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 [[xenharmonic/24edo|24-tone equal temperament]], a common and standard tuning of [[xenharmonic/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 [[xenharmonic/96edo|96-edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri. 72-tone equal temperament approximates [[xenharmonic/11-limit|11-limit just intonation]] exceptionally well, is consistent in the [[xenharmonic/17-limit|17-limit]], and is the ninth [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|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 [[xenharmonic/Gamelismic clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[xenharmonic/Marvel family#Prodigy|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 -- [[xenharmonic/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 || || =Intervals= || degrees || cents value || approximate ratios (11-limit) || || 0 || 0 || 1/1 || || 1 || 16.667 || 81/80 || || 2 || 33.333 || 45/44 || || 3 || 50 || 33/32 || || 4 || 66.667 || 25/24 || || 5 || 83.333 || 21/20 || || 6 || 100 || 35/33 || || 7 || 116.667 || 15/14 || || 8 || 133.333 || 27/25 || || 9 || 150 || 12/11 || || 10 || 166.667 || 11/10 || || 11 || 183.333 || 10/9 || || 12 || 200 || 9/8 || || 13 || 216.667 || 25/22 || || 14 || 233.333 || 8/7 || || 15 || 250 || 81/70 || || 16 || 266.667 || 7/6 || || 17 || 283.333 || 33/28 || || 18 || 300 || 25/21 || || 19 || 316.667 || 6/5 || || 20 || 333.333 || 40/33 || || 21 || 350 || 11/9 || || 22 || 366.667 || 99/80 || || 23 || 383.333 || 5/4 || || 24 || 400 || 44/35 || || 25 || 416.667 || 14/11 || || 26 || 433.333 || 9/7 || || 27 || 450 || 35/27 || || 28 || 466.667 || 21/16 || || 29 || 483.333 || 33/25 || || 30 || 500 || 4/3 || || 31 || 516.667 || 27/20 || || 32 || 533.333 || 15/11 || || 33 || 550 || 11/8 || || 34 || 566.667 || 25/18 || || 35 || 583.333 || 7/5 || || 36 || 600 || 99/70 || || 37 || 616.667 || 10/7 || || 38 || 633.333 || 36/25 || || 39 || 650 || 16/11 || || 40 || 666.667 || 22/15 || || 41 || 683.333 || 40/27 || || 42 || 700 || 3/2 || || 43 || 716.667 || 50/33 || || 44 || 733.333 || 32/21 || || 45 || 750 || 54/35 || || 46 || 766.667 || 14/9 || || 47 || 783.333 || 11/7 || || 48 || 800 || 35/22 || || 49 || 816.667 || 8/5 || || 50 || 833.333 || 81/50 || || 51 || 850 || 18/11 || || 52 || 866.667 || 33/20 || || 53 || 883.333 || 5/3 || || 54 || 900 || 27/16 || || 55 || 916.667 || 56/33 || || 56 || 933.333 || 12/7 || || 57 || 950 || 121/70 || || 58 || 966.667 || 7/4 || || 59 || 983.333 || 44/25 || || 60 || 1000 || 16/9 || || 61 || 1016.667 || 9/5 || || 62 || 1033.333 || 20/11 || || 63 || 1050 || 11/6 || || 64 || 1066.667 || 50/27 || || 65 || 1083.333 || 15/8 || || 66 || 1100 || 66/35 || || 67 || 1116.667 || 21/11 || || 68 || 1133.333 || 27/14 || || 69 || 1150 || 35/18 || || 70 || 1166.667 || 49/25 || || 71 || 1183.333 || 99/50 || || 72 || 1200 || 2/1 || = = =Linear temperaments= ||~ Periods per octave ||~ Generator ||~ Names || || 1 || 1\72 || [[xenharmonic/quincy|quincy]] || || 1 || 5\72 || [[marvolo]] || || 1 || 7\72 || [[xenharmonic/miracle|miracle]]/benediction/manna || || 1 || 11\72 || || || 1 || 13\72 || || || 1 || 17\72 || [[xenharmonic/neominor|neominor]] || || 1 || 19\72 || [[xenharmonic/catakleismic|catakleismic]] || || 1 || 23\72 || || || 1 || 25\72 || [[xenharmonic/sqrtphi|sqrtphi]] || || 1 || 29\72 || || || 1 || 31\72 || [[xenharmonic/marvo|marvo]]/zarvo || || 1 || 35\72 || [[xenharmonic/cotritone|cotritone]] || || 2 || 1\72 || || || 2 || 5\72 || [[xenharmonic/harry|harry]] || || 2 || 7\72 || || || 2 || 11\72 || [[xenharmonic/unidec|unidec]]/hendec || || 2 || 13\72 || [[xenharmonic/wizard|wizard]]/lizard/gizzard || || 2 || 17\72 || || || 3 || 1\72 || || || 3 || 5\72 || [[xenharmonic/tritikleismic|tritikleismic]] || || 3 || 7\72 || || || 3 || 11\72 || [[xenharmonic/mirkat|mirkat]] || || 4 || 1\72 || [[xenharmonic/quadritikleismic|quadritikleismic]] || || 4 || 5\72 || || || 4 || 7\72 || || || 6 || 1\72 || || || 6 || 5\72 || || || 8 || 1\72 || [[xenharmonic/octoid|octoid]] || || 8 || 2\72 || [[xenharmonic/octowerck|octowerck]] || || 8 || 4\72 || || || 9 || 1\72 || || || 9 || 3\72 || [[xenharmonic/ennealimmal|ennealimmal]]/ennealimmic || || 12 || 1\72 || [[xenharmonic/compton|compton]] || || 18 || 1\72 || [[xenharmonic/hemiennealimmal|hemiennealimmal]] || || 24 || 1\72 || [[xenharmonic/hours|hours]] || || 36 || 1\72 || || =Z function= 72edo is the ninth [[xenharmonic/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 [[xenharmonic/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:xenharmonic/plot72.png]] =Music= [[http://www.archive.org/details/Kotekant|Kotekant]] [[http://www.archive.org/download/Kotekant/kotekant.mp3|play]] by [[xenharmonic/Gene Ward Smith|Gene Ward Smith]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3|Twinkle canon – 72 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3|Lazy Sunday]] by [[Jake Freivald]] in the [[lazysunday]] scale. [[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3|June Gloom #9]] by Prent Rodgers =Scales= [[xenharmonic/smithgw72a|smithgw72a]], [[xenharmonic/smithgw72b|smithgw72b]], [[xenharmonic/smithgw72c|smithgw72c]], [[xenharmonic/smithgw72d|smithgw72d]], [[xenharmonic/smithgw72e|smithgw72e]], [[xenharmonic/smithgw72f|smithgw72f]], [[xenharmonic/smithgw72g|smithgw72g]], [[xenharmonic/smithgw72h|smithgw72h]], [[xenharmonic/smithgw72i|smithgw72i]], [[xenharmonic/smithgw72j|smithgw72j]] [[xenharmonic/blackjack|blackjack]], [[xenharmonic/miracle_8|miracle_8]], [[xenharmonic/miracle_10|miracle_10]], [[xenharmonic/miracle_12|miracle_12]], [[xenharmonic/miracle_12a|miracle_12a]], [[xenharmonic/miracle_24hi|miracle_24hi]], [[xenharmonic/miracle_24lo|miracle_24lo]] [[xenharmonic/keenanmarvel|keenanmarvel]], [[xenharmonic/xenakis_chrome|xenakis_chrome]], [[xenharmonic/xenakis_diat|xenakis_diat]], [[xenharmonic/xenakis_schrome|xenakis_schrome]] [[xenharmonic/genus24255et72|Euler(24255) genus in 72 equal]] [[JuneGloom]] =External links= * [[http://en.wikipedia.org/wiki/72_tone_equal_temperament|Wikipedia article on 72edo]] * [[http://orthodoxwiki.org/Byzantine_Chant|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://www.myspace.com/dawier|Danny Wier, composer and musician who specializes in 72-edo]]
Original HTML content:
<html><head><title>72edo</title></head><body><!-- ws:start:WikiTextTocRule:16:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#toc2"> </a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: -->
<!-- ws:end:WikiTextTocRule:25 --><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="http://xenharmonic.wikispaces.com/24edo">24-tone equal temperament</a>, a common and standard tuning of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/96edo">96-edo</a>), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.<br />
<br />
72-tone equal temperament approximates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/11-limit">11-limit just intonation</a> exceptionally well, is consistent in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17-limit">17-limit</a>, and is the ninth <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">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="http://xenharmonic.wikispaces.com/Gamelismic%20clan">miracle temperament</a>, especially the 11-limit version, and the related rank three temperament <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20family#Prodigy">prodigy</a>, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><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="http://xenharmonic.wikispaces.com/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 />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1>
<table class="wiki_table">
<tr>
<td>degrees<br />
</td>
<td>cents value<br />
</td>
<td>approximate ratios (11-limit)<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>16.667<br />
</td>
<td>81/80<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>33.333<br />
</td>
<td>45/44<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>50<br />
</td>
<td>33/32<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>66.667<br />
</td>
<td>25/24<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>83.333<br />
</td>
<td>21/20<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>100<br />
</td>
<td>35/33<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>116.667<br />
</td>
<td>15/14<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>133.333<br />
</td>
<td>27/25<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>150<br />
</td>
<td>12/11<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>166.667<br />
</td>
<td>11/10<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>183.333<br />
</td>
<td>10/9<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>200<br />
</td>
<td>9/8<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>216.667<br />
</td>
<td>25/22<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>233.333<br />
</td>
<td>8/7<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>250<br />
</td>
<td>81/70<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>266.667<br />
</td>
<td>7/6<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>283.333<br />
</td>
<td>33/28<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>300<br />
</td>
<td>25/21<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>316.667<br />
</td>
<td>6/5<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>333.333<br />
</td>
<td>40/33<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>350<br />
</td>
<td>11/9<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>366.667<br />
</td>
<td>99/80<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>383.333<br />
</td>
<td>5/4<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>400<br />
</td>
<td>44/35<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>416.667<br />
</td>
<td>14/11<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>433.333<br />
</td>
<td>9/7<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>450<br />
</td>
<td>35/27<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>466.667<br />
</td>
<td>21/16<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>483.333<br />
</td>
<td>33/25<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>500<br />
