25edo: Difference between revisions

Revision as of 16:37, 8 August 2011

IMPORTED REVISION FROM WIKISPACES

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This revision was by author xenwolf and made on 2011-08-08 16:37:31 UTC.

The original revision id was 244898861.

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Original Wikitext content:

[[toc|flat]]
=<span style="color: #006b2e;">25 tone equal temperament</span>= 

25EDO divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It is a good way to tune the [[Blackwood temperament]], which takes the very sharp fifths of [[5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5_4|5/4]]) and 7 ([[7_4|7/4]]?).

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8_7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]]. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.

If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.

=Music= 
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Rapoport/StudyInFives.mp3|Study in Fives]] by [[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29|Paul Rapoport]]

=Intervals= 

|| Degrees || Cents value ||= Approximate
Ratios* ||
|| 0 || 0 ||= 1/1 ||
|| 1 || 48 ||= 33/32, 39/38, 34/33 ||
|| 2 || 96 ||= 17/16, 20/19, 18/17 ||
|| 3 || 144 ||= 12/11, 38/35 ||
|| 4 || 192 ||= 9/8, 10/9, 19/17 ||
|| 5· || 240 ||= 8/7 ||
|| 6 || 288 ||= 19/16, 20/17 ||
|| 7 || 336 ||= 39/32, 17/14, 40/33 ||
|| 8· || 384 ||= 5/4 ||
|| 9 || 432 ||= 9/7, 32/25, 50/39 ||
|| 10 || 480 ||= 33/25, 25/19 ||
|| 11· || 528 ||= 31/21, 34/25 ||
|| 12 || 576 ||= 7/5, 39/28 ||
|| 13 || 624 ||= 10/7, 56/39 ||
|| 14· || 672 ||= 42/31, 25/17 ||
|| 15 || 720 ||= 50/33, 38/25 ||
|| 16 || 768 ||= 14/9, 25/16, 39/25 ||
|| 17· || 816 ||= 8/5 ||
|| 18 || 864 ||= 64/39, 28/17, 33/20 ||
|| 19 || 912 ||= 32/19, 17/10 ||
|| 20· || 960 ||= 7/4 ||
|| 21 || 1008 ||= 16/9, 9/5, 34/19 ||
|| 22 || 1056 ||= 11/6, 35/19 ||
|| 23 || 1104 ||= 32/17, 17/9, 19/10 ||
|| 24 || 1152 ||= 33/17, 64/33, 76/39 ||
|| 25·· || 1200 ||= 2/1 ||
*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.
=Commas= 
25 EDO tempers out the following commas. (Note: This assumes the val < 25 40 58 70 86 93 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 ||< | 8 -5 > ||> 90.22 ||= Limma ||= Pythagorean Minor 2nd ||=   ||
||= 3125/3072 ||< | -10 -1 5 > ||> 29.61 ||= Small Diesis ||= Magic Comma ||=   ||
||= 6719816/6714445 ||< | 38 -2 -15 > ||> 1.38 ||= Hemithirds Comma ||=   ||=   ||
||= 16807/16384 || | -14 0 0 5 > ||> 44.13 ||   ||   ||   ||
||= 49/48 ||< | -4 -1 0 2 > ||> 35.70 ||= Slendro Diesis ||=   ||=   ||
||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 ||< | 0 -2 5 -3 > ||> 21.18 ||= Gariboh ||=   ||=   ||
||= 50421/50000 ||< | -4 1 -5 5 > ||> 14.52 ||= Trimyna ||=   ||=   ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||=   ||=   ||
||= 3136/3125 ||< | 6 0 -5 2 > ||> 6.08 ||= Hemimean ||=   ||=   ||
||= 65625/65536 ||< | -16 1 5 1 > ||> 2.35 ||= Horwell ||=   ||=   ||
||= 100/99 ||< | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||=   ||
||= 91/90 ||< | -1 -2 -1 1 0 1 > ||> 19.13 ||= Superleap ||=   ||=   ||
||= 676/675 ||< | 2 -3 -2 0 0 2 > ||> 2.56 ||= Parizeksma ||=   ||=   ||

=A 25edo keyboard= 

[[image:mm25.PNG]]

Original HTML content:

