Clipper: Difference between revisions

Latest revision as of 01:46, 8 February 2026

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 is known as didacus temperament.

Scales

See also: Category:Clippers

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

View • Talk • EditScale galleries
JI scales	12-tone JI • Combination product set • Constant structure • Harry Partch-related • Maximal harmony epimorphic • MOS transversal • Non-octave JI • Wakalix • Z-polygon transversal • Other JI Full list: Category:Just intonation scales
Tempered scales	11-tone MOS • 12-tone tempered • Chromatic pair • Clipper • Double mode • Essentially tempered • Fantasy detemper • Marvel woo • Meantone • Min ambiguity • MOS cradle • Negri-9 • Neutral third • Non-octave tempered • Scalesmith systematic • Ternary • Other tempered Full list: Category:Tempered scales
Scales in EDOs	in 10edo • 11 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 33 • 34 • 35 • 36 • 37 • 38 • 40 • 41 • 42 • 43 • 46 • 49 • 53 • 72 • 80
All other scale gallery pages are included in Category:Lists of scales

@@ 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>2012-02-02 18:29:29 UTC</tt>.<br>
-: The original revision id was <tt>297977016</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 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 is known as [[didacus]] 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
+== Scales ==
-/250000.
+{{See also|Category:Clippers}}
+[[clipper1029|clipper(1029/1024)]], 7 notes, 2.3.7
-=Scales=
 [[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
-=Links=
+[[clipper6144|clipper(6144/6125)]], 23 notes, 7-limit
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11429]]
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11432]]
+[[clipper65625|clipper(65625/65536)]], 23 notes, 7-limit
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11439]]
-[[http://tech.groups.yahoo.com/group/tuning-math/message/11441]]
+[[clipper5120|clipper(5120/5103)]], 27 notes, 7-limit
-</pre></div>
-<h4>Original HTML content:</h4>
+[[clipper4000|clipper(4000/3993)]], 31 notes, 2.3.5.11
-<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;
+[[clipper245|clipper(245/243)]], 35 notes, 7-limit
-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;
-&lt;br /&gt;
+== Links ==
-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;
+[http://tech.groups.yahoo.com/group/tuning-math/message/11429 http://tech.groups.yahoo.com/group/tuning-math/message/11429]
-/250000.&lt;br /&gt;
-&lt;br /&gt;
+[http://tech.groups.yahoo.com/group/tuning-math/message/11432 http://tech.groups.yahoo.com/group/tuning-math/message/11432]
-&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="/clipper81"&gt;clipper(81/80)&lt;/a&gt;, 9 notes, 5-limit&lt;br /&gt;
+[http://tech.groups.yahoo.com/group/tuning-math/message/11439 http://tech.groups.yahoo.com/group/tuning-math/message/11439]
-&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="/clipper126"&gt;clipper(126/125)&lt;/a&gt;, 23 notes, 7-limit&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;
-&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;
+{{Navbox scale gallery}}
-&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;
+[[Category:Euler-Fokker genera]]
-&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;
+[[Category:Regular temperament theory]]
-&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>
+[[Category:Lists of scales]]
+[[Category:Todo:clarify]]

Clipper: Difference between revisions

Latest revision as of 01:46, 8 February 2026

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