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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:xenwolf|xenwolf]] and made on <tt>2011-06-29 10:08:59 UTC</tt>.<br>
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| : The original revision id was <tt>239316131</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">**129edo** is the [[equal division of the octave]] into 129 parts of 9.302 [[cent]]s each. It provides the [[optimal patent val]] for the 11-limit rank three [[Didymus rank three family|clio temperament]]. It [[tempering out|tempers out]] 81/80 in the [[5-limit]]; 1029/1024 and 1728/1715 in the [[7-limit]]; 176/175 and 540/539 in the [[11-limit]]; 507/500, 676/675 and 847/845 in the [[13-limit]]; 221/220 in the [[17-limit]]; 171/170 and 286/285 in the [[19-limit]].
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| The factorization of 129 is [[3edo|3]] and [[43edo|43]]</pre></div>
| | 129edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. The [[patent val]] is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[43edo]]. It is the last patent val that [[tempering out|tempers out]] [[81/80]] so as to [[support]] [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. |
| <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>129edo</title></head><body><strong>129edo</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into 129 parts of 9.302 <a class="wiki_link" href="/cent">cent</a>s each. It provides the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the 11-limit rank three <a class="wiki_link" href="/Didymus%20rank%20three%20family">clio temperament</a>. It <a class="wiki_link" href="/tempering%20out">tempers out</a> 81/80 in the <a class="wiki_link" href="/5-limit">5-limit</a>; 1029/1024 and 1728/1715 in the <a class="wiki_link" href="/7-limit">7-limit</a>; 176/175 and 540/539 in the <a class="wiki_link" href="/11-limit">11-limit</a>; 507/500, 676/675 and 847/845 in the <a class="wiki_link" href="/13-limit">13-limit</a>; 221/220 in the <a class="wiki_link" href="/17-limit">17-limit</a>; 171/170 and 286/285 in the <a class="wiki_link" href="/19-limit">19-limit</a>. <br />
| | === Odd harmonics === |
| <br />
| | {{Harmonics in equal|129}} |
| The factorization of 129 is <a class="wiki_link" href="/3edo">3</a> and <a class="wiki_link" href="/43edo">43</a></body></html></pre></div>
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| | === Subsets and supersets === |
| | Since 129 factors into {{factorization|129}}, 129edo contains [[3edo]] and [[43edo]] as its subsets. [[258edo]], which doubles it, provides a good correction for the 3rd and 5th harmonics. |
| | |
| | == Instruments == |
| | * [[Lumatone mapping for 129edo]] |
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| | [[Category:Clio]] |
Latest revision as of 22:57, 13 July 2025
| Prime factorization
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3 × 43
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| Step size
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9.30233 ¢
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| Fifth
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75\129 (697.674 ¢) (→ 25\43)
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| Semitones (A1:m2)
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9:12 (83.72 ¢ : 111.6 ¢)
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| Dual sharp fifth
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76\129 (706.977 ¢)
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| Dual flat fifth
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75\129 (697.674 ¢) (→ 25\43)
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| Dual major 2nd
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22\129 (204.651 ¢)
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| Consistency limit
|
3
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| Distinct consistency limit
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3
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129 equal divisions of the octave (abbreviated 129edo or 129ed2), also called 129-tone equal temperament (129tet) or 129 equal temperament (129et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 129 equal parts of about 9.3 ¢ each. Each step represents a frequency ratio of 21/129, or the 129th root of 2.
129edo is inconsistent to the 5-odd-limit and both harmonics 3 and 5 are about halfway between its steps. The patent val is enfactored in the 5-limit, with the same tuning as 43edo. It is the last patent val that tempers out 81/80 so as to support meantone and its higher-limit expansions. It also tempers out 1029/1024 and 1728/1715 in the 7-limit; 176/175 and 540/539 in the 11-limit; 507/500, 676/675 and 847/845 in the 13-limit; 221/220 in the 17-limit; 171/170 and 286/285 in the 19-limit. It provides the optimal patent val for the 11-limit rank-3 clio temperament.
Odd harmonics
Approximation of odd harmonics in 129edo
| Harmonic
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3
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5
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7
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9
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11
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13
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15
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17
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19
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21
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23
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| Error
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Absolute (¢)
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-4.28
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+4.38
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-1.38
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+0.74
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-2.48
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-3.32
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+0.10
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-2.63
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+0.16
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+3.64
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+4.28
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| Relative (%)
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-46.0
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+47.1
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-14.9
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+8.0
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-26.7
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-35.7
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+1.1
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-28.3
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+1.7
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+39.1
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+46.1
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Steps
(reduced)
|
204
(75)
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300
(42)
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362
(104)
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409
(22)
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446
(59)
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477
(90)
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504
(117)
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527
(11)
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548
(32)
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567
(51)
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584
(68)
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Subsets and supersets
Since 129 factors into 3 × 43, 129edo contains 3edo and 43edo as its subsets. 258edo, which doubles it, provides a good correction for the 3rd and 5th harmonics.
Instruments