Gammic family
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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 11 [[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 [[171edo]], [[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. [[POTE tuning|POTE generator]]: 35.096 Map: [<1 1 2|, <0 20 11|] EDOs: [[34edo|34]], 103, 137, 171, 547, 718, 889, 1607 7-limit Commas: 4375/4374, 6591796875/6576668672 [[POTE tuning|POTE generator]]: 35.090 Map: [<1 1 2 0|, <0 20 11 96|] EDOs: 171, 1402, 1573, 1744, 1915 ===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]]. [[POTE tuning|POTE generator]]: 582.452 Map: [<1 21 13 13|, <0 -40 -22 -21|] Generators: 2, 7/5 EDOs: [[35edo|35]], [[68edo|68]], 103, 171, 1094, 1265, 1436, 1607, 1778 11-limit Commas: 385/384, 1375/1372, 2465529759/2441406250 [[POTE tuning|POTE generator]]: 582.475 Map: [1 21 13 13 2|, <0 -40 -22 -21 3|] Generators: 2, 7/5 EDOs: 35, 68, 103, 171, 274, 445
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 11 <a class="wiki_link" href="/generator">generator</a> 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 <a class="wiki_link" href="/171edo">171edo</a>, <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 <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a> if used for it.<br /> <br /> Because 171 is such a strong <a class="wiki_link" href="/7-limit">7-limit</a> 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 /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 35.096<br /> <br /> Map: [<1 1 2|, <0 20 11|]<br /> EDOs: <a class="wiki_link" href="/34edo">34</a>, 103, 137, 171, 547, 718, 889, 1607<br /> <br /> 7-limit<br /> Commas: 4375/4374, 6591796875/6576668672<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 35.090<br /> <br /> Map: [<1 1 2 0|, <0 20 11 96|]<br /> EDOs: 171, 1402, 1573, 1744, 1915<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 <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a>. <br /> <br /> Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the <a class="wiki_link" href="/11-limit">11-limit</a>, 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 <a class="wiki_link" href="/val">val</a> <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>.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 582.452<br /> <br /> Map: [<1 21 13 13|, <0 -40 -22 -21|]<br /> Generators: 2, 7/5<br /> EDOs: <a class="wiki_link" href="/35edo">35</a>, <a class="wiki_link" href="/68edo">68</a>, 103, 171, 1094, 1265, 1436, 1607, 1778<br /> <br /> 11-limit<br /> Commas: 385/384, 1375/1372, 2465529759/2441406250<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 582.475<br /> <br /> Map: [1 21 13 13 2|, <0 -40 -22 -21 3|]<br /> Generators: 2, 7/5<br /> EDOs: 35, 68, 103, 171, 274, 445</body></html>