72edo: Difference between revisions
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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: | : This revision was by author [[User:guest|guest]] and made on <tt>2012-05-30 11:56:31 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>340989126</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]] | ||
---- | ---- | ||
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. | 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 [[96edo|96-edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri. | 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 [[11-limit|11-limit just intonation]] exceptionally well, is consistent in the [[17-limit]], and is the ninth [[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-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 [[Gamelismic clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel family#Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic. | 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= | =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). | 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 || | || Overtones in "Mode 8": || 8 || || 9 || || 10 || || 11 || || 12 || || 13 || || 14 || || 15 || || 16 || | ||
| Line 53: | Line 53: | ||
|| 20 || 333.333 || 17/14 || | || 20 || 333.333 || 17/14 || | ||
|| 21 || 350 || 11/9 || | || 21 || 350 || 11/9 || | ||
|| 22 || 366.667 || | || 22 || 366.667 || 16/13, 21/17, 26/21 || | ||
|| 23 || 383.333 || 5/4 || | || 23 || 383.333 || 5/4 || | ||
|| 24 || 400 || || | || 24 || 400 || || | ||
| Line 81: | Line 81: | ||
|| 48 || 800 || || | || 48 || 800 || || | ||
|| 49 || 816.667 || 8/5 || | || 49 || 816.667 || 8/5 || | ||
|| 50 || 833.333 || | || 50 || 833.333 || 13/8, 21/13, 34/21 || | ||
|| 51 || 850 || 18/11 || | || 51 || 850 || 18/11 || | ||
|| 52 || 866.667 || 28/17 || | || 52 || 866.667 || 28/17 || | ||
| Line 107: | Line 107: | ||
=Linear temperaments= | =Linear temperaments= | ||
||~ Periods per octave ||~ Generator ||~ Names || | ||~ Periods per octave ||~ Generator ||~ Names || | ||
|| 1 || 1\72 || [[quincy]] || | || 1 || 1\72 || [[xenharmonic/quincy|quincy]] || | ||
|| 1 || 5\72 || || | || 1 || 5\72 || || | ||
|| 1 || 7\72 || [[miracle]]/benediction/manna || | || 1 || 7\72 || [[xenharmonic/miracle|miracle]]/benediction/manna || | ||
|| 1 || 11\72 || || | || 1 || 11\72 || || | ||
|| 1 || 13\72 || || | || 1 || 13\72 || || | ||
|| 1 || 17\72 || [[neominor]] || | || 1 || 17\72 || [[xenharmonic/neominor|neominor]] || | ||
|| 1 || 19\72 || [[catakleismic]] || | || 1 || 19\72 || [[xenharmonic/catakleismic|catakleismic]] || | ||
|| 1 || 23\72 || || | || 1 || 23\72 || || | ||
|| 1 || 25\72 || [[sqrtphi]] || | || 1 || 25\72 || [[xenharmonic/sqrtphi|sqrtphi]] || | ||
|| 1 || 29\72 || || | || 1 || 29\72 || || | ||
|| 1 || 31\72 || [[marvo]]/zarvo || | || 1 || 31\72 || [[xenharmonic/marvo|marvo]]/zarvo || | ||
|| 1 || 35\72 || [[cotritone]] || | || 1 || 35\72 || [[xenharmonic/cotritone|cotritone]] || | ||
|| 2 || 1\72 || || | || 2 || 1\72 || || | ||
|| 2 || 5\72 || [[harry]] || | || 2 || 5\72 || [[xenharmonic/harry|harry]] || | ||
|| 2 || 7\72 || || | || 2 || 7\72 || || | ||
|| 2 || 11\72 || [[unidec]]/hendec || | || 2 || 11\72 || [[xenharmonic/unidec|unidec]]/hendec || | ||
|| 2 || 13\72 || [[wizard]]/lizard/gizzard || | || 2 || 13\72 || [[xenharmonic/wizard|wizard]]/lizard/gizzard || | ||
|| 2 || 17\72 || || | || 2 || 17\72 || || | ||
|| 3 || 1\72 || || | || 3 || 1\72 || || | ||
|| 3 || 5\72 || [[tritikleismic]] || | || 3 || 5\72 || [[xenharmonic/tritikleismic|tritikleismic]] || | ||
|| 3 || 7\72 || || | || 3 || 7\72 || || | ||
|| 3 || 11\72 || [[mirkat]] || | || 3 || 11\72 || [[xenharmonic/mirkat|mirkat]] || | ||
|| 4 || 1\72 || [[quadritikleismic]] || | || 4 || 1\72 || [[xenharmonic/quadritikleismic|quadritikleismic]] || | ||
|| 4 || 5\72 || || | || 4 || 5\72 || || | ||
|| 4 || 7\72 || || | || 4 || 7\72 || || | ||
|| 6 || 1\72 || || | || 6 || 1\72 || || | ||
|| 6 || 5\72 || || | || 6 || 5\72 || || | ||
|| 8 || 1\72 || [[octoid]] || | || 8 || 1\72 || [[xenharmonic/octoid|octoid]] || | ||
|| 8 || 2\72 || [[octowerck]] || | || 8 || 2\72 || [[xenharmonic/octowerck|octowerck]] || | ||
|| 8 || 4\72 || || | || 8 || 4\72 || || | ||
|| 9 || 1\72 || || | || 9 || 1\72 || || | ||
|| 9 || 3\72 || [[ennealimmal]]/ennealimmic || | || 9 || 3\72 || [[xenharmonic/ennealimmal|ennealimmal]]/ennealimmic || | ||
|| 12 || 1\72 || [[compton]] || | || 12 || 1\72 || [[xenharmonic/compton|compton]] || | ||
|| 18 || 1\72 || [[hemiennealimmal]] || | || 18 || 1\72 || [[xenharmonic/hemiennealimmal|hemiennealimmal]] || | ||
|| 24 || 1\72 || [[hours]] || | || 24 || 1\72 || [[xenharmonic/hours|hours]] || | ||
|| 36 || 1\72 || || | || 36 || 1\72 || || | ||
=Z function= | =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. | 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:plot72.png]] | [[image:xenharmonic/plot72.png]] | ||
=Music= | =Music= | ||
[[http://www.archive.org/details/Kotekant|Kotekant]] [[http://www.archive.org/download/Kotekant/kotekant.mp3|play]] by [[Gene Ward Smith]] | [[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/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3|Twinkle canon – 72 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]] | ||
