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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | [[File:13edt.png|thumb|alt=13edt.png|A plot of the [[the Riemann zeta function and tuning#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]] |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-19 15:53:33 UTC</tt>.<br>
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| : The original revision id was <tt>596162028</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">The 13 equal division of 3, the tritave, divides it into 13 equal parts of 146.304 cents each, corresponding to 8.202 edo. An alternative name for it is the [[Bohlen-Pierce]] scale. In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as [[Sensamagic clan#Bohpier|bohpier temperament]]. It is less impressive in higher p-limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 [[26edt]], [[39edt]] and [[52edt]] come to the fore.
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| Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak edt.
| | '''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave. |
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| [[image:13edt.png]] | | It provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as [[Sensamagic_clan#Bohpier|bohpier temperament]]. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore. |
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| ==Intervals==
| | 13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]]. |
| ||~ Steps ||~ Cents ||~ BP nonatonic degree ||~ Corresponding JI intervals ||~ Comments ||~ Generator for... ||
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| || 1 || 146.3 || P1/m2 || 27/25~49/45 || || ||
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| || 2 || 292.6 || M2/m3 || 25/21 || || [[Siirius]] ||
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| || 3 || 438.9 || M3 || 9/7 || || [[Bohlen-Pierce|Linear BP]] ||
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| || 4 || 585.2 || m4 || 7/5 || || [[Canopus]] ||
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| || 5 || 731.5 || M4/m5 || 75/49 || False 3/2 || false Father ||
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| || 6 || 877.8 || M5 || 5/3 || || [[Arcturus]] ||
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| || 7 || 1024.1 || m6 || 9/5 || || Arcturus ||
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| || 8 || 1170.4 || M6/m7 || 49/25 || False 2/1 || false Father ||
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| || 9 || 1316.7 || M7 || 15/7 || || Canopus ||
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| || 10 || 1463.0 || m8 || 7/3 || || Linear BP ||
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| || 11 || 1609.3 || M8/m9 || 63/25 || || Sirius ||
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| || 12 || 1755.7 || M9 || 25/9~135/49 || || ||
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| || 13 || 1902.0 || P10 || 3/1 || Tritave || ||</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>13edt</title></head><body>The 13 equal division of 3, the tritave, divides it into 13 equal parts of 146.304 cents each, corresponding to 8.202 edo. An alternative name for it is the <a class="wiki_link" href="/Bohlen-Pierce">Bohlen-Pierce</a> scale. In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as <a class="wiki_link" href="/Sensamagic%20clan#Bohpier">bohpier temperament</a>. It is less impressive in higher p-limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 <a class="wiki_link" href="/26edt">26edt</a>, <a class="wiki_link" href="/39edt">39edt</a> and <a class="wiki_link" href="/52edt">52edt</a> come to the fore.<br />
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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 terms of which 13edt is the fourth no-twos zeta peak edt.<br />
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| <!-- ws:start:WikiTextLocalImageRule:200:&lt;img src=&quot;/file/view/13edt.png/250612880/13edt.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/13edt.png/250612880/13edt.png" alt="13edt.png" title="13edt.png" /><!-- ws:end:WikiTextLocalImageRule:200 --><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals</h2>
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| <table class="wiki_table">
| | In the [[no-2]] [[3/1]]-[[equave]]-[[7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]]. |
| <tr>
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| <th>Steps<br />
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| </th>
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| <th>Cents<br />
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| </th>
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| <th>BP nonatonic degree<br />
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| </th>
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| <th>Corresponding JI intervals<br />
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| </th>
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| <th>Comments<br />
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| </th>
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| <th>Generator for...<br />
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| </th>
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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>146.3<br />
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| </td>
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| <td>P1/m2<br />
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| </td>
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| <td>27/25~49/45<br />
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| </td>
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| <td><br />
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| </td>
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| <td><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>292.6<br />
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| </td>
