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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | __FORCETOC__ |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | =<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>= |
| : This revision was by author [[User:guest|guest]] and made on <tt>2012-06-04 14:05:06 UTC</tt>.<br>
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| : The original revision id was <tt>342518720</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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]]
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| =<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>=
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| =Properties= | | =Properties= |
| The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. 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. | | The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. 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|19edo]] but has an [[Optimal_patent_val|optimal patent val]] of [[303edo|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. |
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| =Intervals of 15edt= | | =Intervals of 15edt= |
| || Degrees || Cents || Approximate Ratios ||
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| || 0 || 0 || <span style="color: #660000;">[[1_1|1/1]]</span> ||
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| || 1 || 126.797 || [[14_13|14/13]], [[15_14|15/14]], [[16_15|16/15]], 29/27 ||
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| || 2 || 253.594 || [[15_13|15/13]] ||
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| || 3 || 380.391 || <span style="color: #660000;">[[5_4|5/4]]</span> ||
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| || 4 || 507.188 || [[4_3|4/3]] ||
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| || 5 || 633.985 || [[13_9|13/9]] ||
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| || 6 || 760.782 || <span style="color: #660000;">[[14_9|14/9]]</span> ||
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| || 7 || 887.579 || [[5_3|5/3]] ||
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| || 8 || 1014.376 || [[9_5|9/5]] ||
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| || 9 || 1141.173 || <span style="color: #660000;">[[27_14|27/14]]</span> ||
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| || 10 || 1267.970 || [[27_13|27/13]] ||
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| || 11 || 1394.767 || [[9_4|9/4]] ([[9_8|9/8]] plus an octave) ||
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| || 12 || 1521.564 || [[12_5|12/5]] (<span style="color: #660000;">[[6_5|6/5]]</span> plus an octave) ||
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| || 13 || 1648.361 || [[13_5|13/5]] ([[13_10|13/10]] plus an octave) ||
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| || 14 || 1775.158 || [[14_5|14/5]] ([[7_5|7/5]] plus an octave) ||
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| || 15 || 1901.955 || [[3_1|3/1]] ||
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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... | | {| class="wikitable" |
| | |- |
| | | | Degrees |
| | | | Cents |
| | | | Approximate Ratios |
| | |- |
| | | | 0 |
| | | | 0 |
| | | | <span style="color: #660000;">[[1/1|1/1]]</span> |
| | |- |
| | | | 1 |
| | | | 126.797 |
| | | | [[14/13|14/13]], [[15/14|15/14]], [[16/15|16/15]], 29/27 |
| | |- |
| | | | 2 |
| | | | 253.594 |
| | | | [[15/13|15/13]] |
| | |- |
| | | | 3 |
| | | | 380.391 |
| | | | <span style="color: #660000;">[[5/4|5/4]]</span> |
| | |- |
| | | | 4 |
| | | | 507.188 |
| | | | [[4/3|4/3]] |
| | |- |
| | | | 5 |
| | | | 633.985 |
| | | | [[13/9|13/9]] |
| | |- |
| | | | 6 |
| | | | 760.782 |
| | | | <span style="color: #660000;">[[14/9|14/9]]</span> |
| | |- |
| | | | 7 |
| | | | 887.579 |
| | | | [[5/3|5/3]] |
| | |- |
| | | | 8 |
| | | | 1014.376 |
| | | | [[9/5|9/5]] |
| | |- |
| | | | 9 |
| | | | 1141.173 |
| | | | <span style="color: #660000;">[[27/14|27/14]]</span> |
| | |- |
| | | | 10 |
| | | | 1267.970 |
| | | | [[27/13|27/13]] |
| | |- |
| | | | 11 |
| | | | 1394.767 |
| | | | [[9/4|9/4]] ([[9/8|9/8]] plus an octave) |
| | |- |
| | | | 12 |
| | | | 1521.564 |
| | | | [[12/5|12/5]] (<span style="color: #660000;">[[6/5|6/5]]</span> plus an octave) |
| | |- |
| | | | 13 |
| | | | 1648.361 |
| | | | [[13/5|13/5]] ([[13/10|13/10]] plus an octave) |
| | |- |
| | | | 14 |
| | | | 1775.158 |
| | | | [[14/5|14/5]] ([[7/5|7/5]] plus an octave) |
| | |- |
| | | | 15 |
| | | | 1901.955 |
| | | | [[3/1|3/1]] |
| | |} |
| | |
| | 15edt contains 4 intervals from [[5edt|5edt]] and 2 intervals from [[3edt|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... |
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| 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 the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs. |
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| 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. |
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| =Z function= | | =Z function= |
| Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos Z function]] in the vicinity of 15edt: | | Below is a plot of the [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos Z function]] in the vicinity of 15edt: |
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| [[image:15edt.png]] | | [[File:15edt.png|alt=15edt.png|15edt.png]] |
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| Music: | | Music: |
| http://www.youtube.com/watch?v=bC_Pc4jKm2k</pre></div>
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| <h4>Original HTML content:</h4>
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| <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><!-- 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="#x15 Equal Divisions of the Tritave">15 Equal Divisions of the Tritave</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Properties">Properties</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals of 15edt">Intervals of 15edt</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
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| <!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x15 Equal Divisions of the Tritave"></a><!-- ws:end:WikiTextHeadingRule:0 --><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>
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties</h1>
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| The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. 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 126.773, very close to 15edt.<br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals of 15edt"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals of 15edt</h1>
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| <table class="wiki_table">
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| <tr>
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| <td>Degrees<br />
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| </td>
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| <td>Cents<br />
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| </td>
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| <td>Approximate Ratios<br />
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| </td>
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| </tr>
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| <tr>
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| <td>0<br />
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| </td>
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| <td>0<br />
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| </td>
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| <td><span style="color: #660000;"><a class="wiki_link" href="/1_1">1/1</a></span><br />
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| </td>
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| </tr>
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| <tr>
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| <td>1<br />
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| </td>
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| <td>126.797<br />
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| </td>
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| <td><a class="wiki_link" href="/14_13">14/13</a>, <a class="wiki_link" href="/15_14">15/14</a>, <a class="wiki_link" href="/16_15">16/15</a>, 29/27<br />
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| </td>
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| </tr>
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| <tr>
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| <td>2<br />
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| </td>
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| <td>253.594<br />
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| </td>