</td>
<td>4/3<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>516.667<br />
</td>
<td>27/20<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>533.333<br />
</td>
<td>15/11<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>550<br />
</td>
<td>11/8<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>566.667<br />
</td>
<td>25/18<br />
</td>
</tr>
<tr>
<td>35<br />
</td>
<td>583.333<br />
</td>
<td>7/5<br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>600<br />
</td>
<td>99/70<br />
</td>
</tr>
<tr>
<td>37<br />
</td>
<td>616.667<br />
</td>
<td>10/7<br />
</td>
</tr>
<tr>
<td>38<br />
</td>
<td>633.333<br />
</td>
<td>36/25<br />
</td>
</tr>
<tr>
<td>39<br />
</td>
<td>650<br />
</td>
<td>16/11<br />
</td>
</tr>
<tr>
<td>40<br />
</td>
<td>666.667<br />
</td>
<td>22/15<br />
</td>
</tr>
<tr>
<td>41<br />
</td>
<td>683.333<br />
</td>
<td>40/27<br />
</td>
</tr>
<tr>
<td>42<br />
</td>
<td>700<br />
</td>
<td>3/2<br />
</td>
</tr>
<tr>
<td>43<br />
</td>
<td>716.667<br />
</td>
<td>50/33<br />
</td>
</tr>
<tr>
<td>44<br />
</td>
<td>733.333<br />
</td>
<td>32/21<br />
</td>
</tr>
<tr>
<td>45<br />
</td>
<td>750<br />
</td>
<td>54/35<br />
</td>
</tr>
<tr>
<td>46<br />
</td>
<td>766.667<br />
</td>
<td>14/9<br />
</td>
</tr>
<tr>
<td>47<br />
</td>
<td>783.333<br />
</td>
<td>11/7<br />
</td>
</tr>
<tr>
<td>48<br />
</td>
<td>800<br />
</td>
<td>35/22<br />
</td>
</tr>
<tr>
<td>49<br />
</td>
<td>816.667<br />
</td>
<td>8/5<br />
</td>
</tr>
<tr>
<td>50<br />
</td>
<td>833.333<br />
</td>
<td>81/50<br />
</td>
</tr>
<tr>
<td>51<br />
</td>
<td>850<br />
</td>
<td>18/11<br />
</td>
</tr>
<tr>
<td>52<br />
</td>
<td>866.667<br />
</td>
<td>33/20<br />
</td>
</tr>
<tr>
<td>53<br />
</td>
<td>883.333<br />
</td>
<td>5/3<br />
</td>
</tr>
<tr>
<td>54<br />
</td>
<td>900<br />
</td>
<td>27/16<br />
</td>
</tr>
<tr>
<td>55<br />
</td>
<td>916.667<br />
</td>
<td>56/33<br />
</td>
</tr>
<tr>
<td>56<br />
</td>
<td>933.333<br />
</td>
<td>12/7<br />
</td>
</tr>
<tr>
<td>57<br />
</td>
<td>950<br />
</td>
<td>121/70<br />
</td>
</tr>
<tr>
<td>58<br />
</td>
<td>966.667<br />
</td>
<td>7/4<br />
</td>
</tr>
<tr>
<td>59<br />
</td>
<td>983.333<br />
</td>
<td>44/25<br />
</td>
</tr>
<tr>
<td>60<br />
</td>
<td>1000<br />
</td>
<td>16/9<br />
</td>
</tr>
<tr>
<td>61<br />
</td>
<td>1016.667<br />
</td>
<td>9/5<br />
</td>
</tr>
<tr>
<td>62<br />
</td>
<td>1033.333<br />
</td>
<td>20/11<br />
</td>
</tr>
<tr>
<td>63<br />
</td>
<td>1050<br />
</td>
<td>11/6<br />
</td>
</tr>
<tr>
<td>64<br />
</td>
<td>1066.667<br />
</td>
<td>50/27<br />
</td>
</tr>
<tr>
<td>65<br />
</td>
<td>1083.333<br />
</td>
<td>15/8<br />
</td>
</tr>
<tr>
<td>66<br />
</td>
<td>1100<br />
</td>
<td>66/35<br />
</td>
</tr>
<tr>
<td>67<br />
</td>
<td>1116.667<br />
</td>
<td>21/11<br />
</td>
</tr>
<tr>
<td>68<br />
</td>
<td>1133.333<br />
</td>
<td>27/14<br />
</td>
</tr>
<tr>
<td>69<br />
</td>
<td>1150<br />
</td>
<td>35/18<br />
</td>
</tr>
<tr>
<td>70<br />
</td>
<td>1166.667<br />
</td>
<td>49/25<br />
</td>
</tr>
<tr>
<td>71<br />
</td>
<td>1183.333<br />
</td>
<td>99/50<br />
</td>
</tr>
<tr>
<td>72<br />
</td>
<td>1200<br />
</td>
<td>2/1<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><!-- ws:end:WikiTextHeadingRule:4 --> </h1>
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->Linear temperaments</h1>
<table class="wiki_table">
<tr>
<th>Periods per octave<br />
</th>
<th>Generator<br />
</th>
<th>Names<br />
</th>
</tr>
<tr>
<td>1<br />
</td>
<td>1\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/quincy">quincy</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>5\72<br />
</td>
<td><a class="wiki_link" href="/marvolo">marvolo</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>7\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle">miracle</a>/benediction/manna<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>11\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>13\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>17\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/neominor">neominor</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>19\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/catakleismic">catakleismic</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>23\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>25\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/sqrtphi">sqrtphi</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>29\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>31\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/marvo">marvo</a>/zarvo<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>35\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/cotritone">cotritone</a><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>1\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>5\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/harry">harry</a><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>7\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>11\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/unidec">unidec</a>/hendec<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>13\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/wizard">wizard</a>/lizard/gizzard<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>17\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>1\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>5\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/tritikleismic">tritikleismic</a><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>7\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>11\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/mirkat">mirkat</a><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>1\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/quadritikleismic">quadritikleismic</a><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>5\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>7\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>1\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>5\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>1\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/octoid">octoid</a><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>2\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/octowerck">octowerck</a><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>4\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>1\72<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>3\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/ennealimmal">ennealimmal</a>/ennealimmic<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>1\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/compton">compton</a><br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>1\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/hemiennealimmal">hemiennealimmal</a><br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>1\72<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/hours">hours</a><br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>1\72<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:8 -->Z function</h1>
72edo is the ninth <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/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:1277:<img src="http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="" title="" /> --><img src="http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" /><!-- ws:end:WikiTextLocalImageRule:1277 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:10 -->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="http://xenharmonic.wikispaces.com/Gene%20Ward%20Smith">Gene Ward Smith</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3" rel="nofollow">Twinkle canon – 72 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3" rel="nofollow">Lazy Sunday</a> by <a class="wiki_link" href="/Jake%20Freivald">Jake Freivald</a> in the <a class="wiki_link" href="/lazysunday">lazysunday</a> scale.<br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3" rel="nofollow">June Gloom #9</a> by Prent Rodgers<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:12 -->Scales</h1>
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72a">smithgw72a</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72b">smithgw72b</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72c">smithgw72c</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72d">smithgw72d</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72e">smithgw72e</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72f">smithgw72f</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72g">smithgw72g</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72h">smithgw72h</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72i">smithgw72i</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72j">smithgw72j</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/blackjack">blackjack</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_8">miracle_8</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_10">miracle_10</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_12">miracle_12</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_12a">miracle_12a</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_24hi">miracle_24hi</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_24lo">miracle_24lo</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/keenanmarvel">keenanmarvel</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/xenakis_chrome">xenakis_chrome</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/xenakis_diat">xenakis_diat</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/xenakis_schrome">xenakis_schrome</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/genus24255et72">Euler(24255) genus in 72 equal</a><br />
<a class="wiki_link" href="/JuneGloom">JuneGloom</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:14 -->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://orthodoxwiki.org/Byzantine_Chant" 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://www.myspace.com/dawier" rel="nofollow" target="_blank">Danny Wier, composer and musician who specializes in 72-edo</a></li></ul></body></html>