<html><head><title>25edo</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#x25 tone equal temperament">25 tone equal temperament</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#A 25edo keyboard">A 25edo keyboard</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: -->
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x25 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #006b2e;">25 tone equal temperament</span></h1>
 <br />
25EDO divides the <a class="wiki_link" href="/octave">octave</a> in 25 equal steps of exact size 48 <a class="wiki_link" href="/cent">cent</a>s each. It is a good way to tune the <a class="wiki_link" href="/Blackwood%20temperament">Blackwood temperament</a>, which takes the very sharp fifths of <a class="wiki_link" href="/5EDO">5EDO</a> as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (<a class="wiki_link" href="/5_4">5/4</a>) and 7 (<a class="wiki_link" href="/7_4">7/4</a>?).<br />
<br />
25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> tuning. Looking just at 2, 5, and 7, it equates five <a class="wiki_link" href="/8_7">8/7</a>s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a <a class="wiki_link" href="/128_125">128/125</a> <a class="wiki_link" href="/diesis">diesis</a> and two <a class="wiki_link" href="/septimal%20tritones">septimal tritones</a> of <a class="wiki_link" href="/7_5">7/5</a> with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is <a class="wiki_link" href="/50EDO">50EDO</a>. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for <a class="wiki_link" href="/mavila">mavila</a> temperament.<br />
<br />
If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the <a class="wiki_link" href="/k%2AN%20subgroups">2*25 subgroup</a> 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:2 -->Music</h1>
 <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Rapoport/StudyInFives.mp3" rel="nofollow">Study in Fives</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29" rel="nofollow">Paul Rapoport</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1>
 <br />


<table class="wiki_table">
    <tr>
        <td>Degrees<br />
</td>
        <td>Cents value<br />
</td>
        <td style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>48<br />
</td>
        <td style="text-align: center;">33/32, 39/38, 34/33<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>96<br />
</td>
        <td style="text-align: center;">17/16, 20/19, 18/17<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>144<br />
</td>
        <td style="text-align: center;">12/11, 38/35<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>192<br />
</td>
        <td style="text-align: center;">9/8, 10/9, 19/17<br />
</td>
    </tr>
    <tr>
        <td>5·<br />
</td>
        <td>240<br />
</td>
        <td style="text-align: center;">8/7<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>288<br />
</td>
        <td style="text-align: center;">19/16, 20/17<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>336<br />
</td>
        <td style="text-align: center;">39/32, 17/14, 40/33<br />
</td>
    </tr>
    <tr>
        <td>8·<br />
</td>
        <td>384<br />
</td>
        <td style="text-align: center;">5/4<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>432<br />
</td>
        <td style="text-align: center;">9/7, 32/25, 50/39<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>480<br />
</td>
        <td style="text-align: center;">33/25, 25/19<br />
</td>
    </tr>
    <tr>
        <td>11·<br />
</td>
        <td>528<br />
</td>
        <td style="text-align: center;">31/21, 34/25<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>576<br />
</td>
        <td style="text-align: center;">7/5, 39/28<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>624<br />
</td>
        <td style="text-align: center;">10/7, 56/39<br />
</td>
    </tr>
    <tr>
        <td>14·<br />
</td>
        <td>672<br />
</td>
        <td style="text-align: center;">42/31, 25/17<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>720<br />
</td>
        <td style="text-align: center;">50/33, 38/25<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>768<br />
</td>
        <td style="text-align: center;">14/9, 25/16, 39/25<br />
</td>
    </tr>
    <tr>
        <td>17·<br />
</td>
        <td>816<br />
</td>
        <td style="text-align: center;">8/5<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>864<br />
</td>
        <td style="text-align: center;">64/39, 28/17, 33/20<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>912<br />
</td>
        <td style="text-align: center;">32/19, 17/10<br />
</td>
    </tr>
    <tr>
        <td>20·<br />
</td>
        <td>960<br />
</td>
        <td style="text-align: center;">7/4<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>1008<br />
</td>
        <td style="text-align: center;">16/9, 9/5, 34/19<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>1056<br />
</td>
        <td style="text-align: center;">11/6, 35/19<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>1104<br />
</td>
        <td style="text-align: center;">32/17, 17/9, 19/10<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>1152<br />
</td>
        <td style="text-align: center;">33/17, 64/33, 76/39<br />
</td>
    </tr>
    <tr>
        <td>25··<br />
</td>
        <td>1200<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
    </tr>
</table>

