Xenharmonic Wiki

53edo

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[[toc|flat]]
=Theory= 
The famous //53 equal division// divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, and is the [[optimal patent val]] for [[Nuwell family|Big Brother]] temperament, which tempers out both, as well as 11-limit [[Semicomma family|orwell temperament]], which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for [[Marvel family|athene temperament]]. It is the eighth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]] and the 16th [[prime numbers|prime]] edo, following [[47edo]] and coming before [[59edo]].

53EDO has also found a certain dissemination as an EDO tuning for [[Arabic, Turkish, Persian|Arabic/Turkish/Persian music]] .

[[http://en.wikipedia.org/wiki/53_equal_temperament|Wikipeda article about 53edo]]

=Just Approximation= 
53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
||~ interval ||~ size ||~ diff ||
|| perfect fifth ||= 31 || −0.07 cents ||
|| major third ||= 17 || −1.40 cents ||
|| minor third ||= 14 || +1.34 cents ||
|| major tone ||= 9 || −0.14 cents ||
|| major tone ||= 8 || −1.27 cents ||
|| diat. semitone ||= 5 || +1.48 cents ||

One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.

The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO can be considered an extended Pythagorean tuning using the notes. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. The 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.

=Intervals= 
|| degrees of 53edo || cents value || generator for ||
|| 0 || 0.00 ||   ||
|| 1 || 22.64 ||   ||
|| 2 || 45.28 || [[Quartonic]] ||
|| 3 || 67.92 ||   ||
|| 4 || 90.57 ||   ||
|| 5 || 113.21 ||   ||
|| 6 || 135.85 ||   ||
|| 7 || 158.49 || [[Hemikleismic]] ||
|| 8 || 181.13 ||   ||
|| 9 || 203.77 ||   ||
|| 10 || 226.42 ||   ||
|| 11 || 249.06 || [[Hemischis]] ||
|| 12 || 271.70 || [[Orwell]] ||
|| 13 || 294.34 ||   ||
|| 14 || 316.98 || [[Hanson]]/[[Catakleismic]] ||
|| 15 || 339.62 || [[Amity]]/[[Hitchcock]] ||
|| 16 || 362.26 ||   ||
|| 17 || 384.91 ||   ||
|| 18 || 407.55 ||   ||
|| 19 || 430.19 ||   ||
|| 20 || 452.83 ||   ||
|| 21 || 475.47 || [[Vulture]]/[[Buzzard]] ||
|| 22 || 498.11 ||   ||
|| 23 || 520.75 ||   ||
|| 24 || 543.40 ||   ||
|| 25 || 566.04 || [[Tricot]] ||
|| 26 || 588.68 ||   ||
|| 27 || 611.32 ||   ||
|| 28 || 633.96 ||   ||
|| 29 || 656.60 ||   ||
|| 30 || 679.25 ||   ||
|| 31 || 701.89 || [[Helmholtz]]/[[Garibaldi]] ||
|| 32 || 724.53 ||   ||
|| 33 || 747.17 ||   ||
|| 34 || 769.81 ||   ||
|| 35 || 792.45 ||   ||
|| 36 || 815.09 ||   ||
|| 37 || 837.74 ||   ||
|| 38 || 860.38 ||   ||
|| 39 || 883.02 ||   ||
|| 40 || 905.66 ||   ||
|| 41 || 928.30 ||   ||
|| 42 || 950.94 ||   ||
|| 43 || 973.58 ||   ||
|| 44 || 996.23 ||   ||
|| 45 || 1018.87 ||   ||
|| 46 || 1041.51 ||   ||
|| 47 || 1064.15 ||   ||
|| 48 || 1086.79 ||   ||
|| 49 || 1109.43 ||   ||
|| 50 || 1132.08 ||   ||
|| 51 || 1154.72 ||   ||
|| 52 || 1177.36 ||   ||


=Compositions= 
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3|Bach WTC1 Prelude 1 in 53]] by Bach and [[Mykhaylo Khramov]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3|Bach WTC1 Fugue 1 in 53]] by Bach and Mykhaylo Khramov
[[http://www.geocities.com/Bernalorg/Excerpts/n53.wav|53edo guitar study]] by Novaro <-- broken link?
[[http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html|Whisper Song in 53EDO]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prent/sing53-c5-slow.mp3|play]] by [[Prent Rodgers]]
[[http://www.archive.org/details/TrioInOrwell|Trio in Orwell]] [[http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3|play]] by [[Gene Ward Smith]]
[[http://www.akjmusic.com/audio/desert_prayer.mp3|Desert Prayer]] by [[http://www.akjmusic.com|Aaron Krister Johnson]]

Original HTML content:

<html><head><title>53edo</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="#Theory">Theory</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Just Approximation">Just Approximation</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Compositions">Compositions</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="Theory"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theory</h1>
 The famous <em>53 equal division</em> divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a <a class="wiki_link" href="/5-limit">5-limit</a> system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, and is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for <a class="wiki_link" href="/Nuwell%20family">Big Brother</a> temperament, which tempers out both, as well as 11-limit <a class="wiki_link" href="/Semicomma%20family">orwell temperament</a>, which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for <a class="wiki_link" href="/Marvel%20family">athene temperament</a>. It is the eighth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a> and the 16th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/47edo">47edo</a> and coming before <a class="wiki_link" href="/59edo">59edo</a>.<br />
<br />
53EDO has also found a certain dissemination as an EDO tuning for <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">Arabic/Turkish/Persian music</a> .<br />
<br />
<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/53_equal_temperament" rel="nofollow">Wikipeda article about 53edo</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Just Approximation"></a><!-- ws:end:WikiTextHeadingRule:2 -->Just Approximation</h1>
 53edo provides excellent approximations for the classic 5-limit <a class="wiki_link" href="/just">just</a> chords and scales, such as the Ptolemy-Zarlino &quot;just major&quot; scale.<br />


