72edo: Difference between revisions
Wikispaces>spt3125 **Imported revision 615832209 - Original comment: added link (deorphaning); reformatted commas** |
Wikispaces>TallKite **Imported revision 621464699 - Original comment: ** |
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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:TallKite|TallKite]] and made on <tt>2017-11-11 21:21:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>621464699</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
| Line 27: | Line 27: | ||
ampersand = 34171875/33554432 = |-25 7 6> | ampersand = 34171875/33554432 = |-25 7 6> | ||
graviton = 129140163/128000000 = |-13 17 -6> | graviton = 129140163/128000000 = |-13 17 -6> | ||
ennealimma = 7629394531250/7625597484987 = |1 -27 18> | ennealimma = 7629394531250/7625597484987 = |1 -27 18> || | ||
|| | |||
||~ 7-limit ||~ 11-limit ||~ 13-limit || | ||~ 7-limit ||~ 11-limit ||~ 13-limit || | ||
| Line 39: | Line 38: | ||
19683/19600 | 19683/19600 | ||
420175/419904 | 420175/419904 | ||
250047/250000 | 250047/250000 || ....................... | ||
|| ....................... | |||
243/242 | 243/242 | ||
385/384 | 385/384 | ||
| Line 49: | Line 47: | ||
4000/3993 | 4000/3993 | ||
6250/6237 | 6250/6237 | ||
9801/9800 | 9801/9800 || ....................... | ||
|| ....................... | |||
169/168 | 169/168 | ||
325/324 | 325/324 | ||
| Line 62: | Line 59: | ||
1716/1715 | 1716/1715 | ||
2080/2079 | 2080/2079 | ||
6656/6655 | 6656/6655 || | ||
|| | |||
=Temperaments= | =Temperaments= | ||
It provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit. | It provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit. | ||
| Line 166: | Line 162: | ||
|| 71 || 1183.333 || 99/50 ||= v8 ||= down octave ||= Dv || | || 71 || 1183.333 || 99/50 ||= v8 ||= down octave ||= Dv || | ||
|| 72 || 1200 || 2/1 ||= P8 ||= perfect octave ||= D || | || 72 || 1200 || 2/1 ||= P8 ||= perfect octave ||= D || | ||
Combining ups and downs notation with [[Kite's color notation|color notation]], qualities can be loosely associated with colors: | |||
||~ quality ||~ color ||~ monzo format ||~ examples || | |||
||= double-down minor ||= blue ||= {a, b, 0, 1} ||= 7/6, 7/4 || | |||
||= minor ||= fourthward white ||= {a, b}, b < -1 ||= 32/27, 16/9 || | |||
||= upminor ||= green ||= {a, b, -1} ||= 6/5, 9/5 || | |||
||= mid ||= jade ||= {a, b, 0, 0, 1} ||= 11/9, 11/6 || | |||
||= " ||= amber ||= {a, b, 0, 0, -1} ||= 12/11, 18/11 || | |||
||= downmajor ||= yellow ||= {a, b, 1} ||= 5/4, 5/3 || | |||
||= major ||= fifthward white ||= {a, b}, b > 1 ||= 9/8, 27/16 || | |||
||= double-up major ||= red ||= {a, b, 0, -1} ||= 9/7, 12/7 || | |||
All 72edo chords can be named using ups and downs. Here are the blue, green, jade, yellow and red triads: | |||
||~ color of the 3rd ||~ JI chord ||~ notes as edosteps ||~ notes of C chord ||~ written name ||~ spoken name || | |||
||= blue ||= 6:7:9 ||= 0-16-42 ||= C Ebvv G ||= C.vvm ||= C double-down minor || | |||
||= green ||= 10:12:15 ||= 0-19-42 ||= C Eb^ G ||= C.^m ||= C upminor || | |||
||= jade ||= 18:22:27 ||= 0-21-42 ||= C Ev<span style="font-size: 90%; vertical-align: super;">3</span> G ||= C~ ||= C mid || | |||
||= yellow ||= 4:5:6 ||= 0-23-42 ||= C Ev G ||= C.v ||= C downmajor or C dot down || | |||
||= red ||= 14:18:27 ||= 0-26-42 ||= C E^^ G ||= C.^^ ||= C double-upmajor or C dot double-up || | |||
For a more complete list, see [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]]. | |||
=Linear temperaments= | =Linear temperaments= | ||
| Line 235: | Line 248: | ||
<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:18:&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:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Temperaments">Temperaments</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <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:18:&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:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Temperaments">Temperaments</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | ||
<!-- ws:end:WikiTextTocRule:28 --><hr /> | <!-- ws:end:WikiTextTocRule:28 --><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=" | 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="https://xenharmonic.wikispaces.com/24edo">24-tone equal temperament</a>, a common and standard tuning of <a class="wiki_link" href="https://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=" | Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with <a class="wiki_link" href="https://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=" | 72-tone equal temperament approximates <a class="wiki_link" href="https://xenharmonic.wikispaces.com/11-limit">11-limit just intonation</a> exceptionally well, is consistent in the <a class="wiki_link" href="https://xenharmonic.wikispaces.com/17-limit">17-limit</a>, and is the ninth <a class="wiki_link" href="https://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=" | 72 is an excellent tuning for <a class="wiki_link" href="https://xenharmonic.wikispaces.com/Gamelismic%20clan">miracle temperament</a>, especially the 11-limit version, and the related rank three temperament <a class="wiki_link" href="https://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="Commas"></a><!-- ws:end:WikiTextHeadingRule:0 -->Commas</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:0 -->Commas</h1> | ||
