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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET|Consistency=41|Distinct consistency=41}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-23 01:30:08 UTC</tt>.<br>
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| : The original revision id was <tt>510766294</tt>.<br>
| | 31920edo is distinctly [[consistent]] through the 41-odd-limit, with a smaller 41-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller distinctly consistent division. Its 3rd harmonic derives from [[665edo]]. It is also [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 15960edo, which is an [[atomic]] tuning, tempering out [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}. |
| : 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>
| | It is also the smallest multiple of [[12edo]] to be [[purely consistent]] in the 31-odd-limit (i.e. all odd harmonics up to and including 31 are approximated with no greater than 25% relative error). |
| <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">The 31920 division divides the octave into 31920 equal parts of 0.03759 cents each, making a cent equal to exactly 26.6 steps of 31920edo. It is distinctly consistent through the 41 limit, and is an atomic temperament, tempering out the Kirnberger atom, |161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30, 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114, 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336, 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064, 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990, 4560, 5320, 6384, 7980, 10640, 15960, 31920. These facts make it a good candidate for an [[interval size measure]], and one step of it may be called an "imp", so that the cent is 26.6 imps, and a 12edo semitone is 2660 imps. A single step of 15edo is 2128 imps, of 19edo 1680 imps, of 84edo 380 imps, of 140edo 228 imps, of 152edo 210 imps, of 190edo 168 imps, and of 665edo 48 imps.</pre></div>
| | The simplest of the commas under the 43-limit it tempers out are 47916/47915, 52480/52479, 58311/58310, 60516/60515, 67600/67599, 68783/68782, 72501/72500, 75141/75140, 76875/76874, 81549/81548, 81796/81795, 82944/82943, 88320/88319, 93093/93092, 93500/93499, 96876/96875 and 98736/98735. |
| <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>31920edo</title></head><body>The 31920 division divides the octave into 31920 equal parts of 0.03759 cents each, making a cent equal to exactly 26.6 steps of 31920edo. It is distinctly consistent through the 41 limit, and is an atomic temperament, tempering out the Kirnberger atom, |161 -84 -12&gt;. It is a very &quot;smooth&quot; number, with many divisors: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30, 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114, 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336, 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064, 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990, 4560, 5320, 6384, 7980, 10640, 15960, 31920. These facts make it a good candidate for an <a class="wiki_link" href="/interval%20size%20measure">interval size measure</a>, and one step of it may be called an &quot;imp&quot;, so that the cent is 26.6 imps, and a 12edo semitone is 2660 imps. A single step of 15edo is 2128 imps, of 19edo 1680 imps, of 84edo 380 imps, of 140edo 228 imps, of 152edo 210 imps, of 190edo 168 imps, and of 665edo 48 imps.</body></html></pre></div>
| | === Prime harmonics === |
| | {{Harmonics in equal|31920|prec=5|intervals=prime|columns=9}} |
| | {{Harmonics in equal|31920|prec=5|intervals=prime|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31920edo (continued)}} |
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| | === Subsets and supersets === |
| | 31920 is a very composite number, with many divisors: 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30, 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114, 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336, 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064, 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990, 4560, 5320, 6384, 7980, 10640, and 15960. These facts make it a good candidate for an [[interval size measure]], and one step of it may be called an [[imp]], so that the cent is 26.6 imps, and a [[12edo]] semitone is 2660 imps. A single step of [[15edo]] is 2128 imps, of [[19edo]] 1680 imps, of [[84edo]] 380 imps, of [[140edo]] 228 imps, of [[152edo]] 210 imps, of [[190edo]] 168 imps, and of 665edo 48 imps. |