<table class="wiki_table">
    <tr>
        <th>interval<br />
</th>
        <th>size<br />
</th>
        <th>diff<br />
</th>
    </tr>
    <tr>
        <td>perfect fifth<br />
</td>
        <td style="text-align: center;">31<br />
</td>
        <td>−0.07 cents<br />
</td>
    </tr>
    <tr>
        <td>major third<br />
</td>
        <td style="text-align: center;">17<br />
</td>
        <td>−1.40 cents<br />
</td>
    </tr>
    <tr>
        <td>minor third<br />
</td>
        <td style="text-align: center;">14<br />
</td>
        <td>+1.34 cents<br />
</td>
    </tr>
    <tr>
        <td>major tone<br />
</td>
        <td style="text-align: center;">9<br />
</td>
        <td>−0.14 cents<br />
</td>
    </tr>
    <tr>
        <td>major tone<br />
</td>
        <td style="text-align: center;">8<br />
</td>
        <td>−1.27 cents<br />
</td>
    </tr>
    <tr>
        <td>diat. semitone<br />
</td>
        <td style="text-align: center;">5<br />
</td>
        <td>+1.48 cents<br />
</td>
    </tr>
</table>

<br />
One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.<br />
<br />
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO can be considered an extended Pythagorean tuning using the notes. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. The 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the <a class="wiki_link" href="/septimal%20kleisma">septimal kleisma</a>, 225/224.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <td>degrees of 53edo<br />
</td>
        <td>cents value<br />
</td>
        <td>generator for<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.00<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>22.64<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>45.28<br />
</td>
        <td><a class="wiki_link" href="/Quartonic">Quartonic</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>67.92<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>90.57<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>113.21<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>135.85<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>158.49<br />
</td>
        <td><a class="wiki_link" href="/Hemikleismic">Hemikleismic</a><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>181.13<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>203.77<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>226.42<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>249.06<br />
</td>
        <td><a class="wiki_link" href="/Hemischis">Hemischis</a><br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>271.70<br />
</td>
        <td><a class="wiki_link" href="/Orwell">Orwell</a><br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>294.34<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>316.98<br />
</td>
        <td><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/Catakleismic">Catakleismic</a><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>339.62<br />
</td>
        <td><a class="wiki_link" href="/Amity">Amity</a>/<a class="wiki_link" href="/Hitchcock">Hitchcock</a><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>362.26<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>384.91<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>407.55<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>430.19<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>452.83<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>475.47<br />
</td>
        <td><a class="wiki_link" href="/Vulture">Vulture</a>/<a class="wiki_link" href="/Buzzard">Buzzard</a><br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>498.11<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>520.75<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>543.40<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>566.04<br />
</td>
        <td><a class="wiki_link" href="/Tricot">Tricot</a><br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>588.68<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>611.32<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>633.96<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>656.60<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>679.25<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>701.89<br />
</td>
        <td><a class="wiki_link" href="/Helmholtz">Helmholtz</a>/<a class="wiki_link" href="/Garibaldi">Garibaldi</a><br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>724.53<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>747.17<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>769.81<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>792.45<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>815.09<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>837.74<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>860.38<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>39<br />
</td>
        <td>883.02<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>40<br />
</td>
        <td>905.66<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>41<br />
</td>
        <td>928.30<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>42<br />
</td>
        <td>950.94<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>43<br />
</td>
        <td>973.58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>44<br />
</td>
        <td>996.23<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>45<br />
</td>
        <td>1018.87<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>46<br />
</td>
        <td>1041.51<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>47<br />
</td>
        <td>1064.15<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>48<br />
</td>
        <td>1086.79<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>49<br />
</td>
        <td>1109.43<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>50<br />
</td>
        <td>1132.08<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>51<br />
</td>
        <td>1154.72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>52<br />
</td>
        <td>1177.36<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:6 -->Compositions</h1>
 <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3" rel="nofollow">Bach WTC1 Prelude 1 in 53</a> by Bach and <a class="wiki_link" href="/Mykhaylo%20Khramov">Mykhaylo Khramov</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3" rel="nofollow">Bach WTC1 Fugue 1 in 53</a> by Bach and Mykhaylo Khramov<br />
<a class="wiki_link_ext" href="http://www.geocities.com/Bernalorg/Excerpts/n53.wav" rel="nofollow">53edo guitar study</a> by Novaro &lt;-- broken link?<br />
<a class="wiki_link_ext" href="http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html" rel="nofollow">Whisper Song in 53EDO</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prent/sing53-c5-slow.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Prent%20Rodgers">Prent Rodgers</a><br />
<a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow">Trio in Orwell</a> <a class="wiki_link_ext" href="http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a><br />
<a class="wiki_link_ext" href="http://www.akjmusic.com/audio/desert_prayer.mp3" rel="nofollow">Desert Prayer</a> by <a class="wiki_link_ext" href="http://www.akjmusic.com" rel="nofollow">Aaron Krister Johnson</a></body></html>
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