72edo: Difference between revisions

Wikispaces>TallKite
**Imported revision 602866126 - Original comment: **
Wikispaces>TallKite
**Imported revision 602866264 - Original comment: **
Line 1: Line 1:
<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:TallKite|TallKite]] and made on <tt>2016-12-28 04:58:46 UTC</tt>.<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-28 05:09:07 UTC</tt>.<br>
: The original revision id was <tt>602866126</tt>.<br>
: The original revision id was <tt>602866264</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>
Line 17: Line 17:


=Commas and Temperaments=  
=Commas and Temperaments=  
72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and the kleisma, 15625/15552, the ampersand, 34171875/33554432 = |-25 7 6&gt;, the graviton, 129140163/128000000 = |-13 17 -6&gt;, the and the ennealimma, 7629394531250/7625597484987 = |1 -27 18&gt; in the 5-limit. The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000. 72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800. For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655. 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.
 
72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and  
the kleisma, 15625/15552,  
the ampersand, 34171875/33554432 = |-25 7 6&gt;,  
the graviton, 129140163/128000000 = |-13 17 -6&gt;, and  
the ennealimma, 7629394531250/7625597484987 = |1 -27 18&gt; in the 5-limit.  
 
The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.
 
72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800.  
 
For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.
 
 
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.


=Harmonic Scale=  
=Harmonic Scale=  
Line 35: Line 49:
|| degrees || cents value || approximate ratios (11-limit) ||||= [[xenharmonic/Ups and Downs Notation|ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] ||
|| degrees || cents value || approximate ratios (11-limit) ||||= [[xenharmonic/Ups and Downs Notation|ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] ||
|| 0 || 0 || 1/1 ||= P1 ||= D ||
|| 0 || 0 || 1/1 ||= P1 ||= D ||
|| 1 || 16.667 || 81/80 ||= ^1 ||= D^ ||
|| 1 || 16.667 || 81/80 ||= ^1 (up unison) ||= D^ ||
|| 2 || 33.333 || 45/44 ||= ^^1 ||= D^^ ||
|| 2 || 33.333 || 45/44 ||= ^^1(double-up unison) ||= D^^ ||
|| 3 || 50 || 33/32 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;1, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m2 || D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Ebv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 3 || 50 || 33/32 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;1, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m2 || D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Ebv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 4 || 66.667 || 25/24 ||= vvm2 ||= Ebvv ||
|| 4 || 66.667 || 25/24 ||= vvm2 ||= Ebvv ||
|| 5 || 83.333 || 21/20 ||= vm2 ||= Ebv ||
|| 5 || 83.333 || 21/20 ||= vm2 (downminor 2nd) ||= Ebv ||
|| 6 || 100 || 35/33 ||= m2 ||= Eb ||
|| 6 || 100 || 35/33 ||= m2 ||= Eb ||
|| 7 || 116.667 || 15/14 ||= ^m2 ||= Eb^ ||
|| 7 || 116.667 || 15/14 ||= ^m2 (upminor 2nd) ||= Eb^ ||
|| 8 || 133.333 || 27/25 ||= v~2 ||= Eb^^ ||
|| 8 || 133.333 || 27/25 ||= v~2 ||= Eb^^ ||
|| 9 || 150 || 12/11 ||= ~2 ||= Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 9 || 150 || 12/11 ||= ~2 (mid 2nd) ||= Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 10 || 166.667 || 11/10 ||= ^~2 ||= Evv ||
|| 10 || 166.667 || 11/10 ||= ^~2 (upmid 2nd) ||= Evv ||
|| 11 || 183.333 || 10/9 ||= vM2 ||= Ev ||
|| 11 || 183.333 || 10/9 ||= vM2 ||= Ev ||
|| 12 || 200 || 9/8 ||= M2 ||= E ||
|| 12 || 200 || 9/8 ||= M2 ||= E ||
Line 67: Line 81:
|| 31 || 516.667 || 27/20 ||= ^4 ||= G^ ||
|| 31 || 516.667 || 27/20 ||= ^4 ||= G^ ||
|| 32 || 533.333 || 15/11 ||= ^^4 ||= G^^ ||
|| 32 || 533.333 || 15/11 ||= ^^4 ||= G^^ ||
|| 33 || 550 || 11/8 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4&lt;span style="font-size: 90%; vertical-align: super;"&gt; &lt;/span&gt; ||= G^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 33 || 550 || 11/8 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4 ||= G^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 34 || 566.667 || 25/18 ||= vvA4 ||= G#vv ||
|| 34 || 566.667 || 25/18 ||= vvA4 ||= G#vv ||
|| 35 || 583.333 || 7/5 ||= vA4, vd5 ||= G#v, Abv ||
|| 35 || 583.333 || 7/5 ||= vA4, vd5 ||= G#v, Abv ||
Line 73: Line 87:
|| 37 || 616.667 || 10/7 ||= ^A4, ^d5 ||= G#^, Ab^ ||
|| 37 || 616.667 || 10/7 ||= ^A4, ^d5 ||= G#^, Ab^ ||
|| 38 || 633.333 || 36/25 ||= ^^d5 ||= Ab^^ ||
|| 38 || 633.333 || 36/25 ||= ^^d5 ||= Ab^^ ||
