218edo: 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:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-12 14:34:47 UTC</tt>.<br>~~

~~: The original revision id was <tt>589260680</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">=218edo=

~~This needs some content, it contains very accurate ratios, here is the collection: 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10, 17/16~~

~~|| fraction: || 3/2 || 4/3 || 5/4 || 8/5 || 5/3 || 6/5 || 7/4 || 8/7 || 10/9 || 9/5 || 9/8 || 16/9 ||~~

~~|| steps in 218edo: || || || || || || || || || || || || ||</pre></div>~~

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

~~This needs some content, it contains very accurate ratios, here is the collection: 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10, 17/16<br />~~

218edo is in[[consistent]] to the [[5-odd-limit]], with [[harmonic]] [[3/1|3]] falling about halfway between its steps. However, it contains very accurate ratios, such as [[7/4]], [[9/7]], [[9/8]], [[10/9]], [[11/10]], [[17/16]], and [[19/16]], which are approximated within 0.55-cent deviation (10% the step size). The suggested [[subgroup]]s are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23.

~~<table class="wiki_table">~~

Commas using the [[13-limit]] patent val:

~~<tr>~~

~~<td>fraction~~:~~<br />~~

~~</td>~~

~~<td>3/2<br />~~

~~</td>~~

~~<td>4/3<br />~~

~~</td>~~

~~<td>5/4<br />~~

~~</td>~~

~~<td>8/5<br />~~

~~</td>~~

~~<td>5/3<br />~~

~~</td>~~

~~<td>6/5<br />~~

~~</td>~~

~~<td>7/4<br />~~

~~</td>~~

~~<td>8/7<br />~~

~~</td>~~

~~<td>10/9<br />~~

~~</td>~~

~~<td>9/5<br />~~

~~</td>~~

~~<td>9/8<br />~~

~~</td>~~

~~<td>16/9<br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>steps in 218edo:<br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~<td><br />~~

~~</td>~~

~~</tr>~~

~~</table>~~

~~&lt~~;/~~body&gt~~;~~&lt~~;/~~html&gt~~;</~~pre><~~/~~div>~~

; [[5-limit]]: 20000/19683, 1220703125/1207959552

; [[7-limit]]: 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640

; [[11-limit]]: 4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576,

; [[13-limit]]: 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512

=== Odd harmonics ===

=== Subsets and supersets ===

Since 218 factors into {{factorization|218}}, 218edo contains [[2edo]] and [[109edo]] as its subsets. [[436edo]], which doubles it, is worth exploring.

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-12 14:34:47 UTC</tt>.<br>
-: The original revision id was <tt>589260680</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">=218edo=
-This needs some content, it contains very accurate ratios, here is the collection: 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10, 17/16
-|| fraction: || 3/2 || 4/3 || 5/4 || 8/5 || 5/3 || 6/5 || 7/4 || 8/7 || 10/9 || 9/5 || 9/8 || 16/9 ||
-|| steps in 218edo: ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||</pre></div>
-<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;218edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x218edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;218edo&lt;/h1&gt;
- This needs some content, it contains very accurate ratios, here is the collection: 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10, 17/16&lt;br /&gt;
+edo is in[[consistent]] to the [[5-odd-limit]], with [[harmonic]] [[3/1|3]] falling about halfway between its steps. However, it contains very accurate ratios, such as [[7/4]], [[9/7]], [[9/8]], [[10/9]], [[11/10]], [[17/16]], and [[19/16]], which are approximated within 0.55-cent deviation (10% the step size). The suggested [[subgroup]]s are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23.
-&lt;table class="wiki_table"&gt;
+Commas using the [[13-limit]] patent val:
-    &lt;tr&gt;
-        &lt;td&gt;fraction:&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;3/2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;4/3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;5/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;8/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;5/3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;6/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;7/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;8/7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;10/9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;9/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;9/8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;16/9&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;steps in 218edo:&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;/body&gt;&lt;/html&gt;</pre></div>
+; [[5-limit]]: 20000/19683, 1220703125/1207959552
+; [[7-limit]]: 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640
+; [[11-limit]]: 4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576,
+; [[13-limit]]: 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512
+=== Odd harmonics ===
+{{Harmonics in equal|218}}
+=== Subsets and supersets ===
+Since 218 factors into {{factorization|218}}, 218edo contains [[2edo]] and [[109edo]] as its subsets. [[436edo]], which doubles it, is worth exploring.