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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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2013-05-12 07:05:36 UTC</tt>.<br>
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| : The original revision id was <tt>430853780</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">=<span style="background-color: #e5e5e5; color: #008693; font-family: 'Times New Roman',Times,serif; font-size: 113%;">38 tone equal temperament</span>=
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| **38edo** divides the octave into 38 equal parts of 31.579 [[cent]]s. Since 38 = 2*19, it can be thought of as two parallel [[19edo]]s. It [[tempering out|tempers out]] the same [[5-limit]] commas as [[19edo|19]], namely 81/80, 3125/3072 and 15625/15552. In the [[7-limit]], we can add 50/49, and tempering out 81/80 and 50/49 gives [[Meantone family|injera temperament]], for which 38 is the [[optimal patent val]]. In the [[11-limit]], we can add 121/120 and 176/175.
| | == Theory == |
| | Since {{nowrap|38 {{=}} 2 × 19}}, it can be thought of as two parallel [[19edo]]s. While the halving of the step size lowers [[consistency]] and leaves it only mediocre in terms of overall [[relative interval error|relative error]], the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] & [[25/22]], (and their inversions) while a single step nears [[55/54]]; the approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. |
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| |
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| ===38-EDO Intervals:===
| | It [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]]. In the [[11-limit]], we can add 121/120 and 176/175. |
| || **Step** || **Size in Cents** ||
| |
| || 0 || 0 ||
| |
| || 1 || 31.579 ||
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| || 2 || 63.158 ||
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| || 3 || 94.737 ||
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| || 4 || 126.316 ||
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| || 5 || 157.895 ||
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| || 6 || 189.474 ||
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| || 7 || 221.053 ||
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| || 8 || 252.632 ||
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| || 9 || 284.211 ||
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| || 10 || 315.789 ||
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| || 11 || 347.368 ||
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| || 12 || 378.947 ||
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| || 13 || 410.526 ||
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| || 14 || 442.105 ||
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| || 15 || 473.684 ||
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| || 16 || 505.263 ||
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| || 17 || 536.842 ||
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| || 18 || 568.421 ||
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| || 19 || 600 ||
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| || 20 || 631.579 ||
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| || 21 || 663.158 ||
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| || 22 || 694.737 ||
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| || 23 || 726.316 ||
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| || 24 || 757.895 ||
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| || 25 || 789.474 ||
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| || 26 || 821.053 ||
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| || 27 || 852.632 ||
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| || 28 || 884.211 ||
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| || 29 || 915.789 ||
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| || 30 || 947.368 ||
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| || 31 || 978.947 ||
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| || 32 || 1010.526 ||
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| || 33 || 1042.105 ||
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| || 34 || 1073.684 ||
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| || 35 || 1105.263 ||
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| || 36 || 1136.842 ||
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| || 37 || 1168.421 ||</pre></div>
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| <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>38edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x38 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="background-color: #e5e5e5; color: #008693; font-family: 'Times New Roman',Times,serif; font-size: 113%;">38 tone equal temperament</span></h1>
