|
|
| Line 1: |
Line 1: |
| <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>
| |
| : The original revision id was <tt>342518720</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">[[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= | | =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. |
|
| |
|
| =Intervals of 15edt= | | =Intervals of 15edt= |
| || 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]] 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... |
|
| |
|
| 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. |
| Line 37: |
Line 84: |
| 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. |
|
| |
|
| =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: |
|
| |
|
| [[image:15edt.png]] | | [[File:15edt.png|alt=15edt.png|15edt.png]] |
|
| |
|
| Music: | | Music: |
| http://www.youtube.com/watch?v=bC_Pc4jKm2k</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><!-- 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: -->
| |
| <!-- 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>
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties</h1>
| |
| 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 />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals of 15edt"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals of 15edt</h1>
| |
|
| |
|
| |
| <table class="wiki_table">
| |
| <tr>
| |
| <td>Degrees<br />
| |
| </td>
| |
| <td>Cents<br />
| |
| </td>
| |
| <td>Approximate Ratios<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/1_1">1/1</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>126.797<br />
| |
| </td>
| |
| <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 />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>253.594<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/15_13">15/13</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>380.391<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/5_4">5/4</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>507.188<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/4_3">4/3</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>633.985<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/13_9">13/9</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>760.782<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/14_9">14/9</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>887.579<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/5_3">5/3</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>1014.376<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/9_5">9/5</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>1141.173<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/27_14">27/14</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>1267.970<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/27_13">27/13</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>1394.767<br />
| |
| </td>
| |
| <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 />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>1521.564<br />
| |
| </td>
| |
| <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 />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>1648.361<br />
| |
| </td>
| |
| <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 />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>1775.158<br />
| |
| </td>
| |
| <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 />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>1901.955<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/3_1">3/1</a><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <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 />
| |
| <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 />
| |
| <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 />
| |
| <!-- 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 15edt:<br />
| |
| <br />
| |
| <!-- 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 />
| |
| <br />
| |
| Music:<br />
| |
| <!-- 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>
| |
15 Equal Divisions of the Tritave
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.
Intervals of 15edt
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 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.
Z function
Below is a plot of the no-twos Z function in the vicinity of 15edt:
Music:
http://www.youtube.com/watch?v=bC_Pc4jKm2k
