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| <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>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-02 23:33:59 UTC</tt>.<br>
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| : The original revision id was <tt>298043896</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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.
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| 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. | | 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 is known as [[didacus]] temperament. |
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| =Scales= | | == Scales == |
| | {{See also|Category:Clippers}} |
| [[clipper1029|clipper(1029/1024)]], 7 notes, 2.3.7 | | [[clipper1029|clipper(1029/1024)]], 7 notes, 2.3.7 |
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| [[clipper81|clipper(81/80)]], 9 notes, 5-limit | | [[clipper81|clipper(81/80)]], 9 notes, 5-limit |
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| [[clipper3125|clipper(3125/3072)]], 11 notes, 5-limit | | [[clipper3125|clipper(3125/3072)]], 11 notes, 5-limit |
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| [[clipper121|clipper(121/120)]], 11 notes, 2.3.5.11 | | [[clipper121|clipper(121/120)]], 11 notes, 2.3.5.11 |
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| [[clipper176|clipper(176/175)]], 11 notes, 2.5.7.11 | | [[clipper176|clipper(176/175)]], 11 notes, 2.5.7.11 |
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| | [[clipper65536|clipper(65536/65219)]], 11 notes, 2.7.11 |
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| | [[clipper144|clipper(144/143)]], 11 notes, 2.3.11.13 |
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| | [[clipper169|clipper(169/168)]], 11 notes, 2.3.7.13 |
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| | [[clipper640|clipper(640/637)]], 11 notes, 2.5.7.13 |
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| [[clipper2048|clipper(2048/2025)]], 14 notes, 5-limit | | [[clipper2048|clipper(2048/2025)]], 14 notes, 5-limit |
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| [[clipper385|clipper(385/384)]], 15 notes, 11-limit | | [[clipper385|clipper(385/384)]], 15 notes, 11-limit |
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| | [[clipper105|clipper(105/104)]], 15 notes, 2.3.5.7.13 |
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| [[clipper225|clipper(225/224)]], 17 notes, 7-limit | | [[clipper225|clipper(225/224)]], 17 notes, 7-limit |
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| [[clipper32805|clipper(32805/32768)]], 17 notes, 5-limit | | [[clipper32805|clipper(32805/32768)]], 17 notes, 5-limit |
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| [[clipper3136|clipper(3136/3125)]], 17 notes, 2.5.7 | | [[clipper3136|clipper(3136/3125)]], 17 notes, 2.5.7 |
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| [[clipper99|clipper(99/98)]], 17 notes, 2.3.7.11 | | [[clipper99|clipper(99/98)]], 17 notes, 2.3.7.11 |
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| [[clipper100|clipper(100/99)]], 17 notes, 2.3.5.11 | | [[clipper100|clipper(100/99)]], 17 notes, 2.3.5.11 |
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| [[clipper243|clipper(243/242)]], 17 notes, 2.3.11 | | [[clipper243|clipper(243/242)]], 17 notes, 2.3.11 |
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| | [[clipper245242|clipper(245/242)]], 17 notes, 2.5.7.11 |
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| [[clipper896|clipper(896/891)]], 19 notes, 2.3.7.11 | | [[clipper896|clipper(896/891)]], 19 notes, 2.3.7.11 |
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| | [[clipper625|clipper(625/624)]], 19 notes, 2.3.5.13 |
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| [[clipper126|clipper(126/125)]], 23 notes, 7-limit | | [[clipper126|clipper(126/125)]], 23 notes, 7-limit |
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| [[clipper6144|clipper(6144/6125)]], 23 notes, 7-limit | | [[clipper6144|clipper(6144/6125)]], 23 notes, 7-limit |
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| | [[clipper65625|clipper(65625/65536)]], 23 notes, 7-limit |
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| [[clipper5120|clipper(5120/5103)]], 27 notes, 7-limit | | [[clipper5120|clipper(5120/5103)]], 27 notes, 7-limit |
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| | [[clipper4000|clipper(4000/3993)]], 31 notes, 2.3.5.11 |
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| [[clipper245|clipper(245/243)]], 35 notes, 7-limit | | [[clipper245|clipper(245/243)]], 35 notes, 7-limit |
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| =Links= | | == 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]]
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| [[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]]
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| </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>
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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 />
| | [http://tech.groups.yahoo.com/group/tuning-math/message/11441 http://tech.groups.yahoo.com/group/tuning-math/message/11441] |
| <br />
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| 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 />
| | {{Navbox scale gallery}} |
| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1>
| | [[Category:Euler-Fokker genera]] |
| <a class="wiki_link" href="/clipper1029">clipper(1029/1024)</a>, 7 notes, 2.3.7<br />
| | [[Category:Regular temperament theory]] |
| <a class="wiki_link" href="/clipper81">clipper(81/80)</a>, 9 notes, 5-limit<br />
| | [[Category:Lists of scales]] |
| <a class="wiki_link" href="/clipper3125">clipper(3125/3072)</a>, 11 notes, 5-limit<br />
| | [[Category:Todo:clarify]] |
| <a class="wiki_link" href="/clipper121">clipper(121/120)</a>, 11 notes, 2.3.5.11<br />
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| <a class="wiki_link" href="/clipper176">clipper(176/175)</a>, 11 notes, 2.5.7.11<br />
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| <a class="wiki_link" href="/clipper2048">clipper(2048/2025)</a>, 14 notes, 5-limit<br />
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| <a class="wiki_link" href="/clipper385">clipper(385/384)</a>, 15 notes, 11-limit<br />
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| <a class="wiki_link" href="/clipper225">clipper(225/224)</a>, 17 notes, 7-limit<br />
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| <a class="wiki_link" href="/clipper32805">clipper(32805/32768)</a>, 17 notes, 5-limit<br />
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| <a class="wiki_link" href="/clipper3136">clipper(3136/3125)</a>, 17 notes, 2.5.7<br />
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| <a class="wiki_link" href="/clipper99">clipper(99/98)</a>, 17 notes, 2.3.7.11<br />
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| <a class="wiki_link" href="/clipper100">clipper(100/99)</a>, 17 notes, 2.3.5.11<br />
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| <a class="wiki_link" href="/clipper243">clipper(243/242)</a>, 17 notes, 2.3.11<br />
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| <a class="wiki_link" href="/clipper896">clipper(896/891)</a>, 19 notes, 2.3.7.11<br />
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| <a class="wiki_link" href="/clipper126">clipper(126/125)</a>, 23 notes, 7-limit<br />
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| <a class="wiki_link" href="/clipper6144">clipper(6144/6125)</a>, 23 notes, 7-limit<br />
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| <a class="wiki_link" href="/clipper5120">clipper(5120/5103)</a>, 27 notes, 7-limit<br />
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| <a class="wiki_link" href="/clipper245">clipper(245/243)</a>, 35 notes, 7-limit<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:2 -->Links</h1>
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| <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 />
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| <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 />
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| <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 />
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| <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></pre></div>
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