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| <strong>38edo</strong> divides the octave into 38 equal parts of 31.579 <a class="wiki_link" href="/cent">cent</a>s. Since 38 = 2*19, it can be thought of as two parallel <a class="wiki_link" href="/19edo">19edo</a>s. It <a class="wiki_link" href="/tempering%20out">tempers out</a> the same <a class="wiki_link" href="/5-limit">5-limit</a> commas as <a class="wiki_link" href="/19edo">19</a>, namely 81/80, 3125/3072 and 15625/15552. In the <a class="wiki_link" href="/7-limit">7-limit</a>, we can add 50/49, and tempering out 81/80 and 50/49 gives <a class="wiki_link" href="/Meantone%20family">injera temperament</a>, for which 38 is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a>. In the <a class="wiki_link" href="/11-limit">11-limit</a>, we can add 121/120 and 176/175.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x38 tone equal temperament--38-EDO Intervals:"></a><!-- ws:end:WikiTextHeadingRule:2 -->38-EDO Intervals:</h3>
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| <table class="wiki_table">
| | Using the [[Warts|38df]] mapping, every [[prime interval]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. In other words, all 19-odd-limit intervals are [[consistency|consistent]] within the 38df [[val]] {{val| 38 60 88 106 131 140 155 161 }}. |
| <tr>
| |
| <td><strong>Step</strong><br />
| |
| </td>
| |
| <td><strong>Size in Cents</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>1<br />
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| </td>
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| <td>31.579<br />
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| </td>
| |
| </tr>
| |
| <tr>
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| <td>2<br />
| |
| </td>
| |
| <td>63.158<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>94.737<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>126.316<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>5<br />
| |
| </td>
| |
| <td>157.895<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>6<br />
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| </td>
| |
| <td>189.474<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>7<br />
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| </td>
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| <td>221.053<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>8<br />
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| </td>
| |
| <td>252.632<br />
| |
| </td>
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| </tr>
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| <tr>
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| <td>9<br />
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| </td>
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| <td>284.211<br />
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| </td>
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| </tr>
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| <tr>
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| <td>10<br />
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| </td>
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| <td>315.789<br />
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| </td>
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| </tr>
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| <tr>
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| <td>11<br />
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| </td>
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| <td>347.368<br />
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| </td>
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| </tr>
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| <tr>
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| <td>12<br />
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| </td>
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| <td>378.947<br />
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| </td>
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| </tr>
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| <tr>
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| <td>13<br />
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| </td>
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| <td>410.526<br />
| |
| </td>
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| </tr>
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| <tr>
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| <td>14<br />
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| </td>
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| <td>442.105<br />
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| </td>
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| </tr>
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| <tr>
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| <td>15<br />
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| </td>
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| <td>473.684<br />
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| </td>
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| </tr>
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| <tr>
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| <td>16<br />
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| </td>
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| <td>505.263<br />
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| </td>
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| </tr>
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| <tr>
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| <td>17<br />
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| </td>
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| <td>536.842<br />