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| <td><a class="wiki_link" href="/15_13">15/13</a><br />
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| </td>
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| </tr>
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| <tr>
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| <td>3<br />
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| </td>
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| <td>380.391<br />
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| </td>
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| <td><span style="color: #660000;"><a class="wiki_link" href="/5_4">5/4</a></span><br />
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| </td>
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| </tr>
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| <td>4<br />
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| </td>
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| <td>507.188<br />
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| </td>
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| <td><a class="wiki_link" href="/4_3">4/3</a><br />
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| </td>
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| </tr>
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| <tr>
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| <td>5<br />
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| </td>
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| <td>633.985<br />
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| </td>
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| <td><a class="wiki_link" href="/13_9">13/9</a><br />
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| </td>
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| </tr>
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| <tr>
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| <td>6<br />
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| </td>
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| <td>760.782<br />
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| </td>
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| <td><span style="color: #660000;"><a class="wiki_link" href="/14_9">14/9</a></span><br />
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| </td>
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| </tr>
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| <tr>
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| <td>7<br />
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| </td>
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| <td>887.579<br />
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| </td>
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| <td><a class="wiki_link" href="/5_3">5/3</a><br />
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| </td>
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| </tr>
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| <tr>
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| <td>8<br />
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| </td>
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| <td>1014.376<br />
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| </td>
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| <td><a class="wiki_link" href="/9_5">9/5</a><br />
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| </td>
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| </tr>
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| <tr>
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| <td>9<br />
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| </td>
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| <td>1141.173<br />
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| </td>
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| <td><span style="color: #660000;"><a class="wiki_link" href="/27_14">27/14</a></span><br />
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| </td>
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| </tr>
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| <tr>
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| <td>10<br />
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| </td>
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| <td>1267.970<br />
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| </td>
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| <td><a class="wiki_link" href="/27_13">27/13</a><br />
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| </td>
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| </tr>
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| <tr>
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| <td>11<br />
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| </td>
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| <td>1394.767<br />
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| </td>
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| <td><a class="wiki_link" href="/9_4">9/4</a> (<a class="wiki_link" href="/9_8">9/8</a> plus an octave)<br />
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| </td>
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| <td>12<br />
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| </td>
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| <td>1521.564<br />
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| </td>
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| <td><a class="wiki_link" href="/12_5">12/5</a> (<span style="color: #660000;"><a class="wiki_link" href="/6_5">6/5</a></span> plus an octave)<br />
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| </td>
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| <td>13<br />
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| </td>
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| <td>1648.361<br />
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| </td>
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| <td><a class="wiki_link" href="/13_5">13/5</a> (<a class="wiki_link" href="/13_10">13/10</a> plus an octave)<br />
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| </td>
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| </tr>
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| <tr>
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| <td>14<br />
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| </td>
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| <td>1775.158<br />
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| </td>
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| <td><a class="wiki_link" href="/14_5">14/5</a> (<a class="wiki_link" href="/7_5">7/5</a> plus an octave)<br />
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| </td>
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| </tr>
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| <tr>
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| <td>15<br />
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| </td>
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| <td>1901.955<br />
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| </td>
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| <td><a class="wiki_link" href="/3_1">3/1</a><br />
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| </td>
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| </tr>
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| </table>
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| <br />
| | [http://www.youtube.com/watch?v=bC_Pc4jKm2k http://www.youtube.com/watch?v=bC_Pc4jKm2k] |
| 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 />
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| <br />
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| 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 />
| |
| <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 />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:6 -->Z function</h1>
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| 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 15edt:<br />
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| <br />
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| <!-- ws:start:WikiTextLocalImageRule:152:&lt;img src=&quot;/file/view/15edt.png/250617832/15edt.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" /><!-- ws:end:WikiTextLocalImageRule:152 --><br />
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| <br />
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| Music:<br />
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| <!-- ws:start:WikiTextUrlRule:259:http://www.youtube.com/watch?v=bC_Pc4jKm2k --><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=bC_Pc4jKm2k" rel="nofollow">http://www.youtube.com/watch?v=bC_Pc4jKm2k</a><!-- ws:end:WikiTextUrlRule:259 --></body></html></pre></div>
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