17edt
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 2011-09-03 22:45:20 UTC.
- The original revision id was 250558494.
- 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:
[[toc|flat]] =Properties= 17edt divides 3, the tritave, into 17 equal parts of 111.880 cents each. It tempers out 245/243 and 16807/15625 in the 7-limit, 77/75 and 1331/1323 in the 11-limit, and 175/169 and 121/117 in the 13-limit. It supports the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written 17&21. =Discussion= 17edt is closely related [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step. If the major chord is defined by degrees 0,8,13 and a major chord is built atop the two chord tones, an hexatonic set of 0,4,8,9,13,16,17 results. Excluding #16 produces a LLsLL pentatonic scale. Continuing the trend by building major chords on the three new chord tones (4, 9, and 16) gives you a set of the notes 0,3,4,7,8,9,12,13,15,16,17. Excluding #15 produces a LsLssLsLs moment of symmetry, which is called "Moll I" or "Delta". ===Intervals=== || degree of 17edt || cents value || cents value octave reduced || || 0 || 0 || || || 1 || 111.9 || || || 2 || 223.8 || || || 3 || 335.6 || || || 4 || 447.5 || || || 5 || 559.4 || || || 6 || 671.3 || || || 7 || 783.2 || || || 8 || 895.1 || || || 9 || 1006.9 || || || 10 || 1118.8 || || || 11 || 1230.7 || 30.7 || || 12 || 1342.6 || 142.6 || || 13 || 1454.5 || 254.5 || || 14 || 1566.3 || 366.3 || || 15 || 1678.2 || 478.2 || || 16 || 1790.1 || 590.1 || || 17 || 1902.0 || 702.0 || || 18 || 2013.9 || 813.9 || || 19 || 2125.8 || 925.8 || || 20 || 2237.6 || 1037.6 || || 21 || 2349..5 || 1149.5 || || 22 || 2461.4 || 61.4 || || 23 || 2573.2 || 173.2 || || 24 || 2685.2 || 285.2 || || 25 || 2797.1 || 397.1 || || 26 || 2908.9 || 508.9 || || 27 || 3020.8 || 620.8 || || 28 || 3132.7 || 732.7 || || 29 || 3244.6 || 844.6 || || 30 || 3356.5 || 956.5 || || 31 || 3468.3 || 1068.3 || || 32 || 3580.2 || 1180.2 || || 33 || 3692.1 || 92.1 || || 34 || 3804.0 || 204.0 ||
Original HTML content:
<html><head><title>17edt</title></head><body><!-- ws:start:WikiTextTocRule:6:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Properties">Properties</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Discussion">Discussion</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
<!-- ws:end:WikiTextTocRule:10 --><br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Properties</h1>
17edt divides 3, the tritave, into 17 equal parts of 111.880 cents each. It tempers out 245/243 and 16807/15625 in the 7-limit, 77/75 and 1331/1323 in the 11-limit, and 175/169 and 121/117 in the 13-limit. It supports the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written 17&21.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Discussion"></a><!-- ws:end:WikiTextHeadingRule:2 -->Discussion</h1>
17edt is closely related <a class="wiki_link" href="/13edt">13edt</a>, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step.<br />
<br />
If the major chord is defined by degrees 0,8,13 and a major chord is built atop the two chord tones, an hexatonic set of 0,4,8,9,13,16,17 results. Excluding #16 produces a LLsLL pentatonic scale. Continuing the trend by building major chords on the three new chord tones (4, 9, and 16) gives you a set of the notes 0,3,4,7,8,9,12,13,15,16,17. Excluding #15 produces a LsLssLsLs moment of symmetry, which is called "Moll I" or "Delta".<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="Discussion--Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h3>
<table class="wiki_table">
<tr>
<td>degree of 17edt<br />
</td>
<td>cents value<br />
</td>
<td>cents value octave reduced<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>111.9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>223.8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>335.6<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>447.5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>559.4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>671.3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>783.2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>895.1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>1006.9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>1118.8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>1230.7<br />
</td>
<td>30.7<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>1342.6<br />
</td>
<td>142.6<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>1454.5<br />
</td>
<td>254.5<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>1566.3<br />
</td>
<td>366.3<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1678.2<br />
</td>
<td>478.2<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>1790.1<br />
</td>
<td>590.1<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>1902.0<br />
</td>
<td>702.0<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>2013.9<br />
</td>
<td>813.9<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>2125.8<br />
</td>
<td>925.8<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>2237.6<br />
</td>
<td>1037.6<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>2349..5<br />
</td>
<td>1149.5<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>2461.4<br />
</td>
<td>61.4<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>2573.2<br />
</td>
<td>173.2<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>2685.2<br />
</td>
<td>285.2<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>2797.1<br />
</td>
<td>397.1<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>2908.9<br />
</td>
<td>508.9<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>3020.8<br />
</td>
<td>620.8<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>3132.7<br />
</td>
<td>732.7<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>3244.6<br />
</td>
<td>844.6<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>3356.5<br />
</td>
<td>956.5<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>3468.3<br />
</td>
<td>1068.3<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>3580.2<br />
</td>
<td>1180.2<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>3692.1<br />
</td>
<td>92.1<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>3804.0<br />
</td>
<td>204.0<br />
</td>
</tr>
</table>
</body></html>