|| 39 || 650 || 16/11 ||= &lt;span style="font-size: 90%; vertical-align: super;"&gt; &lt;/span&gt;v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5 ||= Av&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 39 || 650 || 16/11 ||= v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5 ||= Av&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|| 40 || 666.667 || 22/15 ||= vv5 ||= Avv ||
|| 40 || 666.667 || 22/15 ||= vv5 ||= Avv ||
|| 41 || 683.333 || 40/27 ||= v5 ||= Av ||
|| 41 || 683.333 || 40/27 ||= v5 ||= Av ||
Line 185: Line 199:
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Commas and Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Commas and Temperaments&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Commas and Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Commas and Temperaments&lt;/h1&gt;
  72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and the kleisma, 15625/15552, the ampersand, 34171875/33554432 = |-25 7 6&amp;gt;, the graviton, 129140163/128000000 = |-13 17 -6&amp;gt;, the and the ennealimma, 7629394531250/7625597484987 = |1 -27 18&amp;gt; in the 5-limit. The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000. 72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800. For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655. 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.&lt;br /&gt;
  &lt;br /&gt;
72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and &lt;br /&gt;
the kleisma, 15625/15552, &lt;br /&gt;
the ampersand, 34171875/33554432 = |-25 7 6&amp;gt;, &lt;br /&gt;
the graviton, 129140163/128000000 = |-13 17 -6&amp;gt;, and &lt;br /&gt;
the ennealimma, 7629394531250/7625597484987 = |1 -27 18&amp;gt; in the 5-limit. &lt;br /&gt;
&lt;br /&gt;
The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.&lt;br /&gt;
&lt;br /&gt;
72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800. &lt;br /&gt;
&lt;br /&gt;
For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Harmonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harmonic Scale&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Harmonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harmonic Scale&lt;/h1&gt;
Line 571: Line 599:
         &lt;td&gt;81/80&lt;br /&gt;
         &lt;td&gt;81/80&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;^1&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;^1 (up unison)&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D^&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D^&lt;br /&gt;
Line 583: Line 611:
         &lt;td&gt;45/44&lt;br /&gt;
         &lt;td&gt;45/44&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;^^1&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;^^1(double-up unison)&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D^^&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D^^&lt;br /&gt;
Line 619: Line 647:
         &lt;td&gt;21/20&lt;br /&gt;
         &lt;td&gt;21/20&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;vm2&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;vm2 (downminor 2nd)&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Ebv&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Ebv&lt;br /&gt;
Line 643: Line 671:
         &lt;td&gt;15/14&lt;br /&gt;
         &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;^m2&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;^m2 (upminor 2nd)&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Eb^&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Eb^&lt;br /&gt;
Line 667: Line 695:
         &lt;td&gt;12/11&lt;br /&gt;
         &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;~2&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;~2 (mid 2nd)&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
Line 679: Line 707:
         &lt;td&gt;11/10&lt;br /&gt;
         &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;^~2&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;^~2 (upmid 2nd)&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Evv&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Evv&lt;br /&gt;
Line 955: Line 983:
         &lt;td&gt;11/8&lt;br /&gt;
         &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4&lt;span style="font-size: 90%; vertical-align: super;"&gt; &lt;/span&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;G^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;G^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
Line 1,027: Line 1,055:
         &lt;td&gt;16/11&lt;br /&gt;
         &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;span style="font-size: 90%; vertical-align: super;"&gt; &lt;/span&gt;v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Av&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Av&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;