385/384: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1> of 4.503 cents. Tempering it out leads to the 11-limit rank four [[Keenanismic_family|keenanismic temperament]].

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-07-25 17:50:56 UTC</tt>.<br>~~

~~: The original revision id was <tt>354810156</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">The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1> of 4.503 cents. Tempering it out leads to the 11-limit rank four [[~~Keenanismic family~~|keenanismic temperament]].

The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of [[~~The Seven Limit Symmetrical Lattices~~|7-limit pitch classes]], the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the [[~~The Seven Limit Symmetrical Lattices~~|cubic lattice of 7-limit tetrads]], the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.

The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of [[The_Seven_Limit_Symmetrical_Lattices|7-limit pitch classes]], the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the [[The_Seven_Limit_Symmetrical_Lattices|cubic lattice of 7-limit tetrads]], the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.

EDOs with [[patent val]]s tempering out the keenansima include [[15edo|15]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[37edo|37]], [[41edo|41]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[118edo|118]], [[159edo|159]], [[190edo|190]], [[212edo|212]] and [[284edo|284]].

EDOs with [[Patent_val|patent val]]s tempering out the keenansima include [[15edo|15]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[37edo|37]], [[41edo|41]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[118edo|118]], [[159edo|159]], [[190edo|190]], [[212edo|212]] and [[284edo|284]].

Characteristic of keenanismic tempering are the [[keenanismic tetrads]], 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered [[dyadic chord]]s, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit [[tonality diamond]], and hence regarded as an 11-limit consonance.

Characteristic of keenanismic tempering are the [[keenanismic_tetrads|keenanismic tetrads]], 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered [[Dyadic_chord|dyadic chord]]s, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit [[Tonality_diamond|tonality diamond]], and hence regarded as an 11-limit consonance.

[[~~image~~:keenanismic_tetrads_in_31edo_sym.png~~]]</pre></div>~~

[[File:keenanismic_tetrads_in_31edo_sym.png|alt=keenanismic_tetrads_in_31edo_sym.png|keenanismic_tetrads_in_31edo_sym.png]] [[Category:11-limit]]

~~<h4>Original HTML content:</h4>~~

[[Category:31edo]]

<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>385_384</title></head><body>The keenanisma is the 11-limit comma 385/384 = |~~-7 -1 1 1 1&gt; of 4.503 cents. Tempering it out leads to the 11-limit rank four <a class~~=~~"wiki_link" href="/Keenanismic%20family">keenanismic temperament</a>~~.~~<br />~~

[[Category:comma]]

~~<br />~~

[[Category:keenanismic]]

~~The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2~~. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">7-limit pitch classes</a>, the &quot;deep holes&quot; of the lattice as opposed to the &quot;holes&quot; represented by major and minor tetrads, and in terms of the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">cubic lattice of 7-limit tetrads</a>, the otonal tetrad with root 11 (or 11/8) is represented by [~~-2 0 0]~~: ~~1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an~~ 11-limit ~~projection comma, equaling the [0 1 -1~~] ~~of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.<br />~~

[[Category:tetrad]]

~~<br />~~

EDOs with <a class="wiki_link" href="/patent%20val">patent val</a>s tempering out the keenansima include <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/37edo">37</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/190edo">190</a>, <a class="wiki_link" href="/212edo">212</a> and <a class="wiki_link" href="/284edo">284</a>.<br />

~~<br />~~

~~Characteristic of~~ keenanismic tempering are the <a class="wiki_link" href="/keenanismic%20tetrads">keenanismic tetrads</a>, 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a>s, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, and hence regarded as an 11-limit consonance.<br />

~~<br />~~

<img src="/file/view/keenanismic_tetrads_in_31edo_sym.png/354772590/keenanismic_tetrads_in_31edo_sym.png" alt="keenanismic_tetrads_in_31edo_sym.png" title="keenanismic_tetrads_in_31edo_sym.png" /></body></html></pre></div>