| Line 330: | Line 343: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperaments</h1> | ||
<br /> | <br /> | ||
It provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit.<br /> | It provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit.<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:4 -->Harmonic Scale</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:4 -->Harmonic Scale</h1> | ||
Mode 8 of the harmonic series -- <a class="wiki_link" href=" | Mode 8 of the harmonic series -- <a class="wiki_link" href="https://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 698: | Line 711: | ||
<td>approximate ratios (11-limit)<br /> | <td>approximate ratios (11-limit)<br /> | ||
</td> | </td> | ||
<td colspan="3" style="text-align: center;"><a class="wiki_link" href=" | <td colspan="3" style="text-align: center;"><a class="wiki_link" href="https://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">ups and downs </a><a class="wiki_link" href="https://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,731: | Line 1,744: | ||
</table> | </table> | ||
Combining ups and downs notation with <a class="wiki_link" href="/Kite%27s%20color%20notation">color notation</a>, qualities can be loosely associated with colors:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>quality<br /> | |||
</th> | |||
<th>color<br /> | |||
</th> | |||
<th>monzo format<br /> | |||
</th> | |||
<th>examples<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">double-down minor<br /> | |||
</td> | |||
<td style="text-align: center;">blue<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b, 0, 1}<br /> | |||
</td> | |||
<td style="text-align: center;">7/6, 7/4<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">minor<br /> | |||
</td> | |||
<td style="text-align: center;">fourthward white<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b}, b &lt; -1<br /> | |||
</td> | |||
<td style="text-align: center;">32/27, 16/9<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">upminor<br /> | |||
</td> | |||
<td style="text-align: center;">green<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b, -1}<br /> | |||
</td> | |||
<td style="text-align: center;">6/5, 9/5<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">mid<br /> | |||
</td> | |||
<td style="text-align: center;">jade<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b, 0, 0, 1}<br /> | |||
</td> | |||
<td style="text-align: center;">11/9, 11/6<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">&quot;<br /> | |||
</td> | |||
<td style="text-align: center;">amber<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b, 0, 0, -1}<br /> | |||
</td> | |||
<td style="text-align: center;">12/11, 18/11<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">downmajor<br /> | |||
</td> | |||
<td style="text-align: center;">yellow<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b, 1}<br /> | |||
</td> | |||
<td style="text-align: center;">5/4, 5/3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">major<br /> | |||
</td> | |||
<td style="text-align: center;">fifthward white<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b}, b &gt; 1<br /> | |||
</td> | |||
<td style="text-align: center;">9/8, 27/16<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">double-up major<br /> | |||
</td> | |||
<td style="text-align: center;">red<br /> | |||
</td> | |||
<td style="text-align: center;">{a, b, 0, -1}<br /> | |||
</td> | |||
<td style="text-align: center;">9/7, 12/7<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
All 72edo chords can be named using ups and downs. Here are the blue, green, jade, yellow and red triads:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>color of the 3rd<br /> | |||
</th> | |||
<th>JI chord<br /> | |||
</th> | |||
<th>notes as edosteps<br /> | |||
</th> | |||
<th>notes of C chord<br /> | |||
</th> | |||
<th>written name<br /> | |||
</th> | |||
<th>spoken name<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">blue<br /> | |||
</td> | |||
<td style="text-align: center;">6:7:9<br /> | |||
</td> | |||
<td style="text-align: center;">0-16-42<br /> | |||
</td> | |||
<td style="text-align: center;">C Ebvv G<br /> | |||
</td> | |||
<td style="text-align: center;">C.vvm<br /> | |||
</td> | |||
<td style="text-align: center;">C double-down minor<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">green<br /> | |||
</td> | |||
<td style="text-align: center;">10:12:15<br /> | |||
</td> | |||
<td style="text-align: center;">0-19-42<br /> | |||
</td> | |||
<td style="text-align: center;">C Eb^ G<br /> | |||
