53edo: Difference between revisions

Revision as of 15:57, 10 February 2012

IMPORTED REVISION FROM WIKISPACES

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Original Wikitext content:

[[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: 0, 4, 9, 13, 18, 22, 26/27, 31, 35, 40, 44, 49, 53. The thirds are close to just as well, and therefore 5-limit tuning can closely be approximated e.g. by using 14- and 17- degree intervals which are very close to 5/4 and 6/5 respectively instead of 13- and 18- degrees which are extremely close to the Pythagorean minor(32/27) and major(81/64) thirds. Because 1 degree is very close to both the [[Syntonic Comma|Syntonic]] and [[Pythagorean comma|Pythagorean commas]], 53EDO is very flexible and wolf intervals can be avoided simply by using the note above or below the note in the scale rather than retuning the note in the scale.

=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: 0, 4, 9, 13, 18, 22, 26/27, 31, 35, 40, 44, 49, 53. The thirds are close to just as well, and therefore 5-limit tuning can closely be approximated e.g. by using 14- and 17- degree intervals which are very close to 5/4 and 6/5 respectively instead of 13- and 18- degrees which are extremely close to the Pythagorean minor(32/27) and major(81/64) thirds. Because 1 degree is very close to both the <a class="wiki_link" href="/Syntonic%20Comma">Syntonic</a> and <a class="wiki_link" href="/Pythagorean%20comma">Pythagorean commas</a>, 53EDO is very flexible and wolf intervals can be avoided simply by using the note above or below the note in the scale rather than retuning the note in the scale.<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>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2012-02-10 15:53:48 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2012-02-10 15:57:19 UTC</tt>.<br>
-: The original revision id was <tt>300640796</tt>.<br>
+: The original revision id was <tt>300641834</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>
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 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: 0, 4, 9, 13, 18, 22, 26/27, 31, 35, 40, 44, 49, 53. The thirds are close to just as well, and therefore 5-limit tuning can closely be approximated using the notes: 0, __5__, 9, __14__, __17__, 22, 26/27, 31, __36__, __39__, 44, __48__, 53. Because 1 degree is very close to both the [[Syntonic Comma|Syntonic]] and [[Pythagorean comma|Pythagorean commas]], 53EDO is very flexible and wolf intervals can be avoided simply by using the note above or below the note in the scale rather than retuning the note in the scale.
+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: 0, 4, 9, 13, 18, 22, 26/27, 31, 35, 40, 44, 49, 53. The thirds are close to just as well, and therefore 5-limit tuning can closely be approximated e.g. by using 14- and 17- degree intervals which are very close to 5/4 and 6/5 respectively instead of 13- and 18- degrees which are extremely close to the Pythagorean minor(32/27) and major(81/64) thirds. Because 1 degree is very close to both the [[Syntonic Comma|Syntonic]] and [[Pythagorean comma|Pythagorean commas]], 53EDO is very flexible and wolf intervals can be avoided simply by using the note above or below the note in the scale rather than retuning the note in the scale.
 =Intervals=
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 One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.&lt;br /&gt;
 &lt;br /&gt;
-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: 0, 4, 9, 13, 18, 22, 26/27, 31, 35, 40, 44, 49, 53. The thirds are close to just as well, and therefore 5-limit tuning can closely be approximated using the notes: 0, &lt;u&gt;5&lt;/u&gt;, 9, &lt;u&gt;14&lt;/u&gt;, &lt;u&gt;17&lt;/u&gt;, 22, 26/27, 31, &lt;u&gt;36&lt;/u&gt;, &lt;u&gt;39&lt;/u&gt;, 44, &lt;u&gt;48&lt;/u&gt;, 53. Because 1 degree is very close to both the &lt;a class="wiki_link" href="/Syntonic%20Comma"&gt;Syntonic&lt;/a&gt; and &lt;a class="wiki_link" href="/Pythagorean%20comma"&gt;Pythagorean commas&lt;/a&gt;, 53EDO is very flexible and wolf intervals can be avoided simply by using the note above or below the note in the scale rather than retuning the note in the scale.&lt;br /&gt;
+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: 0, 4, 9, 13, 18, 22, 26/27, 31, 35, 40, 44, 49, 53. The thirds are close to just as well, and therefore 5-limit tuning can closely be approximated e.g. by using 14- and 17- degree intervals which are very close to 5/4 and 6/5 respectively instead of 13- and 18- degrees which are extremely close to the Pythagorean minor(32/27) and major(81/64) thirds. Because 1 degree is very close to both the &lt;a class="wiki_link" href="/Syntonic%20Comma"&gt;Syntonic&lt;/a&gt; and &lt;a class="wiki_link" href="/Pythagorean%20comma"&gt;Pythagorean commas&lt;/a&gt;, 53EDO is very flexible and wolf intervals can be avoided simply by using the note above or below the note in the scale rather than retuning the note in the scale.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals&lt;/h1&gt;

53edo: Difference between revisions

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