17edt: Difference between revisions

Revision as of 21:57, 26 May 2012

IMPORTED REVISION FROM WIKISPACES

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This revision was by author JosephRuhf and made on 2012-05-26 21:57:16 UTC.

The original revision id was 339787378.

The revision comment was:

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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 a very good approximation of 11/9, which is in fact the size of the large step.

=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.

=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:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- 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:&lt;h1&gt; --><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&amp;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:&lt;h1&gt; --><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 a very good approximation of 11/9, which is in fact the size of the large step.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><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:&lt;img src=&quot;/file/view/17edt.png/250611032/17edt.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/17edt.png/250611032/17edt.png" alt="17edt.png" title="17edt.png" /><!-- ws:end:WikiTextLocalImageRule:380 --></body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2012-05-26 21:55:17 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2012-05-26 21:57:16 UTC</tt>.<br>
-: The original revision id was <tt>339787104</tt>.<br>
+: The original revision id was <tt>339787378</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>
@@ Line 8: / Line 8: @@
 <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">[[toc|flat]]
 =Properties=
-edt divides 3, the tritave, into 17 equal parts of 111.880 cents each, corresponding to 10.726 edo. It tempers out 245/243 and [[tel:16807/15625|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&amp;21.
+edt 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&amp;21.
 edt is the sixth [[The Riemann Zeta Function and Tuning#Removing%20primes|zeta peak tritave division]].
@@ Line 38: / Line 38: @@
 || 19 ||   || 2125.8 || 925.8 ||
 || 20 ||   || 2237.6 || 1037.6 ||
-|| 21 ||   || 2349..5 || 1149.5 ||
+|| 21 ||   || 2349.5 || 1149.5 ||
 || 22 ||   || 2461.4 || 61.4 ||
 || 23 ||   || 2573.2 || 173.2 ||
@@ Line 63: / Line 63: @@
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;17edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Properties"&gt;Properties&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Discussion"&gt;Discussion&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Z function"&gt;Z function&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Properties&lt;/h1&gt;
-edt divides 3, the tritave, into 17 equal parts of 111.880 cents each, corresponding to 10.726 edo. It tempers out 245/243 and [[tel:16807/15625|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&amp;amp;21.&lt;br /&gt;
+edt 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&amp;amp;21.&lt;br /&gt;
 &lt;br /&gt;
 edt is the sixth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes"&gt;zeta peak tritave division&lt;/a&gt;.&lt;br /&gt;
@@ Line 299: / Line 299: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2349..5&lt;br /&gt;
+         &lt;td&gt;2349.5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1149.5&lt;br /&gt;

17edt: Difference between revisions

Revision as of 21:57, 26 May 2012

IMPORTED REVISION FROM WIKISPACES

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