37edo

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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.

37 edo is also a very accurate equal tuning for Undecimation Temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a 7L+2s nonatonic MOS, which in 37-edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, making it a variety of Mavila; as well as a 16 note MOS.



[[toc|flat]]
----

=Subgroups= 
37edo offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].

12\37 = 389.2 cents
30\37 = 973.0 cents
17\37 = 551.4 cents
26\37 = 843.2 cents
[6\37edo = 194.6 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:

The flat fifth is 21\37 = 681.1 cents
The sharp fifth is 22\37 = 713.5 cents

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.

Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.

37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below).

=Intervals= 
||~ Degrees of 37edo ||~ Cents Value ||~ Approximate Ratios
of 2.5.7.11.13.27 subgroup ||~ Ratios of 3 with
a sharp 3/2 ||~ Ratios of 3 with
a flat 3/2 ||~ Ratios of 9 with
194.59¢ 9/8 ||
|| 0 || 0.00 || 1/1 ||   ||   ||   ||
|| 1 || 32.43 ||   ||   ||   ||   ||
|| 2 || 64.86 || 28/27, 27/26 ||   ||   ||   ||
|| 3 || 97.30 ||   ||   ||   ||   ||
|| 4 || 129.73 || 14/13 || 13/12 || 12/11 ||   ||
|| 5 || 162.16 || 11/10 || 12/11 || 13/12 ||   ||
|| 6 || 194.59 ||   ||   ||   || 9/8, 10/9 ||
|| 7 || 227.03 || 8/7 ||   ||   ||   ||
|| 8 || 259.46 ||   || 7/6 ||   ||   ||
|| 9 || 291.89 || 13/11, 32/27 ||   || 6/5, 7/6 ||   ||
|| 10 || 324.32 ||   || 6/5 ||   ||   ||
|| 11 || 356.76 || 16/13, 27/22 ||   ||   || 11/9 ||
|| 12 || 389.19 || 5/4 ||   ||   ||   ||
|| 13 || 421.62 || 14/11 ||   ||   || 9/7 ||
|| 14 || 454.05 || 13/10 ||   ||   ||   ||
|| 15 || 486.49 ||   || 4/3 ||   ||   ||
|| 16 || 518.92 || 27/20 ||   || 4/3 ||   ||
|| 17 || 551.35 || 11/8 ||   ||   || 18/13 ||
|| 18 || 583.78 || 7/5 ||   ||   ||   ||
|| 19 || 616.22 || 10/7 ||   ||   ||   ||
|| 20 || 648.65 || 16/11 ||   ||   || 13/9 ||
|| 21 || 681.08 || 40/27 ||   || 3/2 ||   ||
|| 22 || 713.51 ||   || 3/2 ||   ||   ||
|| 23 || 745.95 || 20/13 ||   ||   ||   ||
|| 24 || 778.38 || 11/7 ||   ||   || 14/9 ||
|| 25 || 810.81 || 8/5 ||   ||   ||   ||
|| 26 || 843.24 || 13/8, 44/27 ||   ||   || 18/11 ||
|| 27 || 875.68 ||   || 5/3 ||   ||   ||
|| 28 || 908.11 || 22/13, 27/16 ||   || 5/3, 12/7 ||   ||
|| 29 || 940.54 ||   || 12/7 ||   ||   ||
|| 30 || 972.97 || 7/4 ||   ||   ||   ||
|| 31 || 1005.41 ||   ||   ||   || 16/9, 9/5 ||
|| 32 || 1037.84 || 20/11 || 11/6 || 24/13 ||   ||
|| 33 || 1070.27 || 13/7 || 24/13 || 11/6 ||   ||
|| 34 || 1102.70 ||   ||   ||   ||   ||
|| 35 || 1135.14 || 27/14, 52/27 ||   ||   ||   ||
|| 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 />
37 edo is also a very accurate equal tuning for Undecimation Temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a 7L+2s nonatonic MOS, which in 37-edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, making it a variety of Mavila; as well as a 16 note MOS.<br />
<br />
<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 [and a usable approximation of 9 as well].<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 />
[6\37edo = 194.6 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 />
The flat fifth is 21\37 = 681.1 cents<br />
The sharp fifth is 22\37 = 713.5 cents<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 />
<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 />
Interestingly, the &quot;major thirds&quot; of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.<br />
<br />
37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below).<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>
        <th>Degrees of 37edo<br />
</th>
        <th>Cents Value<br />
</th>
        <th>Approximate Ratios<br />
of 2.5.7.11.13.27 subgroup<br />
</th>
        <th>Ratios of 3 with<br />
a sharp 3/2<br />
</th>
        <th>Ratios of 3 with<br />
a flat 3/2<br />
</th>
        <th>Ratios of 9 with<br />
194.59¢ 9/8<br />
</th>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.00<br />
</td>
        <td>1/1<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>32.43<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>64.86<br />
</td>
        <td>28/27, 27/26<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>97.30<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>129.73<br />
</td>
        <td>14/13<br />
</td>
        <td>13/12<br />
</td>
        <td>12/11<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>162.16<br />
</td>
        <td>11/10<br />
</td>
        <td>12/11<br />
</td>
        <td>13/12<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>194.59<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9/8, 10/9<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>227.03<br />
</td>
        <td>8/7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>259.46<br />
</td>
        <td><br />
</td>
        <td>7/6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>291.89<br />
</td>
        <td>13/11, 32/27<br />
</td>
        <td><br />
</td>
        <td>6/5, 7/6<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>324.32<br />
</td>
        <td><br />
</td>
        <td>6/5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>356.76<br />
</td>
        <td>16/13, 27/22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11/9<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>389.19<br />
</td>
        <td>5/4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>421.62<br />
</td>
        <td>14/11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9/7<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>454.05<br />
</td>
        <td>13/10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>486.49<br />
</td>
        <td><br />
</td>
        <td>4/3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>518.92<br />
</td>
        <td>27/20<br />
</td>
        <td><br />
</td>
        <td>4/3<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>551.35<br />
</td>
        <td>11/8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>18/13<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>583.78<br />
</td>
        <td>7/5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>616.22<br />
</td>
        <td>10/7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>648.65<br />
</td>
        <td>16/11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13/9<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>681.08<br />
</td>
        <td>40/27<br />
</td>
        <td><br />
</td>
        <td>3/2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>713.51<br />
</td>
        <td><br />
</td>
        <td>3/2<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>745.95<br />
</td>
        <td>20/13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>778.38<br />
</td>
        <td>11/7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>14/9<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>810.81<br />
</td>
        <td>8/5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>843.24<br />
</td>
        <td>13/8, 44/27<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>18/11<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>875.68<br />
</td>
        <td><br />
</td>
        <td>5/3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>908.11<br />
</td>
        <td>22/13, 27/16<br />
</td>
        <td><br />
</td>
        <td>5/3, 12/7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>940.54<br />
</td>
        <td><br />
</td>
        <td>12/7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>972.97<br />
</td>
        <td>7/4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>1005.41<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>16/9, 9/5<br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>1037.84<br />
</td>
        <td>20/11<br />
</td>
        <td>11/6<br />
</td>
        <td>24/13<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>1070.27<br />
</td>
        <td>13/7<br />
</td>
        <td>24/13<br />
</td>
        <td>11/6<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>1102.70<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>1135.14<br />
</td>
        <td>27/14, 52/27<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>1167.57<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><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>