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| <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:genewardsmith|genewardsmith]] and made on <tt>2015-08-12 17:26:20 UTC</tt>.<br>
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| : The original revision id was <tt>556582741</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">The 3125 equal division of the octave divides it into 5^5 = 3125 equal parts of exactly 0.384 cents each. It is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower [[Tenney-Euclidean temperament measures#TE simple badness|relative error]]. It is also distinctly consistent through the 15 odd limit. A basis for its 7-limit commas is 78125000/78121827, 645700815/645657712 and 281484423828125/281474976710656; for 11-limit, 151263/151250, 820125/819896, 21437500/21434787 and 117440512/117406179; and for 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489.
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| The fact that 3125 = 5^5 makes curious notations possible based on the symmetric base 5 positional number system, by converting the number to base 5 with digits {-2, -1, 0, 1, 2}.</pre></div>
| | == Theory == |
| <h4>Original HTML content:</h4>
| | 3125et is notable for being an extremely strong [[7-limit]] system. It is also [[consistent]] through the [[15-odd-limit]], and except for [[17/11]], [[19/17]] and their [[octave complement]]s, it is consistent to the [[35-odd-limit]]. |
| <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>3125edo</title></head><body>The 3125 equal division of the octave divides it into 5^5 = 3125 equal parts of exactly 0.384 cents each. It is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">relative error</a>. It is also distinctly consistent through the 15 odd limit. A basis for its 7-limit commas is 78125000/78121827, 645700815/645657712 and 281484423828125/281474976710656; for 11-limit, 151263/151250, 820125/819896, 21437500/21434787 and 117440512/117406179; and for 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489.<br />
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| | A basis for its 7-limit commas is [[78125000/78121827]], [[645700815/645657712]] and 281484423828125/281474976710656. In the [[11-limit]], [[151263/151250]], 820125/819896, 21437500/21434787 and [[quartisma|117440512/117406179]] are tempered out—it should be noted this edo is so far the only one [https://en.xen.wiki/index.php?title=3125edo&oldid=7860 known to have been confirmed] as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five [[33/32]] quartertones and one [[7/6]] subminor third. In the [[13-limit]], [[6656/6655]], [[123201/123200]], [[140625/140608]] and 1399680/1399489 are all tempered out. |
| The fact that 3125 = 5^5 makes curious notations possible based on the symmetric base 5 positional number system, by converting the number to base 5 with digits {-2, -1, 0, 1, 2}.</body></html></pre></div>
| | |
| | === Prime harmonics === |
| | {{Harmonics in equal|3125|columns=11}} |
| | {{Harmonics in equal|3125|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 3125edo (continued)}} |
| | |
| | === Subsets and supersets === |
| | {{nowrap| 3125 {{=}} 5<sup>5</sup> }}, and as such 3125edo is the 5th edo of the form ''n''<sup>''n''</sup>. It has subset edos {{EDOs| 5, 25, 125, and 625 }}. |
| | |
| | == Regular temperament properties == |
| | 3125et is the first equal temperament past [[171edo|171]] with a lower [[Tenney–Euclidean temperament measures #TE simple badness|relative error]]. |
| | |
| | === Rank-2 temperaments === |
| | {| class="wikitable center-all left-5" |
| | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator |
| | |- |
| | ! Periods<br>per 8ve |
| | ! Generator* |
| | ! Cents* |
| | ! Associated<br>ratio* |
| | ! Temperaments |
| | |- |
| | | 1 |
| | | 139\3125 |
| | | 53.376 |
| | | 33/32 |
| | | [[Prequartismic]] |
| | |- |
| | | 1 |
| | | 411\3125 |
| | | 157.824 |
| | | 36756909/33554432 |
| | | [[Hemiegads]] |
| | |- |
| | | 1 |
| | | 577\3125 |
| | | 221.568 |
| | | 8388608/7381125 |
| | | [[Fortune]] |
| | |- |
| | | 1 |
| | | 822\3125 |
| | | 315.648 |
| | | 6/5 |
| | | [[Egads]] |
| | |- |
| | | 1 |
| | | 894\3125 |
| | | 343.296 |
| | | 8000/6561 |
| | | [[Raider]] |
| | |- |
| | | 1 |
| | | 1359\3125 |
| | | 521.856 |
| | | 80275/59392 |
| | | [[Estates general]] |
| | |- |
| | | 1 |
| | | 1412\3125 |
| | | 542.208 |
| | | 16807/12288 |
| | | [[Revopent]] |
| | |} |
| | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct |
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| | == Music == |
| | ; [[Eliora]] |
| | * [https://www.youtube.com/watch?v=mnTF1vBexBg ''Etude for Gamelan in Estates General and Pentonismic''] (2023) |
| | |
| | [[Category:Quartismic]] |
| | [[Category:Listen]] |