</td> | |||
<td style="text-align: center;">C.^m<br /> | |||
</td> | |||
<td style="text-align: center;">C upminor<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">jade<br /> | |||
</td> | |||
<td style="text-align: center;">18:22:27<br /> | |||
</td> | |||
<td style="text-align: center;">0-21-42<br /> | |||
</td> | |||
<td style="text-align: center;">C Ev<span style="font-size: 90%; vertical-align: super;">3</span> G<br /> | |||
</td> | |||
<td style="text-align: center;">C~<br /> | |||
</td> | |||
<td style="text-align: center;">C mid<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">yellow<br /> | |||
</td> | |||
<td style="text-align: center;">4:5:6<br /> | |||
</td> | |||
<td style="text-align: center;">0-23-42<br /> | |||
</td> | |||
<td style="text-align: center;">C Ev G<br /> | |||
</td> | |||
<td style="text-align: center;">C.v<br /> | |||
</td> | |||
<td style="text-align: center;">C downmajor or C dot down<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">red<br /> | |||
</td> | |||
<td style="text-align: center;">14:18:27<br /> | |||
</td> | |||
<td style="text-align: center;">0-26-42<br /> | |||
</td> | |||
<td style="text-align: center;">C E^^ G<br /> | |||
</td> | |||
<td style="text-align: center;">C.^^<br /> | |||
</td> | |||
<td style="text-align: center;">C double-upmajor or C dot double-up<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
For a more complete list, see <a class="wiki_link" href="https://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs">Ups and Downs Notation - Chord names in other EDOs</a>.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Linear temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Linear temperaments</h1> | ||
| Line 1,750: | Line 1,949: | ||
<td>1\72<br /> | <td>1\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/quincy">quincy</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,766: | Line 1,965: | ||
<td>7\72<br /> | <td>7\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/miracle">miracle</a>/benediction/manna<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,790: | Line 1,989: | ||
<td>17\72<br /> | <td>17\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/neominor">neominor</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,798: | Line 1,997: | ||
<td>19\72<br /> | <td>19\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/catakleismic">catakleismic</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,814: | Line 2,013: | ||
<td>25\72<br /> | <td>25\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/sqrtphi">sqrtphi</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,830: | Line 2,029: | ||
<td>31\72<br /> | <td>31\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/marvo">marvo</a>/zarvo<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,838: | Line 2,037: | ||
<td>35\72<br /> | <td>35\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/cotritone">cotritone</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,854: | Line 2,053: | ||
<td>5\72<br /> | <td>5\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/harry">harry</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,870: | Line 2,069: | ||
<td>11\72<br /> | <td>11\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/unidec">unidec</a>/hendec<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,878: | Line 2,077: | ||
<td>13\72<br /> | <td>13\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/wizard">wizard</a>/lizard/gizzard<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,902: | Line 2,101: | ||
<td>5\72<br /> | <td>5\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/tritikleismic">tritikleismic</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,918: | Line 2,117: | ||
<td>11\72<br /> | <td>11\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/mirkat">mirkat</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,926: | Line 2,125: | ||
<td>1\72<br /> | <td>1\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/quadritikleismic">quadritikleismic</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,966: | Line 2,165: | ||
<td>1\72<br /> | <td>1\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/octoid">octoid</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,974: | Line 2,173: | ||
<td>2\72<br /> | <td>2\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/octowerck">octowerck</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,998: | Line 2,197: | ||
<td>3\72<br /> | <td>3\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/ennealimmal">ennealimmal</a>/ennealimmic<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 2,006: | Line 2,205: | ||
<td>1\72<br /> | <td>1\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/compton">compton</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 2,014: | Line 2,213: | ||
<td>1\72<br /> | <td>1\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/hemiennealimmal">hemiennealimmal</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 2,022: | Line 2,221: | ||
