37edo
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Original Wikitext content:
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; as well as a 16 note MOS. [[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 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 />
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; as well as a 16 note MOS.<br />
<br />
<br />
<br />
<!-- ws:start:WikiTextTocRule:8:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- 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:<h1> --><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:<h1> --><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 />
21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6<br />
"minor third" = 10\37 = 324.3 cents<br />
"major third" = 11\37 = 356.8 cents<br />
<br />
22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1<br />
"minor third" = 8\37 = 259.5 cents<br />
"major third" = 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:<h1> --><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:<h1> --><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>