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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>2011-08-27 16:09:24 UTC</tt>.<br>
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| : The original revision id was <tt>248815519</tt>.<br>
| | It is the first merger of [[12edo]] and [[19edo]], and its step size is the difference between 12edo's and 19edo's fifths. The equal temperament [[tempering out|tempers out]] the [[Pythagorean comma]], 531441/524288, in the 3-limit, and [[225/224]] and [[250047/250000]] in the 7-limit, so that it [[support]]s 7-limit [[compton]] temperament and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out 225/224, [[441/440]], 1375/1372 and 4375/4356, so that it supports 11-limit compton. Aside from the Pythagorean comma, the 12-comma, it tempers out the [[enneadeca]] or 19-tone-comma, and this is reflected in the fact that 228 = 12 × 19. |
| : 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>
| | === Odd harmonics === |
| <h4>Original Wikitext content:</h4>
| | {{Harmonics in equal|228}} |
| <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 //228 equal division// divides the octave into 228 equal parts of 5.263 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit, and 225/224 and 240047/240000 in the 7-limit, so that it supports 7-limit [[Pythagorean family|compton temperament]] and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out 225/224, 441/440, 1375/1372 and 4375/4356, so that it supports 11-limit compton. Aside from the Pythagorean comma, the 12-comma, it tempers out the [[enneadeca]] or 19-tone-comma, and this is reflected in the fact that 228 = 12 * 19. </pre></div>
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| <h4>Original HTML content:</h4>
| | [[Category:Compton]] |
| <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>228edo</title></head><body>The <em>228 equal division</em> divides the octave into 228 equal parts of 5.263 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit, and 225/224 and 240047/240000 in the 7-limit, so that it supports 7-limit <a class="wiki_link" href="/Pythagorean%20family">compton temperament</a> and in fact provides the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a>. In the 11-limit it tempers out 225/224, 441/440, 1375/1372 and 4375/4356, so that it supports 11-limit compton. Aside from the Pythagorean comma, the 12-comma, it tempers out the <a class="wiki_link" href="/enneadeca">enneadeca</a> or 19-tone-comma, and this is reflected in the fact that 228 = 12 * 19.</body></html></pre></div>
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| Prime factorization
|
22 × 3 × 19
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| Step size
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5.26316 ¢
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| Fifth
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133\228 (700 ¢) (→ 7\12)
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| Semitones (A1:m2)
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19:19 (100 ¢ : 100 ¢)
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| Dual sharp fifth
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134\228 (705.263 ¢) (→ 67\114)
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| Dual flat fifth
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133\228 (700 ¢) (→ 7\12)
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| Dual major 2nd
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39\228 (205.263 ¢) (→ 13\76)
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| Consistency limit
|
7
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| Distinct consistency limit
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7
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228 equal divisions of the octave (abbreviated 228edo or 228ed2), also called 228-tone equal temperament (228tet) or 228 equal temperament (228et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 228 equal parts of about 5.26 ¢ each. Each step represents a frequency ratio of 21/228, or the 228th root of 2.
It is the first merger of 12edo and 19edo, and its step size is the difference between 12edo's and 19edo's fifths. The equal temperament tempers out the Pythagorean comma, 531441/524288, in the 3-limit, and 225/224 and 250047/250000 in the 7-limit, so that it supports 7-limit compton temperament and in fact provides the optimal patent val. In the 11-limit it tempers out 225/224, 441/440, 1375/1372 and 4375/4356, so that it supports 11-limit compton. Aside from the Pythagorean comma, the 12-comma, it tempers out the enneadeca or 19-tone-comma, and this is reflected in the fact that 228 = 12 × 19.
Odd harmonics
Approximation of odd harmonics in 228edo
| Harmonic
|
3
|
5
|
7
|
9
|
11
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13
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15
|
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 (¢)
|
-1.96
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-2.10
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-0.40
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+1.35
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+1.31
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+1.58
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+1.20
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+0.31
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+2.49
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-2.36
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-1.96
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| Relative (%)
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-37.1
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-40.0
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-7.7
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+25.7
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+25.0
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+30.0
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+22.9
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+5.8
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+47.3
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-44.8
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-37.2
|
Steps
(reduced)
|
361
(133)
|
529
(73)
|
640
(184)
|
723
(39)
|
789
(105)
|
844
(160)
|
891
(207)
|
932
(20)
|
969
(57)
|
1001
(89)
|
1031
(119)
|