| |
| </td>
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| </tr>
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| <tr>
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| <td>18<br />
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| </td>
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| <td>568.421<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>19<br />
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| </td>
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| <td>600<br />
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| </td>
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| </tr>
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| <tr>
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| <td>20<br />
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| </td>
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| <td>631.579<br />
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| </td>
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| </tr>
| |
| <tr>
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| <td>21<br />
| |
| </td>
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| <td>663.158<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>22<br />
| |
| </td>
| |
| <td>694.737<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
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| </td>
| |
| <td>726.316<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>24<br />
| |
| </td>
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| <td>757.895<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>789.474<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>26<br />
| |
| </td>
| |
| <td>821.053<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>27<br />
| |
| </td>
| |
| <td>852.632<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>28<br />
| |
| </td>
| |
| <td>884.211<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>915.789<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>947.368<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>31<br />
| |
| </td>
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| <td>978.947<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>32<br />
| |
| </td>
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| <td>1010.526<br />
| |
| </td>
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| </tr>
| |
| <tr>
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| <td>33<br />
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| </td>
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| <td>1042.105<br />
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| </td>
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| </tr>
| |
| <tr>
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| <td>34<br />
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| </td>
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| <td>1073.684<br />
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| </td>
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| </tr>
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| <tr>
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| <td>35<br />
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| </td>
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| <td>1105.263<br />
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| </td>
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| </tr>
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| <tr>
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| <td>36<br />
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| </td>
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| <td>1136.842<br />
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| </td>
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| </tr>
| |
| <tr>
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| <td>37<br />
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| </td>
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| <td>1168.421<br />
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| </td>
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| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | The harmonic series from 1 to 20 is approximated within 38df by the sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }} |
| | |
| | [[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]] |
| | |
| | === Prime harmonics === |
| | {{Harmonics in equal|38}} |
| | |
| | == Intervals == |
| | {| class="wikitable center-1 right-2" |
| | |- |
| | ! Step |
| | ! Cents |
| | ! 19-odd-limit ratios,<br>in 38df val |
| | ! colspan="3" | [[Ups and downs notation]]* |
| | ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2) |
| | |- |
| | | 0 |
| | | 0.0 |
| | | |
| | | Perfect 1sn |
| | | P1 |
| | | D |
| | |- |
| | | 1 |
| | | 31.6 |
| | | |
| | | Up 1sn |
| | | ^1 |
| | | ^D |
| | |- |
| | | 2 |
| | | 63.2 |
| | | |
| | | Aug 1sn, dim 2nd |
| | | A1, d2 |
| | | D# |
| | |- |
| | | 3 |
| | | 94.7 |
| | | [[20/19]], [[19/18]], [[18/17]], [[17/16]] |
| | | Upaug 1sn, downminor 2nd |
| | | ^A1, vm2 |
| | | ^D#, vEb |
| | |- |
| | | 4 |
| | | 126.3 |
| | | [[16/15]], [[15/14]], [[14/13]], [[13/12]] |
| | | Minor 2nd |
| | | m2 |
| | | Eb |
| | |- |
| | | 5 |
| | | 157.9 |
| | | [[12/11]], [[11/10]] |
| | | Mid 2nd |
| | | ~2 |
| | | vE |
| | |- |
