72edo: Difference between revisions

Revision as of 12:44, 5 November 2010

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This revision was by author Andrew_Heathwaite and made on 2010-11-05 12:44:11 UTC.

The original revision id was 176850705.

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Original Wikitext content:

72-tone equal temperament (or 72-edo) divides the octave into 72 steps or //moria//. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic, Turkish, Persian|Arabic]] music, and has itself been used to tune Turkish music.

Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo|96-edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.

72-tone equal temperament approximates 11-limit [[JustIntonation|just intonation]] exceptionally well, and is the ninth [[http://www.research.att.com/%7Enjas/sequences/A117538|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.

72 is an excellent tuning for [[Gamelismic clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.

==Harmonic Scale== 

Mode 8 of the harmonic series -- [[overtone scales|overtones 8 through 16]], octave repeating -- is well-represented in 72edo.

|| Overtones in "Mode 8": || 8 ||   || 9 ||   || 10 ||   || 11 ||   || 12 ||   || 13 ||   || 14 ||   || 15 ||   || 16 ||
|| ...as JI Ratio from 1/1: || 1/1 ||   || 9/8 ||   || 5/4 ||   || 11/8 ||   || 3/2 ||   || 13/8 ||   || 7/4 ||   || 15/8 ||   || 2/1 ||
|| ...in cents: || 0 ||   || 203.9 ||   || 386.3 ||   || 551.3 ||   || 702.0 ||   || 840.5 ||   || 968.8 ||   || 1088.3 ||   || 1200.0 ||
|| Nearest degree of 72edo: || 0 ||   || 12 ||   || 23 ||   || 33 ||   || 42 ||   || 50 ||   || 58 ||   || 65 ||   || 72 ||
|| ...in cents: || 0 ||   || 200.0 ||   || 383.3 ||   || 550.0 ||   || 700.0 ||   || 833.3 ||   || 966.7 ||   || 1083.3 ||   || 1200.0 ||
|| Steps as Freq. Ratio: ||   || 9:8 ||   || 10:9 ||   || 11:10 ||   || 12:11 ||   || 13:12 ||   || 14:13 ||   || 15:14 ||   || 16:15 ||   ||
|| ...in cents: ||   || 203.9 ||   || 182.4 ||   || 165.0 ||   || 150.6 ||   || 138.6 ||   || 128.3 ||   || 119.4 ||   || 111.7 ||   ||
|| Nearest degree of 72edo: ||   || 12 ||   || 11 ||   || 10 ||   || 9 ||   || 8 ||   || 8 ||   || 7 ||   || 7 ||   ||
|| ...in cents: ||   || 200.0 ||   || 183.3 ||   || 166.7 ||   || 150.0 ||   || 133.3 ||   || 133.3 ||   || 116.7 ||   || 116.7 ||   ||


==Music== 
[[http://www.archive.org/details/Kotekant|Kotekant]]

==External links== 
* [[http://en.wikipedia.org/wiki/72_tone_equal_temperament|Wikipedia article on 72edo]]
* [[http://en.wikipedia.org/wiki/72_tone_equal_temperament|OrthodoxWiki Article on Byzantine chant, which uses 72edo]]
* [[http://en.wikipedia.org/wiki/Joe_Maneri|Wikipedia article on Joe Maneri (1927-2009)]]
* [[http://members.aon.at/ekmelischemusik/|Gesellschaft für Ekmelische Musik]], a group of composers and researchers dedicated to 72edo music
* [[http://www.72note.com/|Rick Tagawa's 72edo site]], including theory and composers' list
* [[http://dannywier.ucoz.com|Danny Wier, composer and musician who specializes in 72-edo]]

Original HTML content:

