37edo: Difference between revisions

Revision as of 13:09, 30 July 2011

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This revision was by author Sarzadoce and made on 2011-07-30 13:09:25 UTC.

The original revision id was 243514701.

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37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a variant of [[Porcupine family|porcupine temperament]]. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth.

[[toc|flat]]
----

=Subgroups= 
37edo offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13:

12\37 = 389.2 cents
30\37 = 973.0 cents
17\37 = 551.4 cents
26\37 = 843.2 cents

This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111et. In fact, on the larger [[k*N subgroups|3*37 subgroup]] 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111et, it tempers out the same commas.

=The Two Fifths= 
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:

21\37 = 681.1 cents
22\37 = 713.5 cents

37edo thus has the distinction of being the first [[edo]] which occupies two spaces on the syntonic spectrum.

21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6
"minor third" = 10\37 = 324.3 cents
"major third" = 11\37 = 356.8 cents

22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1
"minor third" = 8\37 = 259.5 cents
"major third" = 14\37 = 454.1 cents
If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The Biosphere|Biome]] temperament.

37edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions.

=Intervals= 
|| degrees of 37edo || cents value ||
|| 0 || 0.00 ||
|| 1 || 32.43 ||
|| 2 || 64.86 ||
|| 3 || 97.30 ||
|| 4 || 129.73 ||
|| 5 || 162.16 ||
|| 6 || 194.59 ||
|| 7 || 227.03 ||
|| 8 || 259.46 ||
|| 9 || 291.89 ||
|| 10 || 324.32 ||
|| 11 || 356.76 ||
|| 12 || 389.19 ||
|| 13 || 421.62 ||
|| 14 || 454.05 ||
|| 15 || 486.49 ||
|| 16 || 518.92 ||
|| 17 || 551.35 ||
|| 18 || 583.78 ||
|| 19 || 616.22 ||
|| 20 || 648.65 ||
|| 21 || 681.08 ||
|| 22 || 713.51 ||
|| 23 || 745.95 ||
|| 24 || 778.38 ||
|| 25 || 810.81 ||
|| 26 || 843.24 ||
|| 27 || 875.68 ||
|| 28 || 908.11 ||
|| 29 || 940.54 ||
|| 30 || 972.97 ||
|| 31 || 1005.41 ||
|| 32 || 1037.84 ||
|| 33 || 1070.27 ||
|| 34 || 1102.70 ||
|| 35 || 1135.14 ||
|| 36 || 1167.57 ||

=Scales= 

[[roulette6]]
[[roulette7]]
[[roulette13]]
[[roulette19]]

Original HTML content:

<html><head><title>37edo</title></head><body>37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a variant of <a class="wiki_link" href="/Porcupine%20family">porcupine temperament</a>. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth.<br />
<br />
<!-- ws:start:WikiTextTocRule:8:&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:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Subgroups">Subgroups</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#The Two Fifths">The Two Fifths</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
<!-- ws:end:WikiTextTocRule:13 --><hr />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Subgroups"></a><!-- ws:end:WikiTextHeadingRule:0 -->Subgroups</h1>
 37edo offers close approximations to <a class="wiki_link" href="/OverToneSeries">harmonics</a> 5, 7, 11, and 13:<br />
<br />
12\37 = 389.2 cents<br />
30\37 = 973.0 cents<br />
17\37 = 551.4 cents<br />
26\37 = 843.2 cents<br />
<br />
This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111et. In fact, on the larger <a class="wiki_link" href="/k%2AN%20subgroups">3*37 subgroup</a> 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111et, it tempers out the same commas.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="The Two Fifths"></a><!-- ws:end:WikiTextHeadingRule:2 -->The Two Fifths</h1>
 The just <a class="wiki_link" href="/perfect%20fifth">perfect fifth</a> of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:<br />
<br />
21\37 = 681.1 cents<br />
22\37 = 713.5 cents<br />
<br />
37edo thus has the distinction of being the first <a class="wiki_link" href="/edo">edo</a> which occupies two spaces on the syntonic spectrum.<br />
<br />
21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6<br />
&quot;minor third&quot; = 10\37 = 324.3 cents<br />
&quot;major third&quot; = 11\37 = 356.8 cents<br />
<br />
22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1<br />
&quot;minor third&quot; = 8\37 = 259.5 cents<br />
&quot;major third&quot; = 14\37 = 454.1 cents<br />
If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of <a class="wiki_link" href="/The%20Biosphere">Biome</a> temperament.<br />
<br />
37edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <td>degrees of 37edo<br />
</td>
        <td>cents value<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.00<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>32.43<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>64.86<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>97.30<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>129.73<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>162.16<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>194.59<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>227.03<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>259.46<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>291.89<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>324.32<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>356.76<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>389.19<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>421.62<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>454.05<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>486.49<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>518.92<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>551.35<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>583.78<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>616.22<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>648.65<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>681.08<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>713.51<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>745.95<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>778.38<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>810.81<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>843.24<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>875.68<br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>908.11<br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>940.54<br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>972.97<br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>1005.41<br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>1037.84<br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>1070.27<br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>1102.70<br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>1135.14<br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>1167.57<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->Scales</h1>
 <br />
<a class="wiki_link" href="/roulette6">roulette6</a><br />
<a class="wiki_link" href="/roulette7">roulette7</a><br />
<a class="wiki_link" href="/roulette13">roulette13</a><br />
<a class="wiki_link" href="/roulette19">roulette19</a></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>2011-07-30 12:04:07 UTC</tt>.<br>
+: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2011-07-30 13:09:25 UTC</tt>.<br>
-: The original revision id was <tt>243509555</tt>.<br>
+: The original revision id was <tt>243514701</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>
 <h4>Original Wikitext content:</h4>
-<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">37edo is the scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a [[Porcupine family|porcupine temperament]] tuning. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, giving a temperament where three minor whole tones make up a fifth.
+<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">37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a variant of [[Porcupine family|porcupine temperament]]. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth.
 [[toc|flat]]
@@ Line 36: / Line 36: @@
 "minor third" = 8\37 = 259.5 cents
 "major third" = 14\37 = 454.1 cents
-If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variety of [[The Biosphere|Biome]] temperament.
+If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The Biosphere|Biome]] temperament.
 edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions.
@@ Line 87: / Line 87: @@
 [[roulette19]]</pre></div>
 <h4>Original HTML content:</h4>
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;37edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;37edo is the scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a &lt;a class="wiki_link" href="/Porcupine%20family"&gt;porcupine temperament&lt;/a&gt; tuning. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, giving a temperament where three minor whole tones make up a fifth.&lt;br /&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;37edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a variant of &lt;a class="wiki_link" href="/Porcupine%20family"&gt;porcupine temperament&lt;/a&gt;. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Subgroups"&gt;Subgroups&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#The Two Fifths"&gt;The Two Fifths&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Scales"&gt;Scales&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
@@ Line 117: / Line 117: @@
 &amp;quot;minor third&amp;quot; = 8\37 = 259.5 cents&lt;br /&gt;
 &amp;quot;major third&amp;quot; = 14\37 = 454.1 cents&lt;br /&gt;
-If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variety of &lt;a class="wiki_link" href="/The%20Biosphere"&gt;Biome&lt;/a&gt; temperament.&lt;br /&gt;
+If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of &lt;a class="wiki_link" href="/The%20Biosphere"&gt;Biome&lt;/a&gt; temperament.&lt;br /&gt;
 &lt;br /&gt;
 edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions.&lt;br /&gt;

37edo: Difference between revisions

Revision as of 13:09, 30 July 2011

IMPORTED REVISION FROM WIKISPACES

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