*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1>
 25 EDO tempers out the following commas. (Note: This assumes the val &lt; 25 40 58 70 86 93 |.)<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
        <th>Name 3<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">256/243<br />
</td>
        <td style="text-align: left;">| 8 -5 &gt;<br />
</td>
        <td style="text-align: right;">90.22<br />
</td>
        <td style="text-align: center;">Limma<br />
</td>
        <td style="text-align: center;">Pythagorean Minor 2nd<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3125/3072<br />
</td>
        <td style="text-align: left;">| -10 -1 5 &gt;<br />
</td>
        <td style="text-align: right;">29.61<br />
</td>
        <td style="text-align: center;">Small Diesis<br />
</td>
        <td style="text-align: center;">Magic Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6719816/6714445<br />
</td>
        <td style="text-align: left;">| 38 -2 -15 &gt;<br />
</td>
        <td style="text-align: right;">1.38<br />
</td>
        <td style="text-align: center;">Hemithirds Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16807/16384<br />
</td>
        <td>| -14 0 0 5 &gt;<br />
</td>
        <td style="text-align: right;">44.13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">49/48<br />
</td>
        <td style="text-align: left;">| -4 -1 0 2 &gt;<br />
</td>
        <td style="text-align: right;">35.70<br />
</td>
        <td style="text-align: center;">Slendro Diesis<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64/63<br />
</td>
        <td style="text-align: left;">| 6 -2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">27.26<br />
</td>
        <td style="text-align: center;">Septimal Comma<br />
</td>
        <td style="text-align: center;">Archytas' Comma<br />
</td>
        <td style="text-align: center;">Leipziger Komma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3125/3087<br />
</td>
        <td style="text-align: left;">| 0 -2 5 -3 &gt;<br />
</td>
        <td style="text-align: right;">21.18<br />
</td>
        <td style="text-align: center;">Gariboh<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50421/50000<br />
</td>
        <td style="text-align: left;">| -4 1 -5 5 &gt;<br />
</td>
        <td style="text-align: right;">14.52<br />
</td>
        <td style="text-align: center;">Trimyna<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: left;">| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: right;">8.43<br />
</td>
        <td style="text-align: center;">Gamelisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3136/3125<br />
</td>
        <td style="text-align: left;">| 6 0 -5 2 &gt;<br />
</td>
        <td style="text-align: right;">6.08<br />
</td>
        <td style="text-align: center;">Hemimean<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">65625/65536<br />
</td>
        <td style="text-align: left;">| -16 1 5 1 &gt;<br />
</td>
        <td style="text-align: right;">2.35<br />
</td>
        <td style="text-align: center;">Horwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">100/99<br />
</td>
        <td style="text-align: left;">| 2 -2 2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">17.40<br />
</td>
        <td style="text-align: center;">Ptolemisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td style="text-align: left;">| 4 0 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.86<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td style="text-align: left;">| -1 -2 -1 1 0 1 &gt;<br />
</td>
        <td style="text-align: right;">19.13<br />
</td>
        <td style="text-align: center;">Superleap<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">676/675<br />
</td>
        <td style="text-align: left;">| 2 -3 -2 0 0 2 &gt;<br />
</td>
        <td style="text-align: right;">2.56<br />
</td>
        <td style="text-align: center;">Parizeksma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="A 25edo keyboard"></a><!-- ws:end:WikiTextHeadingRule:8 -->A 25edo keyboard</h1>
 <br />
<!-- ws:start:WikiTextLocalImageRule:461:&lt;img src=&quot;/file/view/mm25.PNG/179204243/mm25.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/mm25.PNG/179204243/mm25.PNG" alt="mm25.PNG" title="mm25.PNG" /><!-- ws:end:WikiTextLocalImageRule:461 --></body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-07 15:26:38 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-08-08 16:37:31 UTC</tt>.<br>
-: The original revision id was <tt>244722873</tt>.<br>
+: The original revision id was <tt>244898861</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 11: / Line 11: @@
 EDO divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It is a good way to tune the [[Blackwood temperament]], which takes the very sharp fifths of [[5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5_4|5/4]]) and 7 ([[7_4|7/4]]?).
-EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8_7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]]. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.
+EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8_7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]]. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.
 If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.
@@ Line 77: / Line 77: @@
 EDO divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; in 25 equal steps of exact size 48 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It is a good way to tune the &lt;a class="wiki_link" href="/Blackwood%20temperament"&gt;Blackwood temperament&lt;/a&gt;, which takes the very sharp fifths of &lt;a class="wiki_link" href="/5EDO"&gt;5EDO&lt;/a&gt; as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;) and 7 (&lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;?).&lt;br /&gt;
 &lt;br /&gt;
-EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; tuning. Looking just at 2, 5, and 7, it equates five &lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a &lt;a class="wiki_link" href="/128_125"&gt;128/125&lt;/a&gt; &lt;a class="wiki_link" href="/diesis"&gt;diesis&lt;/a&gt; and two &lt;a class="wiki_link" href="/septimal%20tritones"&gt;septimal tritones&lt;/a&gt; of &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is &lt;a class="wiki_link" href="/50EDO"&gt;50EDO&lt;/a&gt;. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.&lt;br /&gt;
+EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; tuning. Looking just at 2, 5, and 7, it equates five &lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a &lt;a class="wiki_link" href="/128_125"&gt;128/125&lt;/a&gt; &lt;a class="wiki_link" href="/diesis"&gt;diesis&lt;/a&gt; and two &lt;a class="wiki_link" href="/septimal%20tritones"&gt;septimal tritones&lt;/a&gt; of &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is &lt;a class="wiki_link" href="/50EDO"&gt;50EDO&lt;/a&gt;. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for &lt;a class="wiki_link" href="/mavila"&gt;mavila&lt;/a&gt; temperament.&lt;br /&gt;
 &lt;br /&gt;
 If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;2*25 subgroup&lt;/a&gt; 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.&lt;br /&gt;

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