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| <td>M2/m3<br />
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| </td>
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| <td>25/21<br />
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| </td>
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| <td><br />
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| </td>
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| <td><a class="wiki_link" href="/Siirius">Siirius</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>438.9<br />
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| </td>
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| <td>M3<br />
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| </td>
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| <td>9/7<br />
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| </td>
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| <td><br />
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| </td>
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| <td><a class="wiki_link" href="/Bohlen-Pierce">Linear BP</a><br />
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| </td>
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| </tr>
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| <tr>
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| <td>4<br />
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| </td>
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| <td>585.2<br />
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| </td>
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| <td>m4<br />
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| </td>
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| <td>7/5<br />
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| </td>
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| <td><br />
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| </td>
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| <td><a class="wiki_link" href="/Canopus">Canopus</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>731.5<br />
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| </td>
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| <td>M4/m5<br />
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| </td>
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| <td>75/49<br />
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| </td>
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| <td>False 3/2<br />
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| </td>
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| <td>false Father<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>877.8<br />
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| </td>
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| <td>M5<br />
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| </td>
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| <td>5/3<br />
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| </td>
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| <td><br />
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| </td>
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| <td><a class="wiki_link" href="/Arcturus">Arcturus</a><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>1024.1<br />
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| </td>
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| <td>m6<br />
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| </td>
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| <td>9/5<br />
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| </td>
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| <td><br />
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| </td>
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| <td>Arcturus<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>1170.4<br />
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| </td>
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| <td>M6/m7<br />
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| </td>
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| <td>49/25<br />
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| </td>
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| <td>False 2/1<br />
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| </td>
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| <td>false Father<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>1316.7<br />
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| </td>
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| <td>M7<br />
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| </td>
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| <td>15/7<br />
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| </td>
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| <td><br />
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| </td>
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| <td>Canopus<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>1463.0<br />
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| </td>
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| <td>m8<br />
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| </td>
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| <td>7/3<br />
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| </td>
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| <td><br />
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| </td>
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| <td>Linear BP<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>1609.3<br />
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| </td>
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| <td>M8/m9<br />
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| </td>
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| <td>63/25<br />
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| </td>
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| <td><br />