=Scales= | =Scales= | ||
[[smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]] | [[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]] | ||
[[blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]] | [[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]] | ||
[[keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]] | [[xenharmonic/keenanmarvel|keenanmarvel]], [[xenharmonic/xenakis_chrome|xenakis_chrome]], [[xenharmonic/xenakis_diat|xenakis_diat]], [[xenharmonic/xenakis_schrome|xenakis_schrome]] | ||
[[genus24255et72|Euler(24255) genus in 72 equal]] | [[xenharmonic/genus24255et72|Euler(24255) genus in 72 equal]] | ||
=External links= | =External links= | ||
| Line 169: | Line 169: | ||
<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"><html><head><title>72edo</title></head><body><!-- ws:start:WikiTextTocRule:16:&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: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: --> | <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"><html><head><title>72edo</title></head><body><!-- ws:start:WikiTextTocRule:16:&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: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 /> | <!-- 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="/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 /> | 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 /> | <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 /> | 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 /> | <br /> | ||
72-tone equal temperament approximates <a class="wiki_link" href="/11-limit">11-limit just intonation</a> exceptionally well, is consistent in the <a class="wiki_link" href="/17-limit">17-limit</a>, and is the ninth <a class="wiki_link" href="/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 /> | 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 /> | <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 <a class="wiki_link" href="/Marvel%20family#Prodigy">prodigy</a>, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.<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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->Harmonic Scale</h1> | <!-- 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 /> | 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 /> | <br /> | ||
| Line 721: | Line 721: | ||
<td>366.667<br /> | <td>366.667<br /> | ||
</td> | </td> | ||
<td><br /> | <td>16/13, 21/17, 26/21<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 945: | Line 945: | ||
<td>833.333<br /> | <td>833.333<br /> | ||
</td> | </td> | ||
<td><br /> | <td>13/8, 21/13, 34/21<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,144: | Line 1,144: | ||
<td>1\72<br /> | <td>1\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/quincy">quincy</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/quincy">quincy</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,160: | Line 1,160: | ||
<td>7\72<br /> | <td>7\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/miracle">miracle</a>/benediction/manna<br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle">miracle</a>/benediction/manna<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,184: | Line 1,184: | ||
<td>17\72<br /> | <td>17\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/neominor">neominor</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/neominor">neominor</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,192: | Line 1,192: | ||
<td>19\72<br /> | <td>19\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/catakleismic">catakleismic</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/catakleismic">catakleismic</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,208: | Line 1,208: | ||
<td>25\72<br /> | <td>25\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/sqrtphi">sqrtphi</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/sqrtphi">sqrtphi</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,224: | Line 1,224: | ||
<td>31\72<br /> | <td>31\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/marvo">marvo</a>/zarvo<br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/marvo">marvo</a>/zarvo<br /> | ||
</td> | </td> | ||
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<td>35\72<br /> | <td>35\72<br /> | ||
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<td><a class="wiki_link" href="/cotritone">cotritone</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/cotritone">cotritone</a><br /> | ||
</td> | </td> | ||
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<td>5\72<br /> | <td>5\72<br /> | ||
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<td><a class="wiki_link" href="/harry">harry</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/harry">harry</a><br /> | ||
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</tr> | </tr> | ||
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<td>11\72<br /> | <td>11\72<br /> | ||
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<td><a class="wiki_link" href="/unidec">unidec</a>/hendec<br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/unidec">unidec</a>/hendec<br /> | ||