| | | 6 |
| | | 189.5 |
| | | [[10/9]], [[19/17]], [[9/8]] |
| | | Major 2nd |
| | | M2 |
| | | E |
| | |- |
| | | 7 |
| | | 221.1 |
| | | [[17/15]] |
| | | Upmajor 2nd |
| | | ^M2 |
| | | ^E |
| | |- |
| | | 8 |
| | | 252.6 |
| | | [[8/7]], [[15/13]], [[22/19]], [[7/6]] |
| | | Aug 2nd, Dim 3rd |
| | | A2, d3 |
| | | E#, Fb |
| | |- |
| | | 9 |
| | | 284.2 |
| | | [[20/17]], [[13/11]], [[19/16]] |
| | | Downminor 3rd |
| | | vm3 |
| | | vF |
| | |- |
| | | 10 |
| | | 315.8 |
| | | [[6/5]] |
| | | Minor 3rd |
| | | m3 |
| | | F |
| | |- |
| | | 11 |
| | | 347.4 |
| | | [[17/14]], [[11/9]] |
| | | Mid 3rd |
| | | ~3 |
| | | ^F |
| | |- |
| | | 12 |
| | | 378.9 |
| | | [[16/13]], [[5/4]] |
| | | Major 3rd |
| | | M3 |
| | | F# |
| | |- |
| | | 13 |
| | | 410.5 |
| | | [[24/19]], [[19/15]], [[14/11]] |
| | | Upmajor 3rd, Downdim 4th |
| | | ^M3, vd4 |
| | | ^F#, vGb |
| | |- |
| | | 14 |
| | | 442.1 |
| | | [[9/7]], [[22/17]], [[13/10]] |
| | | Aug 3rd, dim 4th |
| | | A3, d4 |
| | | Gb |
| | |- |
| | | 15 |
| | | 473.7 |
| | | [[17/13]] |
| | | Down 4th |
| | | v4 |
| | | vG |
| | |- |
| | | 16 |
| | | 505.3 |
| | | [[4/3]] |
| | | Perfect 4th |
| | | P4 |
| | | G |
| | |- |
| | | 17 |
| | | 536.8 |
| | | [[19/14]], [[15/11]], [[26/19]], [[11/8]] |
| | | Up 4th |
| | | ^4 |
| | | ^G |
| | |- |
| | | 18 |
| | | 568.4 |
| | | [[18/13]], [[7/5]] |
| | | Aug 4th |
| | | A4 |
| | | G# |
| | |- |
| | | 19 |
| | | 600.0 |
| | | [[24/17]], [[17/12]] |
| | | Upaug 4th, downdim 5th |
| | | ^A4, vd5 |
| | | ^G#, vAb |
| | |- |
| | | 20 |
| | | 631.6 |
| | | [[10/7]], [[13/9]] |
| | | Dim 5th |
| | | d5 |
| | | Ab |
| | |- |
| | | 21 |
| | | 663.2 |
| | | [[16/11]], [[19/13]], [[22/15]], [[28/19]] |
| | | Down 5th |
| | | v5 |
| | | vA |
| | |- |
| | | 22 |
| | | 694.7 |
| | | [[3/2]] |
| | | Perfect 5th |
| | | P5 |
| | | A |
| | |- |
| | | 23 |
| | | 726.3 |
| | | [[26/17]] |
| | | Up 5th |
| | | ^5 |
| | | ^A |
| | |- |
| | | 24 |
| | | 757.9 |
| | | [[20/13]], [[17/11]], [[14/9]] |
| | | Aug 5th, dim 6th |
| | | A5, d6 |
| | | A# |
| | |- |
| | | 25 |
| | | 789.5 |
| | | [[11/7]], [[30/19]], [[19/12]] |
| | | Upaug 5th, downminor 6th |
| | | ^A5, vm6 |
| | | ^A#, vBb |
| | |- |
| | | 26 |
| | | 821.1 |
| | | [[8/5]], [[13/8]] |
| | | Minor 6th |
| | | m6 |
| | | Bb |
| | |- |
| | | 27 |
| | | 852.6 |
| | | [[18/11]], [[28/17]] |
| | | Mid 6th |
| | | ~6 |
| | | vB |
| | |- |
| | | 28 |
| | | 884.2 |
| | | [[5/3]] |
| | | Major 6th |
| | | M6 |
| | | B |
| | |- |
| | | 29 |
| | | 915.8 |
| | | [[32/19]], [[22/13]], [[17/10]] |
| | | Upmajor 6th |
| | | ^M6 |
| | | ^B |
| | |- |
| | | 30 |
| | | 947.4 |
| | | [[12/7]], [[19/11]], [[26/15]], [[7/4]] |
| | | Aug 6th, dim 7th |
| | | A6, d7 |
| | | B#, Cb |
| | |- |
| | | 31 |
| | | 978.9 |
| | | [[30/17]] |
| | | Downminor 7th |
| | | vm7 |
| | | vC |
| | |- |
| | | 32 |
| | | 1010.5 |
| | | [[16/9]], [[34/19]], [[9/5]] |
| | | Minor 7th |
| | | m7 |
| | | C |
| | |- |
| | | 33 |
| | | 1042.1 |
| | | [[20/11]], [[11/6]] |
| | | Mid 7th |
| | | ~7 |
| | | ^C |
| | |- |
| | | 34 |
| | | 1073.7 |
| | | [[24/13]], [[13/7]], [[28/15]], [[15/8]] |
| | | Major 7th |
| | | M7 |
| | | C# |
| | |- |
| | | 35 |
| | | 1105.3 |
| | | [[32/17]], [[17/9]], [[36/19]], [[19/10]] |
| | | Upmajor 7th, Downdim 8ve |
| | | ^M7, vd8 |
| | | ^C#, vDb |
| | |- |
| | | 36 |
| | | 1136.8 |
| | | |
| | | Aug 7th, dim 8ve |
| | | A7, d8 |
| | | Db |
| | |- |
| | | 37 |
| | | 1168.4 |
| | | |
| | | Down 8ve |
| | | v8 |
| | | vD |
| | |- |
| | | 38 |
| | | 1200.0 |
| | | |
| | | Perfect 8ve |
| | | P8 |
| | | D |
| | |} |
| | <nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively |
| | |
| | == Notation == |
| | === Ups and downs notation === |
| | Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. |
| | {{sharpness-sharp2a}} |
| | |
| | === Quarter-tone notation === |
| | Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used: |
| | {{sharpness-sharp2}} |
| | |
| | === Sagittal notation === |
| | This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]]. |
| | |
| | ==== Evo flavor ==== |
| | <imagemap> |
| | File:38-EDO_Evo_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 130 106 [[33/32]] |
| | default [[File:38-EDO_Evo_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Revo flavor ==== |
| | <imagemap> |
| | File:38-EDO_Revo_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 130 106 [[33/32]] |
| | default [[File:38-EDO_Revo_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Evo-SZ flavor ==== |
| | <imagemap> |
| | File:38-EDO_Evo-SZ_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 130 106 [[33/32]] |
| | default [[File:38-EDO_Evo-SZ_Sagittal.svg]] |
| | </imagemap> |
| | |
| | Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation. |
| | |
| | == Approximation to JI == |
| | === Interval mappings === |
| | {{Q-odd-limit intervals}} |
| | |
| | == Instruments == |
| | * [[Lumatone mapping for 38edo]] |
| | * [[Skip fretting system 38 2 11]] |
| | |
| | == Music == |
| | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025) |
| | |
| | ; [[Claudi Meneghin]] |
| | * [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022) |
| | |
| | [[Category:38edo| ]] <!-- Main article --> |
| | [[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> |
| | [[Category:Listen]] |
| | [[Category:Todo:add rank 2 temperaments table]] |