<html><head><title>72edo</title></head><body>72-tone equal temperament (or 72-edo) divides the octave into 72 steps or <em>moria</em>. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of <a class="wiki_link" href="/24edo">24-tone equal temperament</a>, a common and standard tuning of <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">Arabic</a> music, and has itself been used to tune Turkish music.<br />
<br />
Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with <a class="wiki_link" href="/96edo">96-edo</a>), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.<br />
<br />
72-tone equal temperament approximates 11-limit <a class="wiki_link" href="/JustIntonation">just intonation</a> exceptionally well, and is the ninth <a class="wiki_link_ext" href="http://www.research.att.com/%7Enjas/sequences/A117538" rel="nofollow">Zeta integral tuning</a>. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.<br />
<br />
72 is an excellent tuning for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>, especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->Harmonic Scale</h2>
 <br />
Mode 8 of the harmonic series -- <a class="wiki_link" href="/overtone%20scales">overtones 8 through 16</a>, octave repeating -- is well-represented in 72edo.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Overtones in &quot;Mode 8&quot;:<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td>15<br />
</td>
        <td><br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>...as JI Ratio from 1/1:<br />
</td>
        <td>1/1<br />
</td>
        <td><br />
</td>
        <td>9/8<br />
</td>
        <td><br />
</td>
        <td>5/4<br />
</td>
        <td><br />
</td>
        <td>11/8<br />
</td>
        <td><br />
</td>
        <td>3/2<br />
</td>
        <td><br />
</td>
        <td>13/8<br />
</td>
        <td><br />
</td>
        <td>7/4<br />
</td>
        <td><br />
</td>
        <td>15/8<br />
</td>
        <td><br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td><br />
</td>
        <td>203.9<br />
</td>
        <td><br />
</td>
        <td>386.3<br />
</td>
        <td><br />
</td>
        <td>551.3<br />
</td>
        <td><br />
</td>
        <td>702.0<br />
</td>
        <td><br />
</td>
        <td>840.5<br />
</td>
        <td><br />
</td>
        <td>968.8<br />
</td>
        <td><br />
</td>
        <td>1088.3<br />
</td>
        <td><br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 72edo:<br />
</td>
        <td>0<br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td>23<br />
</td>
        <td><br />
</td>
        <td>33<br />
</td>
        <td><br />
</td>
        <td>42<br />
</td>
        <td><br />
</td>
        <td>50<br />
</td>
        <td><br />
</td>
        <td>58<br />
</td>
        <td><br />
</td>
        <td>65<br />
</td>
        <td><br />
</td>
        <td>72<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td><br />
</td>
        <td>200.0<br />
</td>
        <td><br />
</td>
        <td>383.3<br />
</td>
        <td><br />
</td>
        <td>550.0<br />
</td>
        <td><br />
</td>
        <td>700.0<br />
</td>
        <td><br />
</td>
        <td>833.3<br />
</td>
        <td><br />
</td>
        <td>966.7<br />
</td>
        <td><br />
</td>
        <td>1083.3<br />
</td>
        <td><br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Steps as Freq. Ratio:<br />
</td>
        <td><br />
</td>
        <td>9:8<br />
</td>
        <td><br />
</td>
        <td>10:9<br />
</td>
        <td><br />
</td>
        <td>11:10<br />
</td>
        <td><br />
</td>
        <td>12:11<br />
</td>
        <td><br />
</td>
        <td>13:12<br />
</td>
        <td><br />
</td>
        <td>14:13<br />
</td>
        <td><br />
</td>
        <td>15:14<br />
</td>
        <td><br />
</td>
        <td>16:15<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td><br />
</td>
        <td>203.9<br />
</td>
        <td><br />
</td>
        <td>182.4<br />
</td>
        <td><br />
</td>
        <td>165.0<br />
</td>
        <td><br />
</td>
        <td>150.6<br />
</td>
        <td><br />
</td>
        <td>138.6<br />
</td>
        <td><br />
</td>
        <td>128.3<br />
</td>
        <td><br />
</td>
        <td>119.4<br />
</td>
        <td><br />
</td>
        <td>111.7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 72edo:<br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td><br />
</td>
        <td>200.0<br />
</td>
        <td><br />
</td>
        <td>183.3<br />
</td>
        <td><br />
</td>
        <td>166.7<br />
</td>
        <td><br />
</td>
        <td>150.0<br />
</td>
        <td><br />
</td>
        <td>133.3<br />
</td>
        <td><br />
</td>
        <td>133.3<br />
</td>
        <td><br />
</td>
        <td>116.7<br />
</td>
        <td><br />
</td>
        <td>116.7<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Music"></a><!-- ws:end:WikiTextHeadingRule:2 -->Music</h2>
