17edt
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[[toc|flat]] =Properties= 17edt divides 3, the tritave, into 17 equal parts of 111.880 cents each, corresponding to 10.726 edo. 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. 17edt is the sixth [[The Riemann Zeta Function and Tuning#Removing%20primes|zeta peak tritave division]]. =Discussion= 17edt is closely related to [[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 gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to a 16/5 which is only .3 cents flat. =Intervals= || degree of 17edt || note name || cents value || cents value octave reduced || || 0 || C || 0 || || || 1 || Db = B# || 111.9 || || || 2 || Eb = C# || 223.8 || || || 3 || D || 335.6 || || || 4 || E || 447.5 || || || 5 || F = D# || 559.4 || || || 6 || Gb = E# || 671.3 || || || 7 || Hb = F# || 783.2 || || || 8 || G || 895.1 || || || 9 || H || 1006.9 || || || 10 || Jb = G# || 1118.8 || || || 11 || Ab = H# || 1230.7 || 30.7 || || 12 || J || 1342.6 || 142.6 || || 13 || A || 1454.5 || 254.5 || || 14 || Bb = J# || 1566.3 || 366.3 || || 15 || Cb = A# || 1678.2 || 478.2 || || 16 || B || 1790.1 || 590.1 || || 17 || C || 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 || * Notes named so that C D E F G H J A B C = Lambda mode It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents). =Z function= Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos Z function]] in the vicinity of 17edt. [[image:17edt.png]]
Original HTML content:
<html><head><title>17edt</title></head><body><!-- 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="#Properties">Properties</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Discussion">Discussion</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
<!-- ws:end:WikiTextTocRule:13 --><!-- 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, corresponding to 10.726 edo. 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 />
17edt is the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes">zeta peak tritave division</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Discussion"></a><!-- ws:end:WikiTextHeadingRule:2 -->Discussion</h1>
17edt is closely related to <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 gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), <br />
leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to a 16/5 which is only .3 cents flat.<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>degree of 17edt<br />
</td>
<td>note name<br />
</td>
<td>cents value<br />
</td>
<td>cents value octave reduced<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>C<br />
</td>
<td>0<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>Db = B#<br />
</td>
<td>111.9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>Eb = C#<br />
</td>
<td>223.8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>D<br />
</td>
<td>335.6<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>E<br />
</td>
<td>447.5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>F = D#<br />
</td>
<td>559.4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>Gb = E#<br />
</td>
<td>671.3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>Hb = F#<br />
</td>
<td>783.2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>G<br />
</td>
<td>895.1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>H<br />
</td>
<td>1006.9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>Jb = G#<br />
</td>
<td>1118.8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>Ab = H#<br />
</td>
<td>1230.7<br />
</td>
<td>30.7<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>J<br />
</td>
<td>1342.6<br />
</td>
<td>142.6<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>A<br />
</td>
<td>1454.5<br />
</td>
<td>254.5<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>Bb = J#<br />
</td>
<td>1566.3<br />
</td>
<td>366.3<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>Cb = A#<br />
</td>
<td>1678.2<br />
</td>
<td>478.2<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>B<br />
</td>
<td>1790.1<br />
</td>
<td>590.1<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>C<br />
</td>
<td>1902.0<br />
</td>
<td>702.0<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td><br />
</td>
<td>2013.9<br />
</td>
<td>813.9<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td><br />
</td>
<td>2125.8<br />
</td>
<td>925.8<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td><br />
</td>
<td>2237.6<br />
</td>
<td>1037.6<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td><br />
</td>
<td>2349.5<br />
</td>
<td>1149.5<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td><br />
</td>
<td>2461.4<br />
</td>
<td>61.4<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td><br />
</td>
<td>2573.2<br />
</td>
<td>173.2<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td><br />
</td>
<td>2685.2<br />
</td>
<td>285.2<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td><br />
</td>
<td>2797.1<br />
</td>
<td>397.1<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td><br />
</td>
<td>2908.9<br />
</td>
<td>508.9<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td><br />
</td>
<td>3020.8<br />
</td>
<td>620.8<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td><br />
</td>
<td>3132.7<br />
</td>
<td>732.7<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td><br />
</td>
<td>3244.6<br />
</td>
<td>844.6<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td><br />
</td>
<td>3356.5<br />
</td>
<td>956.5<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td><br />
</td>
<td>3468.3<br />
</td>
<td>1068.3<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td><br />
</td>
<td>3580.2<br />
</td>
<td>1180.2<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td><br />
</td>
<td>3692.1<br />
</td>
<td>92.1<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td><br />
</td>
<td>3804.0<br />
</td>
<td>204.0<br />
</td>
</tr>
</table>
<br />
<ul><li>Notes named so that C D E F G H J A B C = Lambda mode</li></ul>It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:6 -->Z function</h1>
Below is a plot of the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes">no-twos Z function</a> in the vicinity of 17edt.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:380:<img src="/file/view/17edt.png/250611032/17edt.png" alt="" title="" /> --><img src="/file/view/17edt.png/250611032/17edt.png" alt="17edt.png" title="17edt.png" /><!-- ws:end:WikiTextLocalImageRule:380 --></body></html>