Xenharmonic Wiki

65edo

Revision as of 17:51, 29 December 2016 by Wikispaces>xenwolf (**Imported revision 602901538 - Original comment: **)

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----

=<span style="color: #750063; font-family: 'Times New Roman',Times,serif; font-size: 113%;">65 tone equal temperament</span>= 

**//65edo//** divides the [[octave]] into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the [[schisma]], 32805/32768, the [[sensipent comma]], 78732/78125, and the [[wuerschmidt comma]]. In the [[7-limit]], there are two different maps; the first is <65 103 151 182|, [[tempering out]] 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[wuerschmidt temperament]] (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals [[3_2|3/2]], [[5_4|5/4]], [[11_8|11/8]] and [[19_16|19/16]] well, so that it does a good job representing the 2.3.5.11.19 [[just intonation subgroup]]. To this one may want to add 13/8 and 17/16, giving the [[19-limit]] no-sevens subgroup 2.3.5.11.13.17.19. Also of interest is the 19-limit [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as [[130edo]].

65edo contains [[13edo]] as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded|Rubble: a Xenuke Unfolded]].

=Intervals= 
||~ [[Degree]] ||~ Size ([[cent|Cents]]) ||
||=  0 ||>    0.0000 ||
||=  1 ||>   18.4615 ||
||=  2 ||>   36.9231 ||
||=  3 ||>   55.3846 ||
||=  4 ||>   73.8462 ||
||=  5 ||>   92.3077 ||
||=  6 ||>  110.7692 ||
||=  7 ||>  129.2308 ||
||=  8 ||>  147.6923 ||
||=  9 ||>  166.1538 ||
||= 10 ||>  184.6154 ||
||= 11 ||>  203.0769 ||
||= 12 ||>  221.5385 ||
||= 13 ||>  240.0000 ||
||= 14 ||>  258.4615 ||
||= 15 ||>  276.9231 ||
||= 16 ||>  295.3846 ||
||= 17 ||>  313.8462 ||
||= 18 ||>  332.3077 ||
||= 19 ||>  350.7692 ||
||= 20 ||>  369.2308 ||
||= 21 ||>  387.6923 ||
||= 22 ||>  406.1538 ||
||= 23 ||>  424.6154 ||
||= 24 ||>  443.0769 ||
||= 25 ||>  461.5385 ||
||= 26 ||>  480.0000 ||
||= 27 ||>  498.4615 ||
||= 28 ||>  516.9231 ||
||= 29 ||>  535.3846 ||
||= 30 ||>  553.8462 ||
||= 31 ||>  572.3077 ||
||= 32 ||>  590.7692 ||
||= 33 ||>  609.2308 ||
||= 34 ||>  627.6923 ||
||= 35 ||>  646.1538 ||
||= 36 ||>  664.6154 ||
||= 37 ||>  683.0769 ||
||= 38 ||>  701.5385 ||
||= 39 ||>  720.0000 ||
||= 40 ||>  738.4615 ||
||= 41 ||>  756.9231 ||
||= 42 ||>  775.3846 ||
||= 43 ||>  793.8462 ||
||= 44 ||>  812.3077 ||
||= 45 ||>  830.7692 ||
||= 46 ||>  849.2308 ||
||= 47 ||>  867.6923 ||
||= 48 ||>  886.1538 ||
||= 49 ||>  904.6154 ||
||= 50 ||>  923.0769 ||
||= 51 ||>  941.5385 ||
||= 52 ||>  960.0000 ||
||= 53 ||>  978.4615 ||
||= 54 ||>  996.9231 ||
||= 55 ||> 1015.3846 ||
||= 56 ||> 1033.8462 ||
||= 57 ||> 1052.3077 ||
||= 58 ||> 1070.7692 ||
||= 59 ||> 1089.2308 ||
||= 60 ||> 1107.6923 ||
||= 61 ||> 1126.1538 ||
||= 62 ||> 1144.6154 ||
||= 63 ||> 1163.0769 ||
||= 64 ||> 1181.5385 ||
||= 65 ||> 1200.0000 ||

=Scales= 
[[photia7]]
[[photia12]]

Original HTML content:

