Schallert]]

[[http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3|Tetracot Jam]] by [[http://soundcloud.com/dustin-schallert/tetracot-jam|Dustin Schallert]]

[[http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3|Tetracot Pump]] by [[http://soundcloud.com/dustin-schallert/tetracot-pump|Dustin Schallert]] all in [[27edo]]

=Ponens=  

==13-limit==  

=Wollemia=  

==11-limit==  

==13-limit==  

=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.

[[POTE tuning|POTE generator]]: 88.076

==11-limit==  

[[POTE tuning|POTE generator]]: 87.975

See also: [[Chords of octacot]]

==13-limit==  

[[POTE tuning|POTE generator]]: ~22/21 = 88.106

==Octocat==  

==Octopod==  

=Dificot=  

=Dodecacot=  

Badness: 0.1198</pre></div>

Map: [&amp;lt;1 1 1|, &amp;lt;0 4 9|]&lt;br /&gt;

EDOs: 14c, 27, 34, 75, 109, 470b, 579b&lt;br /&gt;

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. Adding 245/243 gives octacot, which splits the generator in half.&lt;br /&gt;

Map: [&amp;lt;1 1 1 5|, &amp;lt;0 4 9 -15|]&lt;br /&gt;

Map: [&amp;lt;1 1 1 5 2|, &amp;lt;0 4 9 -15 10|]&lt;br /&gt;

Map: [&amp;lt;1 1 1 5 2 4|, &amp;lt;0 4 9 -15 10 -2|]&lt;br /&gt;

Map: [&amp;lt;1 1 1 -1|, &amp;lt;0 4 9 26|]&lt;br /&gt;

Badness: 0.0629&lt;br /&gt;

&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3" rel="nofollow"&gt;Tetracot Perc-Sitar&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-perc-sitar" rel="nofollow"&gt;Dustin Schallert&lt;/a&gt;&lt;br /&gt;

Map: [&amp;lt;1 1 1 4 3|, &amp;lt;0 4 9 -8 3|]&lt;br /&gt;

Map: [&amp;lt;1 1 1 4 3 4|, &amp;lt;0 4 9 -8 3 -2|]&lt;br /&gt;

Map: [&amp;lt;1 1 1 0|, &amp;lt;0 4 9 19|]&lt;br /&gt;

Map: [&amp;lt;1 1 1 0 2|, &amp;lt;0 4 9 19 10|]&lt;br /&gt;

EDOs: 27e, 34, 61e&lt;br /&gt;

Map: [&amp;lt;1 1 1 2 2 4|, &amp;lt;0 8 18 11 20 -4|]&lt;br /&gt;

Map: [&amp;lt;1 1 1 2 2 1|, &amp;lt;0 8 18 11 20 37|]&lt;br /&gt;