Revision as of 00:00, 17 July 2018

If c is a comma, then clipper(c) is defined as Euler(Benedetti(c)), tempered by the codimension one temperament tempering out c. Here Euler(N) is the Euler genus, the divisors of the integer N reduced to the octave, and Benedetti(c) is the Benedetti height of c = p/q, which is p*q if p/q is reduced to its lowest terms. Euler(Benedetti(c)) has exactly one interval of size c, which is removed when c is tempered out. Two transversals of clipper(c) are obtained by leaving out either one or the other of the pair of scale intervals separated by c.

Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a normal interval list. This group is characteristic of the comma, and is the group on which tempering by the comma takes place. For instance, Euler(Benedetti(225/224)) generates 2.3.5.7, the full 7-limit group, and tempering it out leads to a rank three temperament, marvel. However, Euler(Benedeti(3136/3125)) generates 2.5.7, and tempering it out generates a rank-two temperament of the 2.5.7 JI subgroup, with mapping [<1 0 -3|, <0 2 5|] and an approximate 28/25 generator, which might be called 7-limit roulette temperament.

Scales

clipper(1029/1024), 7 notes, 2.3.7

clipper(81/80), 9 notes, 5-limit

clipper(3125/3072), 11 notes, 5-limit

clipper(121/120), 11 notes, 2.3.5.11

clipper(176/175), 11 notes, 2.5.7.11

clipper(65536/65219), 11 notes, 2.7.11

clipper(144/143), 11 notes, 2.3.11.13

clipper(169/168), 11 notes, 2.3.7.13

clipper(640/637), 11 notes, 2.5.7.13

clipper(2048/2025), 14 notes, 5-limit

clipper(385/384), 15 notes, 11-limit

clipper(105/104), 15 notes, 2.3.5.7.13

clipper(225/224), 17 notes, 7-limit

clipper(32805/32768), 17 notes, 5-limit

clipper(3136/3125), 17 notes, 2.5.7

clipper(99/98), 17 notes, 2.3.7.11

clipper(100/99), 17 notes, 2.3.5.11

clipper(243/242), 17 notes, 2.3.11

clipper(245/242), 17 notes, 2.5.7.11

clipper(896/891), 19 notes, 2.3.7.11

clipper(625/624), 19 notes, 2.3.5.13

clipper(126/125), 23 notes, 7-limit

clipper(6144/6125), 23 notes, 7-limit

clipper(65625/65536), 23 notes, 7-limit

clipper(5120/5103), 27 notes, 7-limit

clipper(4000/3993), 31 notes, 2.3.5.11

clipper(245/243), 35 notes, 7-limit

Links

http://tech.groups.yahoo.com/group/tuning-math/message/11429

http://tech.groups.yahoo.com/group/tuning-math/message/11432

http://tech.groups.yahoo.com/group/tuning-math/message/11439

http://tech.groups.yahoo.com/group/tuning-math/message/11441

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+If c is a [[Comma|comma]], then clipper(c) is defined as Euler(Benedetti(c)), tempered by the codimension one temperament tempering out c. Here Euler(N) is the [[Euler_genera|Euler genus]], the divisors of the integer N reduced to the octave, and Benedetti(c) is the [[Benedetti_height|Benedetti height]] of c = p/q, which is p*q if p/q is reduced to its lowest terms. Euler(Benedetti(c)) has exactly one interval of size c, which is removed when c is tempered out. Two [[Transversal|transversals]] of clipper(c) are obtained by leaving out either one or the other of the pair of scale intervals separated by c.
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-03 12:20:05 UTC</tt>.<br>
-: The original revision id was <tt>465629208</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<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">If c is a [[comma]], then clipper(c) is defined as Euler(Benedetti(c)), tempered by the codimension one temperament tempering out c. Here Euler(N) is the [[Euler genera|Euler genus]], the divisors of the integer N reduced to the octave, and Benedetti(c) is the [[Benedetti height]] of c = p/q, which is p*q if p/q is reduced to its lowest terms. Euler(Benedetti(c)) has exactly one interval of size c, which is removed when c is tempered out. Two [[Transversal|transversals]] of clipper(c) are obtained by leaving out either one or the other of the pair of scale intervals separated by c.
-Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a [[Normal lists#x-Normal interval lists|normal interval list]]. This group is characteristic of the comma, and is the group on which tempering by the comma takes place. For instance, Euler(Benedetti(225/224)) generates 2.3.5.7, the full 7-limit group, and tempering it out leads to a rank three temperament, marvel. However, Euler(Benedeti(3136/3125)) generates 2.5.7, and tempering it out generates a rank-two temperament of the 2.5.7 [[Just intonation subgroups|JI subgroup]], with mapping [&lt;1 0 -3|, &lt;0 2 5|] and an approximate 28/25 generator, which might be called 7-limit [[Chromatic pairs#Roulette|roulette]] temperament.
+Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a [[Normal_lists#x-Normal interval lists|normal interval list]]. This group is characteristic of the comma, and is the group on which tempering by the comma takes place. For instance, Euler(Benedetti(225/224)) generates 2.3.5.7, the full 7-limit group, and tempering it out leads to a rank three temperament, marvel. However, Euler(Benedeti(3136/3125)) generates 2.5.7, and tempering it out generates a rank-two temperament of the 2.5.7 [[Just_intonation_subgroups|JI subgroup]], with mapping [&lt;1 0 -3|, &lt;0 2 5|] and an approximate 28/25 generator, which might be called 7-limit [[Chromatic_pairs#Roulette|roulette]] temperament.
 =Scales=
 [[clipper1029|clipper(1029/1024)]], 7 notes, 2.3.7
 [[clipper81|clipper(81/80)]], 9 notes, 5-limit
 [[clipper3125|clipper(3125/3072)]], 11 notes, 5-limit
 [[clipper121|clipper(121/120)]], 11 notes, 2.3.5.11
 [[clipper176|clipper(176/175)]], 11 notes, 2.5.7.11
 [[clipper65536|clipper(65536/65219)]], 11 notes, 2.7.11
 [[clipper144|clipper(144/143)]], 11 notes, 2.3.11.13
 [[clipper169|clipper(169/168)]], 11 notes, 2.3.7.13
 [[clipper640|clipper(640/637)]], 11 notes, 2.5.7.13
 [[clipper2048|clipper(2048/2025)]], 14 notes, 5-limit
 [[clipper385|clipper(385/384)]], 15 notes, 11-limit
 [[clipper105|clipper(105/104)]], 15 notes, 2.3.5.7.13
 [[clipper225|clipper(225/224)]], 17 notes, 7-limit
 [[clipper32805|clipper(32805/32768)]], 17 notes, 5-limit
 [[clipper3136|clipper(3136/3125)]], 17 notes, 2.5.7
 [[clipper99|clipper(99/98)]], 17 notes, 2.3.7.11
 [[clipper100|clipper(100/99)]], 17 notes, 2.3.5.11
 [[clipper243|clipper(243/242)]], 17 notes, 2.3.11
 [[clipper245242|clipper(245/242)]], 17 notes, 2.5.7.11
 [[clipper896|clipper(896/891)]], 19 notes, 2.3.7.11
 [[clipper625|clipper(625/624)]], 19 notes, 2.3.5.13
 [[clipper126|clipper(126/125)]], 23 notes, 7-limit
 [[clipper6144|clipper(6144/6125)]], 23 notes, 7-limit
 [[clipper65625|clipper(65625/65536)]], 23 notes, 7-limit
 [[clipper5120|clipper(5120/5103)]], 27 notes, 7-limit
 [[clipper4000|clipper(4000/3993)]], 31 notes, 2.3.5.11
 [[clipper245|clipper(245/243)]], 35 notes, 7-limit
 =Links=
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11429]]
+[http://tech.groups.yahoo.com/group/tuning-math/message/11429 http://tech.groups.yahoo.com/group/tuning-math/message/11429]
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11432]]
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11439]]
+[http://tech.groups.yahoo.com/group/tuning-math/message/11432 http://tech.groups.yahoo.com/group/tuning-math/message/11432]
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11441]]
-</pre></div>
+[http://tech.groups.yahoo.com/group/tuning-math/message/11439 http://tech.groups.yahoo.com/group/tuning-math/message/11439]
-<h4>Original HTML content:</h4>
-<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;Clippers&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If c is a &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;, then clipper(c) is defined as Euler(Benedetti(c)), tempered by the codimension one temperament tempering out c. Here Euler(N) is the &lt;a class="wiki_link" href="/Euler%20genera"&gt;Euler genus&lt;/a&gt;, the divisors of the integer N reduced to the octave, and Benedetti(c) is the &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt; of c = p/q, which is p*q if p/q is reduced to its lowest terms. Euler(Benedetti(c)) has exactly one interval of size c, which is removed when c is tempered out. Two &lt;a class="wiki_link" href="/Transversal"&gt;transversals&lt;/a&gt; of clipper(c) are obtained by leaving out either one or the other of the pair of scale intervals separated by c.&lt;br /&gt;
+[http://tech.groups.yahoo.com/group/tuning-math/message/11441 http://tech.groups.yahoo.com/group/tuning-math/message/11441]
-&lt;br /&gt;
-Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a &lt;a class="wiki_link" href="/Normal%20lists#x-Normal interval lists"&gt;normal interval list&lt;/a&gt;. This group is characteristic of the comma, and is the group on which tempering by the comma takes place. For instance, Euler(Benedetti(225/224)) generates 2.3.5.7, the full 7-limit group, and tempering it out leads to a rank three temperament, marvel. However, Euler(Benedeti(3136/3125)) generates 2.5.7, and tempering it out generates a rank-two temperament of the 2.5.7 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;JI subgroup&lt;/a&gt;, with mapping [&amp;lt;1 0 -3|, &amp;lt;0 2 5|] and an approximate 28/25 generator, which might be called 7-limit &lt;a class="wiki_link" href="/Chromatic%20pairs#Roulette"&gt;roulette&lt;/a&gt; temperament.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scales&lt;/h1&gt;
-&lt;a class="wiki_link" href="/clipper1029"&gt;clipper(1029/1024)&lt;/a&gt;, 7 notes, 2.3.7&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper81"&gt;clipper(81/80)&lt;/a&gt;, 9 notes, 5-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper3125"&gt;clipper(3125/3072)&lt;/a&gt;, 11 notes, 5-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper121"&gt;clipper(121/120)&lt;/a&gt;, 11 notes, 2.3.5.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper176"&gt;clipper(176/175)&lt;/a&gt;, 11 notes, 2.5.7.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper65536"&gt;clipper(65536/65219)&lt;/a&gt;, 11 notes, 2.7.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper144"&gt;clipper(144/143)&lt;/a&gt;, 11 notes, 2.3.11.13&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper169"&gt;clipper(169/168)&lt;/a&gt;, 11 notes, 2.3.7.13&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper640"&gt;clipper(640/637)&lt;/a&gt;, 11 notes, 2.5.7.13&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper2048"&gt;clipper(2048/2025)&lt;/a&gt;, 14 notes, 5-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper385"&gt;clipper(385/384)&lt;/a&gt;, 15 notes, 11-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper105"&gt;clipper(105/104)&lt;/a&gt;, 15 notes, 2.3.5.7.13&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper225"&gt;clipper(225/224)&lt;/a&gt;, 17 notes, 7-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper32805"&gt;clipper(32805/32768)&lt;/a&gt;, 17 notes, 5-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper3136"&gt;clipper(3136/3125)&lt;/a&gt;, 17 notes, 2.5.7&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper99"&gt;clipper(99/98)&lt;/a&gt;, 17 notes, 2.3.7.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper100"&gt;clipper(100/99)&lt;/a&gt;, 17 notes, 2.3.5.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper243"&gt;clipper(243/242)&lt;/a&gt;, 17 notes, 2.3.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper245242"&gt;clipper(245/242)&lt;/a&gt;, 17 notes, 2.5.7.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper896"&gt;clipper(896/891)&lt;/a&gt;, 19 notes, 2.3.7.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper625"&gt;clipper(625/624)&lt;/a&gt;, 19 notes, 2.3.5.13&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper126"&gt;clipper(126/125)&lt;/a&gt;, 23 notes, 7-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper6144"&gt;clipper(6144/6125)&lt;/a&gt;, 23 notes, 7-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper65625"&gt;clipper(65625/65536)&lt;/a&gt;, 23 notes, 7-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper5120"&gt;clipper(5120/5103)&lt;/a&gt;, 27 notes, 7-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper4000"&gt;clipper(4000/3993)&lt;/a&gt;, 31 notes, 2.3.5.11&lt;br /&gt;
-&lt;a class="wiki_link" href="/clipper245"&gt;clipper(245/243)&lt;/a&gt;, 35 notes, 7-limit&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Links&lt;/h1&gt;
-&lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11429" rel="nofollow"&gt;http://tech.groups.yahoo.com/group/tuning-math/message/11429&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11432" rel="nofollow"&gt;http://tech.groups.yahoo.com/group/tuning-math/message/11432&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11439" rel="nofollow"&gt;http://tech.groups.yahoo.com/group/tuning-math/message/11439&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11441" rel="nofollow"&gt;http://tech.groups.yahoo.com/group/tuning-math/message/11441&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Clipper: Difference between revisions

Revision as of 00:00, 17 July 2018

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Clipper: Difference between revisions

Revision as of 00:00, 17 July 2018

Scales

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