53edo: Difference between revisions

Wikispaces>genewardsmith
**Imported revision 311999564 - Original comment: **
Wikispaces>guest
**Imported revision 330095780 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-17 21:39:49 UTC</tt>.<br>
: This revision was by author [[User:guest|guest]] and made on <tt>2012-05-04 13:23:33 UTC</tt>.<br>
: The original revision id was <tt>311999564</tt>.<br>
: The original revision id was <tt>330095780</tt>.<br>
: The revision comment was: <tt></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>
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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<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]]
<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]]
=Theory=  
=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]].
The famous //53 equal division// divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a [[xenharmonic/5-limit|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 [[xenharmonic/optimal patent val|optimal patent val]] for [[xenharmonic/Nuwell family|Big Brother]] temperament, which tempers out both, as well as 11-limit [[xenharmonic/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 [[xenharmonic/Marvel family|athene temperament]]. It is the eighth [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]] and the 16th [[xenharmonic/prime numbers|prime]] edo, following [[xenharmonic/47edo|47edo]] and coming before [[xenharmonic/59edo|59edo]].


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


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


=Just Approximation=  
=Just Approximation=  
53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
53edo provides excellent approximations for the classic 5-limit [[xenharmonic/just|just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
||~ interval ||~ size ||~ diff ||
||~ interval ||~ size ||~ diff ||
|| perfect fifth ||= 31 || −0.07 cents ||
|| perfect fifth ||= 31 || −0.07 cents ||
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|| minor third ||= 14 || +1.34 cents ||
|| minor third ||= 14 || +1.34 cents ||
|| major tone ||= 9 || −0.14 cents ||
|| major tone ||= 9 || −0.14 cents ||
|| major tone ||= 8 || −1.27 cents ||
|| minor tone ||= 8 || −1.27 cents ||
|| diat. semitone ||= 5 || +1.48 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.
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 is practically equal to an extended Pythagorean. 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. In addition, 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.
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. 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. In addition, 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 [[xenharmonic/septimal kleisma|septimal kleisma]], 225/224.


=Intervals=  
=Intervals=  
Line 32: Line 32:
|| 0 || 0.00 ||  ||
|| 0 || 0.00 ||  ||
|| 1 || 22.64 ||  ||
|| 1 || 22.64 ||  ||
|| 2 || 45.28 || [[Quartonic]] ||
|| 2 || 45.28 || [[xenharmonic/Quartonic|Quartonic]] ||
|| 3 || 67.92 ||  ||
|| 3 || 67.92 ||  ||
|| 4 || 90.57 ||  ||
|| 4 || 90.57 ||  ||
|| 5 || 113.21 ||  ||
|| 5 || 113.21 ||  ||
|| 6 || 135.85 ||  ||
|| 6 || 135.85 ||  ||
|| 7 || 158.49 || [[Hemikleismic]] ||
|| 7 || 158.49 || [[xenharmonic/Hemikleismic|Hemikleismic]] ||
|| 8 || 181.13 ||  ||
|| 8 || 181.13 ||  ||
|| 9 || 203.77 ||  ||
|| 9 || 203.77 ||  ||
|| 10 || 226.42 ||  ||
|| 10 || 226.42 ||  ||
|| 11 || 249.06 || [[Hemischis]] ||
|| 11 || 249.06 || [[xenharmonic/Hemischis|Hemischis]] ||
|| 12 || 271.70 || [[Orwell]] ||
|| 12 || 271.70 || [[xenharmonic/Orwell|Orwell]] ||
|| 13 || 294.34 ||  ||
|| 13 || 294.34 ||  ||
|| 14 || 316.98 || [[Hanson]]/[[Catakleismic]] ||
|| 14 || 316.98 || [[xenharmonic/Hanson|Hanson]]/[[xenharmonic/Catakleismic|Catakleismic]] ||
|| 15 || 339.62 || [[Amity]]/[[Hitchcock]] ||
|| 15 || 339.62 || [[xenharmonic/Amity|Amity]]/[[xenharmonic/Hitchcock|Hitchcock]] ||
|| 16 || 362.26 ||  ||
|| 16 || 362.26 ||  ||
|| 17 || 384.91 ||  ||
|| 17 || 384.91 ||  ||
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|| 19 || 430.19 ||  ||
|| 19 || 430.19 ||  ||
|| 20 || 452.83 ||  ||
|| 20 || 452.83 ||  ||
|| 21 || 475.47 || [[Vulture]]/[[Buzzard]] ||
|| 21 || 475.47 || [[xenharmonic/Vulture|Vulture]]/[[xenharmonic/Buzzard|Buzzard]] ||
|| 22 || 498.11 ||  ||
|| 22 || 498.11 ||  ||
