Gammic family
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Original Wikitext content:
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>. This temperament, gammic, takes five generator steps to reach 5/4, and 20 to reach 3/2.The generator in question is 1990656/1953125 = |13 5 -9>, 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 171-et, [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of <<1 -8 -15|| is plainly much less complex than gammic with wedgie <<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 <<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. ===Neptune=== 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&171 temperament, with wedgie <<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 <<40 22 21 -3 ...|| or 68&103, and 171 can still be used as a tuning, with val <171 271 397 480 591|. An article on Neptune as an analog of miracle can be found [[http://tech.groups.yahoo.com/group/tuning-math/message/6001|here]].
Original HTML content:
<html><head><title>Gammic family</title></head><body>The <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a> 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>. This temperament, gammic, takes five generator steps to reach 5/4, and 20 to reach 3/2.The generator in question is 1990656/1953125 = |13 5 -9>, 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 171-et, <a class="wiki_link" href="/Schismatic%20family">schismatic</a> temperament makes for a natural comparison. Schismatic, with a wedgie of <<1 -8 -15|| is plainly much less complex than gammic with wedgie <<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 <a class="wiki_link" href="/34edo">34edo</a> 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.<br /> <br /> 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 <<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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h3> --><h3 id="toc0"><a name="x--Neptune"></a><!-- ws:end:WikiTextHeadingRule:0 -->Neptune</h3> 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&171 temperament, with wedgie <<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. <a class="wiki_link" href="/171edo">171edo</a> 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. <br /> <br /> 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 <<40 22 21 -3 ...|| or 68&103, and 171 can still be used as a tuning, with val <171 271 397 480 591|.<br /> <br /> An article on Neptune as an analog of miracle can be found <a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/6001" rel="nofollow">here</a>.</body></html>