 <a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow">Kotekant</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-External links"></a><!-- ws:end:WikiTextHeadingRule:4 -->External links</h2>
 <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/72_tone_equal_temperament" rel="nofollow">Wikipedia article on 72edo</a></li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/72_tone_equal_temperament" rel="nofollow">OrthodoxWiki Article on Byzantine chant, which uses 72edo</a></li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Joe_Maneri" rel="nofollow">Wikipedia article on Joe Maneri (1927-2009)</a></li><li><a class="wiki_link_ext" href="http://members.aon.at/ekmelischemusik/" rel="nofollow">Gesellschaft für Ekmelische Musik</a>, a group of composers and researchers dedicated to 72edo music</li><li><a class="wiki_link_ext" href="http://www.72note.com/" rel="nofollow">Rick Tagawa's 72edo site</a>, including theory and composers' list</li><li><a class="wiki_link_ext" href="http://dannywier.ucoz.com" rel="nofollow">Danny Wier, composer and musician who specializes in 72-edo</a></li></ul></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:genewardsmith|genewardsmith]] and made on <tt>2010-09-21 23:22:18 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-11-05 12:44:11 UTC</tt>.<br>
-: The original revision id was <tt>164459673</tt>.<br>
+: The original revision id was <tt>176850705</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>
@@ Line 10: / Line 10: @@
 Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo|96-edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
--tone equal temperament approximates 11-limit [[JustIntonation|just intonation]] exceptionally well, and is the ninth [[http://www.research.att.com/~njas/sequences/A117538|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.
+-tone equal temperament approximates 11-limit [[JustIntonation|just intonation]] exceptionally well, and is the ninth [[http://www.research.att.com/%7Enjas/sequences/A117538|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.
 is an excellent tuning for [[Gamelismic clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
-==Music==
+==Harmonic Scale==
+Mode 8 of the harmonic series -- [[overtone scales|overtones 8 through 16]], octave repeating -- is well-represented in 72edo.
+|| Overtones in "Mode 8": || 8 ||   || 9 ||   || 10 ||   || 11 ||   || 12 ||   || 13 ||   || 14 ||   || 15 ||   || 16 ||
+|| ...as JI Ratio from 1/1: || 1/1 ||   || 9/8 ||   || 5/4 ||   || 11/8 ||   || 3/2 ||   || 13/8 ||   || 7/4 ||   || 15/8 ||   || 2/1 ||
+|| ...in cents: || 0 ||   || 203.9 ||   || 386.3 ||   || 551.3 ||   || 702.0 ||   || 840.5 ||   || 968.8 ||   || 1088.3 ||   || 1200.0 ||
+|| Nearest degree of 72edo: || 0 ||   || 12 ||   || 23 ||   || 33 ||   || 42 ||   || 50 ||   || 58 ||   || 65 ||   || 72 ||
+|| ...in cents: || 0 ||   || 200.0 ||   || 383.3 ||   || 550.0 ||   || 700.0 ||   || 833.3 ||   || 966.7 ||   || 1083.3 ||   || 1200.0 ||
+|| Steps as Freq. Ratio: ||   || 9:8 ||   || 10:9 ||   || 11:10 ||   || 12:11 ||   || 13:12 ||   || 14:13 ||   || 15:14 ||   || 16:15 ||   ||
+|| ...in cents: ||   || 203.9 ||   || 182.4 ||   || 165.0 ||   || 150.6 ||   || 138.6 ||   || 128.3 ||   || 119.4 ||   || 111.7 ||   ||
+|| Nearest degree of 72edo: ||   || 12 ||   || 11 ||   || 10 ||   || 9 ||   || 8 ||   || 8 ||   || 7 ||   || 7 ||   ||
+|| ...in cents: ||   || 200.0 ||   || 183.3 ||   || 166.7 ||   || 150.0 ||   || 133.3 ||   || 133.3 ||   || 116.7 ||   || 116.7 ||   ||
+==Music==
 [[http://www.archive.org/details/Kotekant|Kotekant]]
@@ Line 29: / Line 44: @@
 Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with &lt;a class="wiki_link" href="/96edo"&gt;96-edo&lt;/a&gt;), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.&lt;br /&gt;
 &lt;br /&gt;
--tone equal temperament approximates 11-limit &lt;a class="wiki_link" href="/JustIntonation"&gt;just intonation&lt;/a&gt; exceptionally well, and is the ninth &lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&gt;Zeta integral tuning&lt;/a&gt;. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.&lt;br /&gt;
+-tone equal temperament approximates 11-limit &lt;a class="wiki_link" href="/JustIntonation"&gt;just intonation&lt;/a&gt; exceptionally well, and is the ninth &lt;a class="wiki_link_ext" href="http://www.research.att.com/%7Enjas/sequences/A117538" rel="nofollow"&gt;Zeta integral tuning&lt;/a&gt;. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.&lt;br /&gt;
 &lt;br /&gt;
 is an excellent tuning for &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;miracle temperament&lt;/a&gt;, especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Music&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Harmonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Harmonic Scale&lt;/h2&gt;
-&lt;a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow"&gt;Kotekant&lt;/a&gt;&lt;br /&gt;
+ &lt;br /&gt;