|| 23 || 520.75 ||  ||
|| 23 || 520.75 ||  ||
|| 24 || 543.40 ||  ||
|| 24 || 543.40 ||  ||
|| 25 || 566.04 || [[Tricot]] ||
|| 25 || 566.04 || [[xenharmonic/Tricot|Tricot]] ||
|| 26 || 588.68 ||  ||
|| 26 || 588.68 ||  ||
|| 27 || 611.32 ||  ||
|| 27 || 611.32 ||  ||
Line 61: Line 61:
|| 29 || 656.60 ||  ||
|| 29 || 656.60 ||  ||
|| 30 || 679.25 ||  ||
|| 30 || 679.25 ||  ||
|| 31 || 701.89 || [[Helmholtz]]/[[Garibaldi]] ||
|| 31 || 701.89 || [[xenharmonic/Helmholtz|Helmholtz]]/[[xenharmonic/Garibaldi|Garibaldi]] ||
|| 32 || 724.53 ||  ||
|| 32 || 724.53 ||  ||
|| 33 || 747.17 ||  ||
|| 33 || 747.17 ||  ||
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=Compositions=  
=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/prelude1-53.mp3|Bach WTC1 Prelude 1 in 53]] by Bach and [[xenharmonic/Mykhaylo Khramov|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://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 &lt;-- broken link?
[[http://www.geocities.com/Bernalorg/Excerpts/n53.wav|53edo guitar study]] by Novaro &lt;-- 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://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 [[xenharmonic/Prent Rodgers|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.archive.org/details/TrioInOrwell|Trio in Orwell]] [[http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3|play]] by [[xenharmonic/Gene Ward Smith|Gene Ward Smith]]
[[http://www.akjmusic.com/audio/desert_prayer.mp3|Desert Prayer]] by [[http://www.akjmusic.com|Aaron Krister Johnson]]
[[http://www.akjmusic.com/audio/desert_prayer.mp3|Desert Prayer]] by [[http://www.akjmusic.com/|Aaron Krister Johnson]]
[[@http://andrewheathwaite.bandcamp.com/track/elf-dine-on-ho-ho|Elf Dine on Ho Ho]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2005%20Elf%20Dine%20on%20Ho%20Ho.mp3|play]] and [[@http://andrewheathwaite.bandcamp.com/track/spun|Spun]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3|play]] by [[Andrew Heathwaite]]</pre></div>
[[@http://andrewheathwaite.bandcamp.com/track/elf-dine-on-ho-ho|Elf Dine on Ho Ho]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2005%20Elf%20Dine%20on%20Ho%20Ho.mp3|play]] and [[@http://andrewheathwaite.bandcamp.com/track/spun|Spun]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3|play]] by [[xenharmonic/Andrew Heathwaite|Andrew Heathwaite]]</pre></div>
<h4>Original HTML content:</h4>
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;53edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Theory"&gt;Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Just Approximation"&gt;Just Approximation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;53edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Theory"&gt;Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Just Approximation"&gt;Just Approximation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Theory&lt;/h1&gt;
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Theory&lt;/h1&gt;
  The famous &lt;em&gt;53 equal division&lt;/em&gt; divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; 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 &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/Nuwell%20family"&gt;Big Brother&lt;/a&gt; temperament, which tempers out both, as well as 11-limit &lt;a class="wiki_link" href="/Semicomma%20family"&gt;orwell temperament&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/Marvel%20family"&gt;athene temperament&lt;/a&gt;. It is the eighth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt; and the 16th &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; edo, following &lt;a class="wiki_link" href="/47edo"&gt;47edo&lt;/a&gt; and coming before &lt;a class="wiki_link" href="/59edo"&gt;59edo&lt;/a&gt;.&lt;br /&gt;
  The famous &lt;em&gt;53 equal division&lt;/em&gt; divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit"&gt;5-limit&lt;/a&gt; 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Nuwell%20family"&gt;Big Brother&lt;/a&gt; temperament, which tempers out both, as well as 11-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Semicomma%20family"&gt;orwell temperament&lt;/a&gt;, 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20family"&gt;athene temperament&lt;/a&gt;. It is the eighth &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt; and the 16th &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/prime%20numbers"&gt;prime&lt;/a&gt; edo, following &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/47edo"&gt;47edo&lt;/a&gt; and coming before &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/59edo"&gt;59edo&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
