72edo
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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. 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 || || ==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:<h2> --><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. 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 "Mode 8":<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:<h2> --><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:<h2> --><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>