</td> | </td> | ||
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<td>13\72<br /> | <td>13\72<br /> | ||
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<td><a class="wiki_link" href="/wizard">wizard</a>/lizard/gizzard<br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/wizard">wizard</a>/lizard/gizzard<br /> | ||
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<td><a class="wiki_link" href="/tritikleismic">tritikleismic</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/tritikleismic">tritikleismic</a><br /> | ||
</td> | </td> | ||
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<td><a class="wiki_link" href="/mirkat">mirkat</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/mirkat">mirkat</a><br /> | ||
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<td><a class="wiki_link" href="/quadritikleismic">quadritikleismic</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/quadritikleismic">quadritikleismic</a><br /> | ||
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<td><a class="wiki_link" href="/octoid">octoid</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/octoid">octoid</a><br /> | ||
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<td>2\72<br /> | <td>2\72<br /> | ||
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<td><a class="wiki_link" href="/octowerck">octowerck</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/octowerck">octowerck</a><br /> | ||
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<td><a class="wiki_link" href="/ennealimmal">ennealimmal</a>/ennealimmic<br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/ennealimmal">ennealimmal</a>/ennealimmic<br /> | ||
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<td><a class="wiki_link" href="/compton">compton</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/compton">compton</a><br /> | ||
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<td><a class="wiki_link" href="/hemiennealimmal">hemiennealimmal</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/hemiennealimmal">hemiennealimmal</a><br /> | ||
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<td><a class="wiki_link" href="/hours">hours</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/hours">hours</a><br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:8 -->Z function</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:8 -->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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule:1277:&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:1277 --><br /> | <!-- ws:start:WikiTextLocalImageRule:1277:&lt;img src=&quot;http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:10 -->Music</h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><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="/Gene%20Ward%20Smith">Gene Ward Smith</a><br /> | <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/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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:12 -->Scales</h1> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:12 -->Scales</h1> | ||
<a class="wiki_link" href="/smithgw72a">smithgw72a</a>, <a class="wiki_link" href="/smithgw72b">smithgw72b</a>, <a class="wiki_link" href="/smithgw72c">smithgw72c</a>, <a class="wiki_link" href="/smithgw72d">smithgw72d</a>, <a class="wiki_link" href="/smithgw72e">smithgw72e</a>, <a class="wiki_link" href="/smithgw72f">smithgw72f</a>, <a class="wiki_link" href="/smithgw72g">smithgw72g</a>, <a class="wiki_link" href="/smithgw72h">smithgw72h</a>, <a class="wiki_link" href="/smithgw72i">smithgw72i</a>, <a class="wiki_link" href="/smithgw72j">smithgw72j</a><br /> | <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="/blackjack">blackjack</a>, <a class="wiki_link" href="/miracle_8">miracle_8</a>, <a class="wiki_link" href="/miracle_10">miracle_10</a>, <a class="wiki_link" href="/miracle_12">miracle_12</a>, <a class="wiki_link" href="/miracle_12a">miracle_12a</a>, <a class="wiki_link" href="/miracle_24hi">miracle_24hi</a>, <a class="wiki_link" href="/miracle_24lo">miracle_24lo</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="/keenanmarvel">keenanmarvel</a>, <a class="wiki_link" href="/xenakis_chrome">xenakis_chrome</a>, <a class="wiki_link" href="/xenakis_diat">xenakis_diat</a>, <a class="wiki_link" href="/xenakis_schrome">xenakis_schrome</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="/genus24255et72">Euler(24255) genus in 72 equal</a><br /> | <a class="wiki_link" href="http://xenharmonic.wikispaces.com/genus24255et72">Euler(24255) genus in 72 equal</a><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:14 -->External links</h1> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><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://soundcloud.com/dawiertx" rel="nofollow">Danny Wier, composer and musician who specializes in 72-edo</a></li></ul></body></html></pre></div> | <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://soundcloud.com/dawiertx" rel="nofollow">Danny Wier, composer and musician who specializes in 72-edo</a></li></ul></body></html></pre></div> | ||