53EDO has also found a certain dissemination as an EDO tuning for &lt;a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian"&gt;Arabic/Turkish/Persian music&lt;/a&gt; .&lt;br /&gt;
53EDO has also found a certain dissemination as an EDO tuning for &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/-/Arabic%2C%20Turkish%2C%20Persian%7CArabic/Turkish/Persian%20music"&gt;xenharmonic/Arabic, Turkish, Persian|Arabic/Turkish/Persian music&lt;/a&gt; .&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/53_equal_temperament" rel="nofollow"&gt;Wikipeda article about 53edo&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/53_equal_temperament" rel="nofollow"&gt;Wikipeda article about 53edo&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Just Approximation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Just Approximation&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Just Approximation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Just Approximation&lt;/h1&gt;
  53edo provides excellent approximations for the classic 5-limit &lt;a class="wiki_link" href="/just"&gt;just&lt;/a&gt; chords and scales, such as the Ptolemy-Zarlino &amp;quot;just major&amp;quot; scale.&lt;br /&gt;
  53edo provides excellent approximations for the classic 5-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/just"&gt;just&lt;/a&gt; chords and scales, such as the Ptolemy-Zarlino &amp;quot;just major&amp;quot; scale.&lt;br /&gt;




Line 148: Line 148:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td&gt;major tone&lt;br /&gt;
         &lt;td&gt;minor tone&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
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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;
One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.&lt;br /&gt;
&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 is practically equal to an extended Pythagorean. 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. In addition, 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 &lt;a class="wiki_link" href="/septimal%20kleisma"&gt;septimal kleisma&lt;/a&gt;, 225/224.&lt;br /&gt;
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. 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. In addition, 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/septimal%20kleisma"&gt;septimal kleisma&lt;/a&gt;, 225/224.&lt;br /&gt;
&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;
&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;
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         &lt;td&gt;45.28&lt;br /&gt;
         &lt;td&gt;45.28&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Quartonic"&gt;Quartonic&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Quartonic"&gt;Quartonic&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;158.49&lt;br /&gt;
         &lt;td&gt;158.49&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Hemikleismic"&gt;Hemikleismic&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hemikleismic"&gt;Hemikleismic&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;249.06&lt;br /&gt;
         &lt;td&gt;249.06&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Hemischis"&gt;Hemischis&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hemischis"&gt;Hemischis&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;271.70&lt;br /&gt;
         &lt;td&gt;271.70&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Orwell"&gt;Orwell&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Orwell"&gt;Orwell&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;316.98&lt;br /&gt;
         &lt;td&gt;316.98&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Hanson"&gt;Hanson&lt;/a&gt;/&lt;a class="wiki_link" href="/Catakleismic"&gt;Catakleismic&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hanson"&gt;Hanson&lt;/a&gt;/&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Catakleismic"&gt;Catakleismic&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;339.62&lt;br /&gt;