<td>1\72<br /> | <td>1\72<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href=" | <td><a class="wiki_link" href="https://xenharmonic.wikispaces.com/hours">hours</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 2,037: | Line 2,236: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:10 -->Z function</h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:10 -->Z function</h1> | ||
72edo is the ninth <a class="wiki_link" href=" | 72edo is the ninth <a class="wiki_link" href="https://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="https://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: | <!-- ws:start:WikiTextLocalImageRule:1936:&lt;img src=&quot;https://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="https://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" /><!-- ws:end:WikiTextLocalImageRule:1936 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:12 -->Music</h1> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:12 -->Music</h1> | ||
<a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow">Kotekant</a> <em><a class="wiki_link_ext" href="http://www.archive.org/download/Kotekant/kotekant.mp3" rel="nofollow">play</a></em> by <a class="wiki_link" href=" | <a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow">Kotekant</a> <em><a class="wiki_link_ext" href="http://www.archive.org/download/Kotekant/kotekant.mp3" rel="nofollow">play</a></em> by <a class="wiki_link" href="https://xenharmonic.wikispaces.com/Gene%20Ward%20Smith">Gene Ward Smith</a><br /> | ||
<em><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></em> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br /> | <em><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></em> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br /> | ||
<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3" rel="nofollow">Lazy Sunday</a></em> by <a class="wiki_link" href="/Jake%20Freivald">Jake Freivald</a> in the <a class="wiki_link" href="/lazysunday">lazysunday</a> scale.<br /> | <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3" rel="nofollow">Lazy Sunday</a></em> by <a class="wiki_link" href="/Jake%20Freivald">Jake Freivald</a> in the <a class="wiki_link" href="/lazysunday">lazysunday</a> scale.<br /> | ||
| Line 2,048: | Line 2,247: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->Scales</h1> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->Scales</h1> | ||
<a class="wiki_link" href=" | <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72a">smithgw72a</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72b">smithgw72b</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72c">smithgw72c</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72d">smithgw72d</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72e">smithgw72e</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72f">smithgw72f</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72g">smithgw72g</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72h">smithgw72h</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72i">smithgw72i</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/smithgw72j">smithgw72j</a><br /> | ||
<a class="wiki_link" href=" | <a class="wiki_link" href="https://xenharmonic.wikispaces.com/blackjack">blackjack</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/miracle_8">miracle_8</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/miracle_10">miracle_10</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/miracle_12">miracle_12</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/miracle_12a">miracle_12a</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/miracle_24hi">miracle_24hi</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/miracle_24lo">miracle_24lo</a><br /> | ||
<a class="wiki_link" href=" | <a class="wiki_link" href="https://xenharmonic.wikispaces.com/keenanmarvel">keenanmarvel</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/xenakis_chrome">xenakis_chrome</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/xenakis_diat">xenakis_diat</a>, <a class="wiki_link" href="https://xenharmonic.wikispaces.com/xenakis_schrome">xenakis_schrome</a><br /> | ||
<a class="wiki_link" href=" | <a class="wiki_link" href="https://xenharmonic.wikispaces.com/genus24255et72">Euler(24255) genus in 72 equal</a><br /> | ||
<a class="wiki_link" href="/JuneGloom">JuneGloom</a><br /> | <a class="wiki_link" href="/JuneGloom">JuneGloom</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:16 -->External links</h1> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:16 -->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/" 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://72note.com/site/original.html" 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></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/" 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://72note.com/site/original.html" 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></pre></div> | ||