Clipper: Difference between revisions

Revision as of 23:28, 2 February 2012

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2012-02-02 23:28:18 UTC.

The original revision id was 298043012.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

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 [[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 
250047/250000.

=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
[[clipper2048|clipper(2048/2025)]], 14 notes, 5-limit
[[clipper385|clipper(385/384)]], 15 notes, 11-limit
[[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
[[clipper896|clipper(896/891)]], 19 notes, 2.3.7.11
[[clipper126|clipper(126/125)]], 23 notes, 7-limit
[[clipper6144|clipper(6144/6125)]], 23 notes, 7-limit
[[clipper5120|clipper(5120/5103)]], 27 notes, 7-limit
[[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/11432]]
[[http://tech.groups.yahoo.com/group/tuning-math/message/11439]]
[[http://tech.groups.yahoo.com/group/tuning-math/message/11441]]

Original HTML content:

<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 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 />
250047/250000.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->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="/clipper121">clipper(121/120)</a>, 11 notes, 2.3.5.11<br />
<a class="wiki_link" href="/clipper176">clipper(176/175)</a>, 11 notes, 2.5.7.11<br />
<a class="wiki_link" href="/clipper2048">clipper(2048/2025)</a>, 14 notes, 5-limit<br />
<a class="wiki_link" href="/clipper385">clipper(385/384)</a>, 15 notes, 11-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="/clipper3136">clipper(3136/3125)</a>, 17 notes, 2.5.7<br />
<a class="wiki_link" href="/clipper99">clipper(99/98)</a>, 17 notes, 2.3.7.11<br />
<a class="wiki_link" href="/clipper100">clipper(100/99)</a>, 17 notes, 2.3.5.11<br />
<a class="wiki_link" href="/clipper243">clipper(243/242)</a>, 17 notes, 2.3.11<br />
<a class="wiki_link" href="/clipper896">clipper(896/891)</a>, 19 notes, 2.3.7.11<br />
<a class="wiki_link" href="/clipper126">clipper(126/125)</a>, 23 notes, 7-limit<br />
<a class="wiki_link" href="/clipper6144">clipper(6144/6125)</a>, 23 notes, 7-limit<br />
<a class="wiki_link" href="/clipper5120">clipper(5120/5103)</a>, 27 notes, 7-limit<br />
<a class="wiki_link" href="/clipper245">clipper(245/243)</a>, 35 notes, 7-limit<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:2 -->Links</h1>
<a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11429" rel="nofollow">http://tech.groups.yahoo.com/group/tuning-math/message/11429</a><br />
<a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11432" rel="nofollow">http://tech.groups.yahoo.com/group/tuning-math/message/11432</a><br />
<a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11439" rel="nofollow">http://tech.groups.yahoo.com/group/tuning-math/message/11439</a><br />
<a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11441" rel="nofollow">http://tech.groups.yahoo.com/group/tuning-math/message/11441</a></body></html>

@@ 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 21:01:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-02 23:28:18 UTC</tt>.<br>
-: The original revision id was <tt>298014138</tt>.<br>
+: The original revision id was <tt>298043012</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 17: / Line 17: @@
 [[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
 [[clipper2048|clipper(2048/2025)]], 14 notes, 5-limit
+[[clipper385|clipper(385/384)]], 15 notes, 11-limit
 [[clipper225|clipper(225/224)]], 17 notes, 7-limit
 [[clipper32805|clipper(32805/32768)]], 17 notes, 5-limit
@@ Line 23: / Line 26: @@
 [[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
+[[clipper896|clipper(896/891)]], 19 notes, 2.3.7.11
 [[clipper126|clipper(126/125)]], 23 notes, 7-limit
 [[clipper6144|clipper(6144/6125)]], 23 notes, 7-limit
@@ Line 46: / Line 51: @@
 &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="/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="/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;
@@ Line 52: / Line 60: @@
 &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="/clipper896"&gt;clipper(896/891)&lt;/a&gt;, 19 notes, 2.3.7.11&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;

Clipper: Difference between revisions

Revision as of 23:28, 2 February 2012

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

Navigation menu

Search