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| </td>
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| <td>Sirius<br />
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| </td>
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| </tr>
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| <tr>
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| <td>12<br />
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| </td>
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| <td>1755.7<br />
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| </td>
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| <td>M9<br />
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| </td>
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| <td>25/9~135/49<br />
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| </td>
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| <td><br />
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| </td>
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| <td><br />
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| </td>
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| </tr>
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| <tr>
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| <td>13<br />
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| </td>
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| <td>1902.0<br />
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| </td>
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| <td>P10<br />
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| </td>
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| <td>3/1<br />
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| </td>
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| <td>Tritave<br />
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| </td>
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| <td><br />
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| </td>
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| </tr>
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| </table>
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| </body></html></pre></div>
| | == Theory == |
| | {{Harmonics in equal|13|3|1|prec=2|intervals=odd}} |
| | {{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}} |
| | |
| | * [[Relationship between Bohlen-Pierce and octave-ful temperaments]] |
| | |
| | == Intervals == |
| | {{Main|Intervals of BP}} |
| | |
| | {| class="wikitable center-all right-2 right-3" |
| | |- |
| | ! Steps |
| | ! [[Cent]]s |
| | ! [[Hekt]]s |
| | ! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree |
| | ! Corresponding<br />3.5.7 subgroup<br />intervals |
| | ! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs, {{nowrap|E {{=}} 1/1}}) |
| | |- |
| | | 0 |
| | | 0 |
| | | 0 |
| | | P1 |
| | | 1/1 |
| | | E |
| | |- |
| | | 1 |
| | | 146.3 |
| | | 100 |
| | | A1/m2 |
| | | [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}}) |
| | | F |
| | |- |
| | | 2 |
| | | 292.6 |
| | | 200 |
| | | M2/d3 |
| | | [[25/21]] (−9.2{{c}}) |
| | | F#, Gb |
| | |- |
| | | 3 |
| | | 438.9 |
| | | 300 |
| | | A2/P3/d4 |
| | | [[9/7]] (+3.8{{c}}) |
| | | G |
| | |- |
| | | 4 |
| | | 585.2 |
| | | 400 |
| | | A3/m4/d5 |
| | | [[7/5]] (+2.7{{c}}) |
| | | H |
| | |- |
| | | 5 |
| | | 731.5 |
| | | 500 |
| | | M4/m5 |
| | | [[75/49]] (−5.4{{c}}) |
| | | H#, Jb |
| | |- |
| | | 6 |
| | | 877.8 |
| | | 600 |
| | | A4/M5 |
| | | [[5/3]] (−6.5{{c}}) |
| | | J |
| | |- |
| | | 7 |
| | | 1024.1 |
| | | 700 |
| | | A5/m6/d7 |
| | | [[9/5]] (+6.5{{c}}) |
| | | A |
| | |- |
| | | 8 |
| | | 1170.4 |
| | | 800 |
| | | M6/m7 |
| | | [[49/25]] (+5.4{{c}}) |
| | | A#, Bb |
| | |- |
| | | 9 |
| | | 1316.7 |
| | | 900 |
| | | A6/M7/d8 |
| | | [[15/7]] (−2.7{{c}}) |
| | | B |
| | |- |
| | | 10 |
| | | 1463.0 |
| | | 1000 |
| | | P8/d9 |
| | | [[7/3]] (−3.8{{c}}) |
| | | C |
| | |- |
| | | 11 |
| | | 1609.3 |
| | | 1100 |
| | | A8/m9 |
| | | [[63/25]] (+9.2{{c}}) |
| | | C#, Db |
| | |- |
| | | 12 |
| | | 1755.7 |
| | | 1200 |
| | | M9/d10 |
| | | [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}}) |
| | | D |
| | |- |
| | | 13 |
| | | 1902.0 |
| | | 1300 |
| | | A9/P10 |
| | | [[3/1]] |
| | | E |
| | |} |
| | |
| | == JI approximation == |
| | [[File:13ed3-17-001.svg|alt=alt : Your browser has no SVG support.]] |
| | |
| | == Regular temperament properties == |
| | {| class="wikitable center-4 center-5 center-6" |
| | ! rowspan="2" | Subgroup |
| | ! rowspan="2" | [[Comma list]] |
| | ! rowspan="2" | [[Mapping]] |
| | ! rowspan="2" | Optimal<br>Equave stretch (¢) |
| | ! colspan="2" | Tuning error |
| | |- |
| | ! [[TE error|Absolute]] (¢) |
| | ! [[TE simple badness|Relative]] (%) |
| | |- |
| | | 3.5.7 |
| | | 245/243, 3125/3087 |
| | | [{{val| 13 19 23 }}] (b13) |
| | | +1.393 |
| | | 1.150 |
| | | 0.79 |
| | |} |
| | |
| | === Rank-2 temperaments === |
| | {| class="wikitable center-all right-3 left-5" |
| | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator |
| | |- |
| | ! Periods<br />per tritave |
| | ! Generator<br />(reduced) |
| | ! Cents<br />(reduced) |
| | ! Associated<br />ratio |
| | ! Temperament |
| | |- |
| | | 1 |
| | | 1\13 |
| | | 146.30 |
| | | 49/45 |
| | | [[Procyon]] |
| | |- |
| | | 1 |
| | | 2\13 |
| | | 292.61 |
| | | 25/21 |
| | | [[Sirius]] |
| | |- |
| | | 1 |
| | | 3\13 |
| | | 438.91 |
| | | 9/7 |
| | | [[BPS]] |
| | |- |
| | | 1 |
| | | 4\13 |
| | | 585.22 |
| | | 7/5 |
| | | [[Canopus]] |
| | |- |
| | | 1 |
| | | 5\13 |
| | | 731.63 |
| | | 75/49 |
| | | |
| | |- |
| | | 1 |
| | | 6\13 |
| | | 877.83 |
| | | 5/3 |
| | | [[Arcturus]] |
| | |} |
| | |
| | == See also == |
| | * [[Bohlen-p_et]] |
| | * [[Catalog of 3.5.7 subgroup rank two temperaments]] |
| | * [[No-twos subgroup temperaments#3.5.7 subgroup temperaments]] |
| | * [[19ed5|19ED5]]: relative ED5 |
| | * [[23ed7|23ED7]]: relative ED7 |
| | |
| | [[Category:Tritave]] |
| | [[Category:Macrotonal]] |
| | [[Category:Nonoctave]] |
| | [[Category:Bohlen–Pierce]] |