72edo

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[[toc|flat]]
----
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 [[The Riemann Zeta Function+and Tuning#Zeta EDO lists|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. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).

|| 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 ||   ||

=Z function= 
72edo is the ninth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The Riemann Zeta Function and Tuning#The%20Z%20function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.

[[image:plot72.png]]

=Music= 
[[http://www.archive.org/details/Kotekant|Kotekant]] [[http://www.archive.org/download/Kotekant/kotekant.mp3|play]] by [[Gene Ward Smith]]

=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://www.ekmelic-music.org/en/index.htmmusik/|Ekmelic Music Society/Gesellschaft für Ekmelische Musik]], a group of composers and researchers dedicated to 72edo music
* [[http://sonic-arts.org/tagawa/72edo.htm|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><!-- 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="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
<!-- ws:end:WikiTextTocRule:13 --><hr />
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 [[The Riemann Zeta Function+and Tuning#Zeta EDO lists|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.<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;h1&gt; --><h1 id="toc0"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->Harmonic Scale</h1>
 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. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).<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 />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:2 -->Z function</h1>
 72edo is the ninth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>, as well as being a peak and gap edo, and the maximum value of the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function">Z function</a> in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:373:&lt;img src=&quot;/file/view/plot72.png/219772696/plot72.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" /><!-- ws:end:WikiTextLocalImageRule:373 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:4 -->Music</h1>
 <a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow">Kotekant</a> <a class="wiki_link_ext" href="http://www.archive.org/download/Kotekant/kotekant.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:6 -->External links</h1>
 <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://www.ekmelic-music.org/en/index.htmmusik/" rel="nofollow">Ekmelic Music Society/Gesellschaft für Ekmelische Musik</a>, a group of composers and researchers dedicated to 72edo music</li><li><a class="wiki_link_ext" href="http://sonic-arts.org/tagawa/72edo.htm" 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>