<html><head><title>65edo</title></head><body><hr />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x65 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #750063; font-family: 'Times New Roman',Times,serif; font-size: 113%;">65 tone equal temperament</span></h1>
 <br />
<strong><em>65edo</em></strong> divides the <a class="wiki_link" href="/octave">octave</a> into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the <a class="wiki_link" href="/schisma">schisma</a>, 32805/32768, the <a class="wiki_link" href="/sensipent%20comma">sensipent comma</a>, 78732/78125, and the <a class="wiki_link" href="/wuerschmidt%20comma">wuerschmidt comma</a>. In the <a class="wiki_link" href="/7-limit">7-limit</a>, there are two different maps; the first is &lt;65 103 151 182|, <a class="wiki_link" href="/tempering%20out">tempering out</a> 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is &lt;65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the <a class="wiki_link" href="/5-limit">5-limit</a> over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit <a class="wiki_link" href="/wuerschmidt%20temperament">wuerschmidt temperament</a> (wurschmidt and worschmidt) these two mappings provide.<br />
<br />
65edo approximates the intervals <a class="wiki_link" href="/3_2">3/2</a>, <a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/11_8">11/8</a> and <a class="wiki_link" href="/19_16">19/16</a> well, so that it does a good job representing the 2.3.5.11.19 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>. To this one may want to add 13/8 and 17/16, giving the <a class="wiki_link" href="/19-limit">19-limit</a> no-sevens subgroup 2.3.5.11.13.17.19. Also of interest is the 19-limit <a class="wiki_link" href="/k%2AN%20subgroups">2*65 subgroup</a> 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as <a class="wiki_link" href="/130edo">130edo</a>.<br />
<br />
65edo contains <a class="wiki_link" href="/13edo">13edo</a> as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see <a class="wiki_link_ext" href="https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded" rel="nofollow">Rubble: a Xenuke Unfolded</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <th><a class="wiki_link" href="/Degree">Degree</a><br />
</th>
        <th>Size (<a class="wiki_link" href="/cent">Cents</a>)<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: right;">0.0000<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: right;">18.4615<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: right;">36.9231<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: right;">55.3846<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: right;">73.8462<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: right;">92.3077<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: right;">110.7692<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: right;">129.2308<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: right;">147.6923<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: right;">166.1538<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: right;">184.6154<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: right;">203.0769<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: right;">221.5385<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: right;">240.0000<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: right;">258.4615<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: right;">276.9231<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: right;">295.3846<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: right;">313.8462<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: right;">332.3077<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: right;">350.7692<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: right;">369.2308<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: right;">387.6923<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: right;">406.1538<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: right;">424.6154<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: right;">443.0769<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: right;">461.5385<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: right;">480.0000<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: right;">498.4615<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: right;">516.9231<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: right;">535.3846<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: right;">553.8462<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: right;">572.3077<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: right;">590.7692<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: right;">609.2308<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: right;">627.6923<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: right;">646.1538<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: right;">664.6154<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: right;">683.0769<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: right;">701.5385<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: right;">720.0000<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: right;">738.4615<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">41<br />
</td>
        <td style="text-align: right;">756.9231<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">42<br />
</td>
        <td style="text-align: right;">775.3846<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">43<br />
</td>
        <td style="text-align: right;">793.8462<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">44<br />
</td>
        <td style="text-align: right;">812.3077<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">45<br />
</td>
        <td style="text-align: right;">830.7692<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">46<br />
</td>
        <td style="text-align: right;">849.2308<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">47<br />
</td>
        <td style="text-align: right;">867.6923<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">48<br />
</td>
        <td style="text-align: right;">886.1538<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">49<br />
</td>
        <td style="text-align: right;">904.6154<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50<br />
</td>
        <td style="text-align: right;">923.0769<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">51<br />
</td>
        <td style="text-align: right;">941.5385<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">52<br />
</td>
        <td style="text-align: right;">960.0000<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">53<br />
</td>
        <td style="text-align: right;">978.4615<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">54<br />
</td>
        <td style="text-align: right;">996.9231<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">55<br />
</td>
        <td style="text-align: right;">1015.3846<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">56<br />
</td>
        <td style="text-align: right;">1033.8462<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">57<br />
</td>
        <td style="text-align: right;">1052.3077<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">58<br />
</td>
        <td style="text-align: right;">1070.7692<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">59<br />
</td>
        <td style="text-align: right;">1089.2308<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">60<br />
</td>
        <td style="text-align: right;">1107.6923<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">61<br />
</td>
        <td style="text-align: right;">1126.1538<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">62<br />
</td>
        <td style="text-align: right;">1144.6154<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">63<br />
</td>
        <td style="text-align: right;">1163.0769<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64<br />
</td>
        <td style="text-align: right;">1181.5385<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">65<br />
</td>
        <td style="text-align: right;">1200.0000<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scales</h1>
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