Clipper: Difference between revisions

<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 mis 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 [[http://xenharmonic.wikispaces.com/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.

A comma such that the smallest interval in Euler(Benedetti(c)) is c may be called a //clipper comma//; clipper commas would seem to make for superior clippers, avoiding very small intervals. Some 7-limit clipper commas are 16807/16384, 525/512, 128/125, 49/48, 50/49, 3125/3072, 64/63, 81/80, 2048/2025, 245/243, 2109375/2097152, 1029/1024, 15625/15552, 225/224, 3136/3125, 5120/5103, 6144/6125, 2100875/2097152, 5250987/5242880, 65625/65536, 32805/32768, 703125/702464, 2401/2400, 4375/4374 and

250047/250000.

[[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]]

<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 mis 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;

Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a [[&lt;!-- ws:start:WikiTextUrlRule:22:http://xenharmonic.wikispaces.com/Normal+lists#x-Normal --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Normal+lists#x-Normal"&gt;http://xenharmonic.wikispaces.com/Normal+lists#x-Normal&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:22 --&gt; 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 &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;

A comma such that the smallest interval in Euler(Benedetti(c)) is c may be called a &lt;em&gt;clipper comma&lt;/em&gt;; clipper commas would seem to make for superior clippers, avoiding very small intervals. Some 7-limit clipper commas are 16807/16384, 525/512, 128/125, 49/48, 50/49, 3125/3072, 64/63, 81/80, 2048/2025, 245/243, 2109375/2097152, 1029/1024, 15625/15552, 225/224, 3136/3125, 5120/5103, 6144/6125, 2100875/2097152, 5250987/5242880, 65625/65536, 32805/32768, 703125/702464, 2401/2400, 4375/4374 and &lt;br /&gt;

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&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>