Clipper: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2012-02-02 18:29:29 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-02 19:23:18 UTC</tt>.<br>

: The original revision id was <tt>~~297977016~~</tt>.<br>

: The original revision id was <tt>297989486</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>

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<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 [<1 0 -3|, <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 [<1 0 -3|, <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

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=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

[[clipper2048|clipper(2048/2025)]], 14 notes, 5-limit

[[clipper225|clipper(225/224)]], 17 notes, 7-limit

[[clipper32805|clipper(32805/32768)]], 17 notes, 5-limit

[[clipper126|clipper(126/125)]], 23 notes, 7-limit

[[clipper245|clipper(245/243)]], 35 notes, 7-limit

=Links=

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<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"><html><head><title>Clippers</title></head><body>If c is a <a class="wiki_link" href="/comma">comma</a>, then clipper(c) is defined as Euler(Benedetti(c)), tempered by the codimension one temperament tempering out c. Here Euler(N) is the <a class="wiki_link" href="/Euler%20genera">Euler genus</a>, the divisors of the integer N reduced to the octave, and Benedetti(c) is the <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a> 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 <a class="wiki_link" href="/Transversal">transversals</a> of clipper(c) are obtained by leaving out either one or the other of the pair of scale intervals separated by c.<br />

<br />

Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a ~~[[~~<a href="~~http://xenharmonic.wikispaces.com~~/Normal~~+lists~~#x-Normal">~~http://xenharmonic.wikispaces.com/Normal+lists#x-Normal~~</a>~~ 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 <a class="wiki_link" href="/Just%20intonation%20subgroups">JI subgroup</a>, with mapping [&lt;1 0 -3|, &lt;0 2 5|] and an approximate 28/25 generator, which might be called 7-limit <a class="wiki_link" href="/Chromatic%20pairs#Roulette">roulette</a> temperament.<br />

Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a <a class="wiki_link" href="/Normal%20lists#x-Normal interval lists">normal interval list</a>. 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 <a class="wiki_link" href="/Just%20intonation%20subgroups">JI subgroup</a>, with mapping [&lt;1 0 -3|, &lt;0 2 5|] and an approximate 28/25 generator, which might be called 7-limit <a class="wiki_link" href="/Chromatic%20pairs#Roulette">roulette</a> temperament.<br />

<br />

A comma such that the smallest interval in Euler(Benedetti(c)) is c may be called a <em>clipper comma</em>; 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 <br />

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<br />

<h1 id="toc0"><a name="Scales"></a>Scales</h1>

<a class="wiki_link" href="/clipper1029">clipper(1029/1024)</a>, 7 notes, 2.3.7<br />

<a class="wiki_link" href="/clipper81">clipper(81/80)</a>, 9 notes, 5-limit<br />

<a class="wiki_link" href="/clipper3125">clipper(3125/3072)</a>, 11 notes, 5-limit<br />

<a class="wiki_link" href="/clipper2048">clipper(2048/2025)</a>, 14 notes, 5-limit<br />

<a class="wiki_link" href="/clipper225">clipper(225/224)</a>, 17 notes, 7-limit<br />

<a class="wiki_link" href="/clipper32805">clipper(32805/32768)</a>, 17 notes, 5-limit<br />

<a class="wiki_link" href="/clipper126">clipper(126/125)</a>, 23 notes, 7-limit<br />

<a class="wiki_link" href="/clipper245">clipper(245/243)</a>, 35 notes, 7-limit<br />

<br />

<h1 id="toc1"><a name="Links"></a>Links</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-02-02 18:29:29 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-02 19:23:18 UTC</tt>.<br>
-: The original revision id was <tt>297977016</tt>.<br>
+: The original revision id was <tt>297989486</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>
@@ Line 8: / Line 8: @@
 <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.
+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.
 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
@@ Line 14: / Line 14: @@
 =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
 [[clipper2048|clipper(2048/2025)]], 14 notes, 5-limit
+[[clipper225|clipper(225/224)]], 17 notes, 7-limit
+[[clipper32805|clipper(32805/32768)]], 17 notes, 5-limit
 [[clipper126|clipper(126/125)]], 23 notes, 7-limit
+[[clipper245|clipper(245/243)]], 35 notes, 7-limit
 =Links=
@@ Line 27: / Line 32: @@
 <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;
 &lt;br /&gt;
-Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a [[&lt;!-- ws:start:WikiTextUrlRule:31: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:31 --&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;
+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;
 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;
@@ Line 33: / Line 38: @@
 &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="/clipper2048"&gt;clipper(2048/2025)&lt;/a&gt;, 14 notes, 5-limit&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="/clipper126"&gt;clipper(126/125)&lt;/a&gt;, 23 notes, 7-limit&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;