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-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1&gt; of 4.503 cents. Tempering it out leads to the 11-limit rank four [[Keenanismic_family|keenanismic temperament]].
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-07-25 17:50:56 UTC</tt>.<br>
-: The original revision id was <tt>354810156</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">The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1&gt; of 4.503 cents. Tempering it out leads to the 11-limit rank four [[Keenanismic family|keenanismic temperament]].
-The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of [[The Seven Limit Symmetrical Lattices|7-limit pitch classes]], the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the [[The Seven Limit Symmetrical Lattices|cubic lattice of 7-limit tetrads]], the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.
+The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of [[The_Seven_Limit_Symmetrical_Lattices|7-limit pitch classes]], the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the [[The_Seven_Limit_Symmetrical_Lattices|cubic lattice of 7-limit tetrads]], the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.
-EDOs with [[patent val]]s tempering out the keenansima include [[15edo|15]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[37edo|37]], [[41edo|41]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[118edo|118]], [[159edo|159]], [[190edo|190]], [[212edo|212]] and [[284edo|284]].
+EDOs with [[Patent_val|patent val]]s tempering out the keenansima include [[15edo|15]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[37edo|37]], [[41edo|41]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[118edo|118]], [[159edo|159]], [[190edo|190]], [[212edo|212]] and [[284edo|284]].
-Characteristic of keenanismic tempering are the [[keenanismic tetrads]], 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered [[dyadic chord]]s, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit [[tonality diamond]], and hence regarded as an 11-limit consonance.
+Characteristic of keenanismic tempering are the [[keenanismic_tetrads|keenanismic tetrads]], 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered [[Dyadic_chord|dyadic chord]]s, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit [[Tonality_diamond|tonality diamond]], and hence regarded as an 11-limit consonance.
-[[image:keenanismic_tetrads_in_31edo_sym.png]]</pre></div>
+[[File:keenanismic_tetrads_in_31edo_sym.png|alt=keenanismic_tetrads_in_31edo_sym.png|keenanismic_tetrads_in_31edo_sym.png]]      [[Category:11-limit]]
-<h4>Original HTML content:</h4>
+[[Category:31edo]]
-<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;385_384&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1&amp;gt; of 4.503 cents. Tempering it out leads to the 11-limit rank four &lt;a class="wiki_link" href="/Keenanismic%20family"&gt;keenanismic temperament&lt;/a&gt;.&lt;br /&gt;
+[[Category:comma]]
-&lt;br /&gt;
+[[Category:keenanismic]]
-The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;7-limit pitch classes&lt;/a&gt;, the &amp;quot;deep holes&amp;quot; of the lattice as opposed to the &amp;quot;holes&amp;quot; represented by major and minor tetrads, and in terms of the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;cubic lattice of 7-limit tetrads&lt;/a&gt;, the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.&lt;br /&gt;
+[[Category:tetrad]]
-&lt;br /&gt;
-EDOs with &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt;s tempering out the keenansima include &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/68edo"&gt;68&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/190edo"&gt;190&lt;/a&gt;, &lt;a class="wiki_link" href="/212edo"&gt;212&lt;/a&gt; and &lt;a class="wiki_link" href="/284edo"&gt;284&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-Characteristic of keenanismic tempering are the &lt;a class="wiki_link" href="/keenanismic%20tetrads"&gt;keenanismic tetrads&lt;/a&gt;, 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered &lt;a class="wiki_link" href="/dyadic%20chord"&gt;dyadic chord&lt;/a&gt;s, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt;, and hence regarded as an 11-limit consonance.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:0:&amp;lt;img src=&amp;quot;/file/view/keenanismic_tetrads_in_31edo_sym.png/354772590/keenanismic_tetrads_in_31edo_sym.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/keenanismic_tetrads_in_31edo_sym.png/354772590/keenanismic_tetrads_in_31edo_sym.png" alt="keenanismic_tetrads_in_31edo_sym.png" title="keenanismic_tetrads_in_31edo_sym.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:0 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>