15edt: Difference between revisions

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<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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2011-08-~~22 03~~:25:32 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 12:12:46 UTC</tt>.<br>

: The original revision id was <tt>~~247588139~~</tt>.<br>

: The original revision id was <tt>250618252</tt>.<br>

: The revision comment was: <tt>~~Reverted to Aug 21, 2011 8:22 pm: it was no typo~~</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>

<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">~~=<span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span>=~~

<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]]

=<span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span>=

=Properties=

The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15> in the 5-limit, which is tempered out by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&123, which has a mapping [<1 0 0 0|, <0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of

126.773, very close to 15edt.

=Intervals of 15edt=

|| Degrees || Cents || Approximate Ratios ||

|| 0 || 0 || <span style="color: #660000;">[[1_1|1/1]]</span> ||

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15edt contains 4 intervals from [[5edt]] and 2 intervals from [[3edt]], meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...

15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.

15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and [[image:13edt.png]][[image:15edt.png]][[image:15edt.png]]the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.

15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.

~~15edt approximates~~ the ~~5th~~ and ~~13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale",~~ in ~~which three 13/9 intervals close to a tritave, and another three are set 5/3 away~~.</pre></div>

=Z function=

Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing primes|no-twos Z function]] in the vicinity of 15edt:

[[15edt.png]]</pre></div>

<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>15edt</title></head><body><h1 id="toc0"><a name="x15 Equal Divisions of the Tritave"></a><span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span></h1>

<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>15edt</title></head><body><a href="#x15 Equal Divisions of the Tritave">15 Equal Divisions of the Tritave</a> | <a href="#Properties">Properties</a> | <a href="#Intervals of 15edt">Intervals of 15edt</a> | <a href="#Z function">Z function</a>

<br />

<h1 id="toc0"><a name="x15 Equal Divisions of the Tritave"></a><span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span></h1>

<br />

<h1 id="toc1"><a name="Properties"></a>Properties</h1>

The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&gt; in the 5-limit, which is tempered out by <a class="wiki_link" href="/19edo">19edo</a> but has an <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> of <a class="wiki_link" href="/303edo">303edo</a>. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;123, which has a mapping [&lt;1 0 0 0|, &lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of <br />

126.773, very close to 15edt.<br />

<br />

<h1 id="toc2"><a name="Intervals of 15edt"></a>Intervals of 15edt</h1>

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15edt contains 4 intervals from <a class="wiki_link" href="/5edt">5edt</a> and 2 intervals from <a class="wiki_link" href="/3edt">3edt</a>, meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...<br />

<br />

15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.<br />

15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and <img src="/file/view/13edt.png/250612880/13edt.png" alt="13edt.png" title="13edt.png" /><img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" /><img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" />the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.<br />

<br />

15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS &quot;augmented scale&quot;, in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.<br />

<br />

<h1 id="toc3"><a name="Z function"></a>Z function</h1>

Below is a plot of the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing primes">no-twos Z function</a> in the vicinity of 15edt:<br />

<br />

15edt ~~approximates the 5th and 13th harmonics (and 29th) very well~~. ~~Taking these as consonances one obtains an 3L+3s MOS~~ &~~amp~~;~~quot;augmented scale~~&~~amp~~;~~quot;, in which three 13~~/~~9 intervals close to~~ a ~~tritave, and another three are set 5/3 away.~~</body></html></pre></div>

@@ 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:xenwolf|xenwolf]] and made on <tt>2011-08-22 03:25:32 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 12:12:46 UTC</tt>.<br>
-: The original revision id was <tt>247588139</tt>.<br>
+: The original revision id was <tt>250618252</tt>.<br>
-: The revision comment was: <tt>Reverted to Aug 21, 2011 8:22 pm: it was no typo</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>
 <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">=&lt;span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;"&gt;15 Equal Divisions of the Tritave&lt;/span&gt;=
+<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]]
+=&lt;span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;"&gt;15 Equal Divisions of the Tritave&lt;/span&gt;=
+=Properties=
+The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&gt; in the 5-limit, which is tempered out by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;123, which has a mapping [&lt;1 0 0 0|, &lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of
+.773, very close to 15edt.
+=Intervals of 15edt=
 || Degrees || Cents || Approximate Ratios ||
 || 0 || 0 || &lt;span style="color: #660000;"&gt;[[1_1|1/1]]&lt;/span&gt; ||
@@ Line 28: / Line 35: @@
 edt contains 4 intervals from [[5edt]] and 2 intervals from [[3edt]], meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...
-edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.
+edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and [[image:13edt.png]][[image:15edt.png]][[image:15edt.png]]the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.
+edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.
-edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.</pre></div>
+=Z function=
+Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing primes|no-twos Z function]] in the vicinity of 15edt:
+[[15edt.png]]</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;15edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x15 Equal Divisions of the Tritave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;"&gt;15 Equal Divisions of the Tritave&lt;/span&gt;&lt;/h1&gt;
+<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;15edt&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="#x15 Equal Divisions of the Tritave"&gt;15 Equal Divisions of the Tritave&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Properties"&gt;Properties&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Intervals of 15edt"&gt;Intervals of 15edt&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;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x15 Equal Divisions of the Tritave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;"&gt;15 Equal Divisions of the Tritave&lt;/span&gt;&lt;/h1&gt;
   &lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Properties&lt;/h1&gt;
+The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&amp;gt; in the 5-limit, which is tempered out by &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; but has an &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; of &lt;a class="wiki_link" href="/303edo"&gt;303edo&lt;/a&gt;. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;amp;123, which has a mapping [&amp;lt;1 0 0 0|, &amp;lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of  &lt;br /&gt;
+.773, very close to 15edt.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Intervals of 15edt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals of 15edt&lt;/h1&gt;
@@ Line 178: / Line 197: @@
 edt contains 4 intervals from &lt;a class="wiki_link" href="/5edt"&gt;5edt&lt;/a&gt; and 2 intervals from &lt;a class="wiki_link" href="/3edt"&gt;3edt&lt;/a&gt;, meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...&lt;br /&gt;
 &lt;br /&gt;
-edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.&lt;br /&gt;
+edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and &lt;!-- ws:start:WikiTextLocalImageRule:152:&amp;lt;img src=&amp;quot;/file/view/13edt.png/250612880/13edt.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/13edt.png/250612880/13edt.png" alt="13edt.png" title="13edt.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:152 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:153:&amp;lt;img src=&amp;quot;/file/view/15edt.png/250617832/15edt.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:153 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:154:&amp;lt;img src=&amp;quot;/file/view/15edt.png/250617832/15edt.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:154 --&gt;the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.&lt;br /&gt;
+&lt;br /&gt;
+edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS &amp;quot;augmented scale&amp;quot;, in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Z function"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Z function&lt;/h1&gt;
+Below is a plot of the &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing primes"&gt;no-twos Z function&lt;/a&gt; in the vicinity of 15edt:&lt;br /&gt;
 &lt;br /&gt;
-edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS &amp;quot;augmented scale&amp;quot;, in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;a class="wiki_link" href="/15edt.png"&gt;15edt.png&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>