Gammic family: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-17 04:32:24 UTC</tt>.<br>

: The original revision id was <tt>188883039</tt>.<br>

<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 [[Carlos Gamma]] rank one temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank two microtemperament tempering out |-29 -11 20&gt;. This temperament, gammic, takes 11 generator steps to reach 5/4, and 20 to reach 3/2.The generator in question is 1990656/1953125 = |13 5 -9&gt;, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by [[171edo]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of &lt;&lt;1 -8 -15|| is plainly much less complex than gammic with wedgie &lt;&lt;20 11 -29||, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L33s refinement of the [[34edo]] tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.

Because 171 is such a strong 7-limit system, it is natural to extend gammic to the 7-limit. This we may do by adding 4375/4374 to the comma list, giving a wedgie of &lt;&lt;20 11 96 -29 96 192||. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note MOS is possible.

EDOs: 34, 103, 137, 171, 547, 718, 889, 1607

A more interesting extension is to Neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds 2401/2400 to the gammic comma, and may be described as the 68&amp;171 temperament, with wedgie &lt;&lt;40 22 21 -58 -79 -13||. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending Carlos Gamma.  

Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &lt;&lt;40 22 21 -3 ...|| or 68&amp;103, and 171 can still be used as a tuning, with val &lt;171 271 397 480 591|.

EDOs: 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778

EDOs: 35, 68, 103, 171, 274, 445</pre></div>

<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;Gammic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;a class="wiki_link" href="/Carlos%20Gamma"&gt;Carlos Gamma&lt;/a&gt; rank one temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank two microtemperament tempering out |-29 -11 20&amp;gt;. This temperament, gammic, takes 11 generator steps to reach 5/4, and 20 to reach 3/2.The generator in question is 1990656/1953125 = |13 5 -9&amp;gt;, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt;, &lt;a class="wiki_link" href="/Schismatic%20family"&gt;schismatic&lt;/a&gt; temperament makes for a natural comparison. Schismatic, with a wedgie of &amp;lt;&amp;lt;1 -8 -15|| is plainly much less complex than gammic with wedgie &amp;lt;&amp;lt;20 11 -29||, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L33s refinement of the &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.&lt;br /&gt;

Because 171 is such a strong 7-limit system, it is natural to extend gammic to the 7-limit. This we may do by adding 4375/4374 to the comma list, giving a wedgie of &amp;lt;&amp;lt;20 11 96 -29 96 192||. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note MOS is possible.&lt;br /&gt;

EDOs: 34, 103, 137, 171, 547, 718, 889, 1607&lt;br /&gt;

  A more interesting extension is to Neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds 2401/2400 to the gammic comma, and may be described as the 68&amp;amp;171 temperament, with wedgie &amp;lt;&amp;lt;40 22 21 -58 -79 -13||. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending Carlos Gamma. &lt;br /&gt;

Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &amp;lt;&amp;lt;40 22 21 -3 ...|| or 68&amp;amp;103, and 171 can still be used as a tuning, with val &amp;lt;171 271 397 480 591|.&lt;br /&gt;

EDOs: 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778&lt;br /&gt;