37edo: Difference between revisions
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Wikispaces>Andrew_Heathwaite **Imported revision 204112484 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 215359554 - Original comment: ** |
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| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-30 04:27:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>215359554</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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. It offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13: | <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. | ||
=Subgroups= | |||
37edo offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13: | |||
12\37 = 389.2 cents | 12\37 = 389.2 cents | ||
| Line 13: | Line 16: | ||
26\37 = 843.2 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 | 21\37 = 681.1 cents | ||
| Line 30: | Line 36: | ||
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. | 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 || | || degrees of 37edo || cents value || | ||
|| 0 || 0.00 || | || 0 || 0.00 || | ||
| Line 71: | Line 76: | ||
|| 36 || 1167.57 ||</pre></div> | || 36 || 1167.57 ||</pre></div> | ||
<h4>Original HTML content:</h4> | <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"><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. It offers close approximations to <a class="wiki_link" href="/OverToneSeries">harmonics</a> 5, 7, 11, and 13:<br /> | <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"><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:&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 /> | <br /> | ||
12\37 = 389.2 cents<br /> | 12\37 = 389.2 cents<br /> | ||
| Line 78: | Line 86: | ||
26\37 = 843.2 cents<br /> | 26\37 = 843.2 cents<br /> | ||
<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:&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 /> | <br /> | ||
21\37 = 681.1 cents<br /> | 21\37 = 681.1 cents<br /> | ||
| Line 95: | Line 106: | ||
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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1> | ||
Revision as of 04:27, 30 March 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-03-30 04:27:23 UTC.
- The original revision id was 215359554.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
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>