72edo: Difference between revisions
Wikispaces>TallKite **Imported revision 602866126 - Original comment: ** |
Wikispaces>TallKite **Imported revision 602866264 - 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:TallKite|TallKite]] and made on <tt>2016-12-28 | : 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> | : 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> | ||
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=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>, the graviton, 129140163/128000000 = |-13 17 -6>, | |||
72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and | |||
the kleisma, 15625/15552, | |||
the ampersand, 34171875/33554432 = |-25 7 6>, | |||
the graviton, 129140163/128000000 = |-13 17 -6>, and | |||
the ennealimma, 7629394531250/7625597484987 = |1 -27 18> 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= | ||
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|| 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 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2 || D^<span style="font-size: 90%; vertical-align: super;">3</span>, Ebv<span style="font-size: 90%; vertical-align: super;">3</span> || | || 3 || 50 || 33/32 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2 || D^<span style="font-size: 90%; vertical-align: super;">3</span>, Ebv<span style="font-size: 90%; vertical-align: super;">3</span> || | ||
|| 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<span style="font-size: 90%; vertical-align: super;">3</span> || | || 9 || 150 || 12/11 ||= ~2 (mid 2nd) ||= Ev<span style="font-size: 90%; vertical-align: super;">3</span> || | ||
|| 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 || | ||
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|| 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 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>4 | || 33 || 550 || 11/8 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>4 ||= G^<span style="font-size: 90%; vertical-align: super;">3</span> || | ||
|| 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 || | ||
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|| 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 ||= | || 39 || 650 || 16/11 ||= v<span style="font-size: 90%; vertical-align: super;">3</span>5 ||= Av<span style="font-size: 90%; vertical-align: super;">3</span> || | ||
|| 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 || | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Commas and Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Commas and Temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Commas and Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Commas and Temperaments</h1> | ||
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;, | <br /> | ||
72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and <br /> | |||
the kleisma, 15625/15552, <br /> | |||
the ampersand, 34171875/33554432 = |-25 7 6&gt;, <br /> | |||
the graviton, 129140163/128000000 = |-13 17 -6&gt;, and <br /> | |||
the ennealimma, 7629394531250/7625597484987 = |1 -27 18&gt; in the 5-limit. <br /> | |||
<br /> | |||
The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.<br /> | |||
<br /> | |||
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. <br /> | |||
<br /> | |||
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.<br /> | |||
<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 /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmonic Scale</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmonic Scale</h1> | ||
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<td>81/80<br /> | <td>81/80<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^1<br /> | <td style="text-align: center;">^1 (up unison)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">D^<br /> | <td style="text-align: center;">D^<br /> | ||
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<td>45/44<br /> | <td>45/44<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^^1<br /> | <td style="text-align: center;">^^1(double-up unison)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">D^^<br /> | <td style="text-align: center;">D^^<br /> | ||
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<td>21/20<br /> | <td>21/20<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">vm2<br /> | <td style="text-align: center;">vm2 (downminor 2nd)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Ebv<br /> | <td style="text-align: center;">Ebv<br /> | ||
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<td>15/14<br /> | <td>15/14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^m2<br /> | <td style="text-align: center;">^m2 (upminor 2nd)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Eb^<br /> | <td style="text-align: center;">Eb^<br /> | ||
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<td>12/11<br /> | <td>12/11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">~2<br /> | <td style="text-align: center;">~2 (mid 2nd)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Ev<span style="font-size: 90%; vertical-align: super;">3</span><br /> | <td style="text-align: center;">Ev<span style="font-size: 90%; vertical-align: super;">3</span><br /> | ||
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<td>11/10<br /> | <td>11/10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^~2<br /> | <td style="text-align: center;">^~2 (upmid 2nd)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Evv<br /> | <td style="text-align: center;">Evv<br /> | ||
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<td>11/8<br /> | <td>11/8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>4 | <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">G^<span style="font-size: 90%; vertical-align: super;">3</span><br /> | <td style="text-align: center;">G^<span style="font-size: 90%; vertical-align: super;">3</span><br /> | ||
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<td>16/11<br /> | <td>16/11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;" | <td style="text-align: center;">v<span style="font-size: 90%; vertical-align: super;">3</span>5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Av<span style="font-size: 90%; vertical-align: super;">3</span><br /> | <td style="text-align: center;">Av<span style="font-size: 90%; vertical-align: super;">3</span><br /> | ||