37edo
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Original Wikitext content:
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. =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 subgroup, where it shares the same tuning as 111et. In fact, on the larger 2.27.5.7.11.51.57 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 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 ||
Original HTML content:
<html><head><title>37edo</title></head><body>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 <a class="wiki_link" href="/Porcupine%20family">porcupine temperament</a> 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.<br />
<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 subgroup, where it shares the same tuning as 111et. In fact, on the larger 2.27.5.7.11.51.57 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 />
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 />
"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 />
<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>
</body></html>