         &lt;td&gt;339.62&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Amity"&gt;Amity&lt;/a&gt;/&lt;a class="wiki_link" href="/Hitchcock"&gt;Hitchcock&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Amity"&gt;Amity&lt;/a&gt;/&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hitchcock"&gt;Hitchcock&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;475.47&lt;br /&gt;
         &lt;td&gt;475.47&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Vulture"&gt;Vulture&lt;/a&gt;/&lt;a class="wiki_link" href="/Buzzard"&gt;Buzzard&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vulture"&gt;Vulture&lt;/a&gt;/&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Buzzard"&gt;Buzzard&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;566.04&lt;br /&gt;
         &lt;td&gt;566.04&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Tricot"&gt;Tricot&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tricot"&gt;Tricot&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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         &lt;td&gt;701.89&lt;br /&gt;
         &lt;td&gt;701.89&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="/Helmholtz"&gt;Helmholtz&lt;/a&gt;/&lt;a class="wiki_link" href="/Garibaldi"&gt;Garibaldi&lt;/a&gt;&lt;br /&gt;
         &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Helmholtz"&gt;Helmholtz&lt;/a&gt;/&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Garibaldi"&gt;Garibaldi&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Compositions&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Compositions&lt;/h1&gt;
  &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3" rel="nofollow"&gt;Bach WTC1 Prelude 1 in 53&lt;/a&gt; by Bach and &lt;a class="wiki_link" href="/Mykhaylo%20Khramov"&gt;Mykhaylo Khramov&lt;/a&gt;&lt;br /&gt;
  &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3" rel="nofollow"&gt;Bach WTC1 Prelude 1 in 53&lt;/a&gt; by Bach and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Mykhaylo%20Khramov"&gt;Mykhaylo Khramov&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3" rel="nofollow"&gt;Bach WTC1 Fugue 1 in 53&lt;/a&gt; by Bach and Mykhaylo Khramov&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3" rel="nofollow"&gt;Bach WTC1 Fugue 1 in 53&lt;/a&gt; by Bach and Mykhaylo Khramov&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.geocities.com/Bernalorg/Excerpts/n53.wav" rel="nofollow"&gt;53edo guitar study&lt;/a&gt; by Novaro &amp;lt;-- broken link?&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.geocities.com/Bernalorg/Excerpts/n53.wav" rel="nofollow"&gt;53edo guitar study&lt;/a&gt; by Novaro &amp;lt;-- broken link?&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html" rel="nofollow"&gt;Whisper Song in 53EDO&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prent/sing53-c5-slow.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Prent%20Rodgers"&gt;Prent Rodgers&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html" rel="nofollow"&gt;Whisper Song in 53EDO&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prent/sing53-c5-slow.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Prent%20Rodgers"&gt;Prent Rodgers&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow"&gt;Trio in Orwell&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow"&gt;Trio in Orwell&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/desert_prayer.mp3" rel="nofollow"&gt;Desert Prayer&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.akjmusic.com" rel="nofollow"&gt;Aaron Krister Johnson&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/desert_prayer.mp3" rel="nofollow"&gt;Desert Prayer&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.akjmusic.com/" rel="nofollow"&gt;Aaron Krister Johnson&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/elf-dine-on-ho-ho" rel="nofollow" target="_blank"&gt;Elf Dine on Ho Ho&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2005%20Elf%20Dine%20on%20Ho%20Ho.mp3" rel="nofollow"&gt;play&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/spun" rel="nofollow" target="_blank"&gt;Spun&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/elf-dine-on-ho-ho" rel="nofollow" target="_blank"&gt;Elf Dine on Ho Ho&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2005%20Elf%20Dine%20on%20Ho%20Ho.mp3" rel="nofollow"&gt;play&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/spun" rel="nofollow" target="_blank"&gt;Spun&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>