+Mode 8 of the harmonic series -- &lt;a class="wiki_link" href="/overtone%20scales"&gt;overtones 8 through 16&lt;/a&gt;, octave repeating -- is well-represented in 72edo.&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Overtones in &amp;quot;Mode 8&amp;quot;:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;16&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;...as JI Ratio from 1/1:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3/2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;15/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2/1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;...in cents:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;203.9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;386.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;551.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;702.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;840.5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;968.8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1088.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1200.0&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Nearest degree of 72edo:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;33&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;42&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;58&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;65&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;72&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;...in cents:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;200.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;383.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;550.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;700.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;833.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;966.7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1083.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1200.0&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Steps as Freq. Ratio:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9:8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10:9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11:10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12:11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13:12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14:13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;15:14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;16:15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;...in cents:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;203.9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;182.4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;165.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;150.6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;138.6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;128.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;119.4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;111.7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Nearest degree of 72edo:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;...in cents:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;200.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;183.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;166.7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;150.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;133.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;133.3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;116.7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;116.7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Music&lt;/h2&gt;
+ &lt;a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow"&gt;Kotekant&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-External links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;External links&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-External links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;External links&lt;/h2&gt;
   &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/72_tone_equal_temperament" rel="nofollow"&gt;Wikipedia article on 72edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/72_tone_equal_temperament" rel="nofollow"&gt;OrthodoxWiki Article on Byzantine chant, which uses 72edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Joe_Maneri" rel="nofollow"&gt;Wikipedia article on Joe Maneri (1927-2009)&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://members.aon.at/ekmelischemusik/" rel="nofollow"&gt;Gesellschaft für Ekmelische Musik&lt;/a&gt;, a group of composers and researchers dedicated to 72edo music&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.72note.com/" rel="nofollow"&gt;Rick Tagawa's 72edo site&lt;/a&gt;, including theory and composers' list&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://dannywier.ucoz.com" rel="nofollow"&gt;Danny Wier, composer and musician who specializes in 72-edo&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

72edo: Difference between revisions

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