5L 3s: Difference between revisions
Wikispaces>keenanpepper **Imported revision 399961360 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 540641396 - 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:JosephRuhf|JosephRuhf]] and made on <tt>2015-02-11 14:14:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>540641396</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<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">5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of [[5edo]] = 480¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The spectrum looks like this: | <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">5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of [[5edo]] = 480¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The spectrum looks like this: | ||
|||||||||||||| | ||~ ||~ ||~ ||~ ||~ ||~ ||~ ||~ ||~ <span style="display: block; text-align: center;">tetrachord</span> ||~ <span style="display: block; text-align: center;">g in cents</span> ||~ <span style="display: block; text-align: center;">2g</span> ||~ <span style="display: block; text-align: center;">3g</span> ||~ <span style="display: block; text-align: center;">4g</span> ||~ <span style="display: block; text-align: center;">Comments</span> || | ||
|| 2\5 || || || || || || || || | || 2\5 || || || || || || || || <span style="text-align: center;">1 0 1</span> || 480.000 || 960.000 || 240.00 || <span style="line-height: 15.6000003814697px;">720.000</span> ||= || | ||
|| | || 21\53 || || || || || || || || <span style="text-align: center;">10 1 10</span> || 475.472 || 950.943 || 226.415 || 701.887 ||= <span style="text-align: center;">Vulture/Buzzard is around here</span> || | ||
|| | || 19\48 || || || || || || || || <span style="text-align: center;">9 1 9</span> || 475 || 950 || 225 || 700 ||= || | ||
|| | || 17\43 || || || || || || || || <span style="text-align: center;">8 1 8</span> || 474.419 || 948.837 || 223.256 || 697.674 ||= || | ||
|| | || 15\38 || || || || || || || || <span style="text-align: center;">7 1 7</span> || 473.684 || 947.368 || 221.053 || 694.737 ||= || | ||
|| | || 13\33 || || || || || || || || <span style="text-align: center;">6 1 6</span> || 472.727 || 945.455 || 218.181 || 690.909 ||= || | ||
|| | || 11\28 || || || || || || || || <span style="text-align: center;">5 1 5</span> || 471.429 || 942.857 || 214.286 || 685.714 ||= || | ||
|| | || 9\23 || || || || || || || || <span style="text-align: center;">4 1 4</span> || 469.565 || 939.130 || 208.696 || 678.261 ||= <span style="text-align: center;">L/s = 4</span> || | ||
|| | || || || || || || || || || 3.03 1 3.03 || 466.769 || 933.538 || 200.307 || 667.076 ||= <span style="display: block; text-align: center;">L/s = 3*2^(1/75)</span> || | ||
|| | || 7\18 || || || || || || || || <span style="text-align: center;">3 1 3</span> || 466.667 || 933.333 || 200.000 || 666.667 ||= <span style="text-align: center;">L/s = 3</span> || | ||
|| || || || || || || || || 2.97 1 2.97 || 466.564 || 933.127 || 199.691 || 666.255 ||= <span style="display: block; text-align: center;">L/s = 3/2^(1/75)</span> || | |||
|| || | || || 12\31 || || || || || || || <span style="text-align: center;">5 2 5</span> || 464.516 || 929.032 || 193.549 || 658.065 ||= || | ||
|| | || 5\13 || || || || || || || || <span style="text-align: center;">2 1 2</span> || 461.538 || 923.077 || 184.615 || 646.154 ||= || | ||
|| || | || || 13\34 || || || || || || || <span style="text-align: center;">5 3 5</span> || 458.824 || 917.647 || 176.471 || 635.294 ||= || | ||
|| || || || | || || || || 34\89 || || || || || <span style="text-align: center;">13 8 13</span> || 458.427 || 916.854 || 175.281 || 633.708 ||= || | ||
|| || || || || || | || || || || || || 89\233 || || || <span style="text-align: center;">34 21 34</span> || <span style="line-height: 15.6000003814697px;">458.369</span> || <span style="text-align: center;">916.738</span> || 175.107 || <span style="line-height: 15.6000003814697px;">633.473</span> ||= || | ||
|| || || || || || | || || || || || || || || 233\610 || <span style="text-align: center;">89 55 89</span> || 458.361 || 916.721 || 175.082 || 633.443 ||= <span style="text-align: center;">Golden father</span> || | ||
|| || || || | || || || || || || || 144\377 || || <span style="text-align: center;">55 34 55</span> || 458.355 || 916.711 || 175.066 || 633.422 ||= || | ||
|| || || | || || || || || 55\144 || || || || <span style="text-align: center;">21 13 21</span> || 458.333 || 916.666 || 175 || 633.333 ||= || | ||
|| || || 21\55 || || || || || || <span style="text-align: center;">8 5 8</span> || 458.182 || 916.364 || 174.545 || 632.727 ||= || | |||
|| 3\8 || || || || || || || || | || 8\21 || || || || || || || || <span style="text-align: center;">3 2 3</span> || 457.143 || 914.286 || 171.429 || 628.571 ||= <span style="text-align: center;">Optimum rank range (L/s=3/2) father</span> || | ||
|| 11\29 || || || || || || || || <span style="text-align: center;">4 3 4</span> || 455.172 || 910.345 || 165.517 || 620.690 ||= || | |||
|| 3\8 || || || || || || || || <span style="text-align: center;">1 1 1</span> || 450.000 || 900.000 || 150.000 || 600.000 ||= || | |||
The only notable harmonic entropy minimum is Vulture/[[Hemifamity temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region is a kind of wasteland in terms of harmonious MOSes.</pre></div> | The only notable harmonic entropy minimum is Vulture/[[Hemifamity temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region is a kind of wasteland in terms of harmonious MOSes.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
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<th>Comments<br /> | <th><br /> | ||
</th> | |||
<th><br /> | |||
</th> | |||
<th><span style="display: block; text-align: center;">tetrachord</span><br /> | |||
</th> | |||
<th><span style="display: block; text-align: center;">g in cents</span><br /> | |||
</th> | |||
<th><span style="display: block; text-align: center;">2g</span><br /> | |||
</th> | |||
<th><span style="display: block; text-align: center;">3g</span><br /> | |||
</th> | |||
<th><span style="display: block; text-align: center;">4g</span><br /> | |||
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<th><span style="display: block; text-align: center;">Comments</span><br /> | |||
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<td><br /> | <td><br /> | ||
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<td>< | <td><span style="text-align: center;">1 0 1</span><br /> | ||
</td> | </td> | ||
<td>480.000<br /> | <td>480.000<br /> | ||
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<td>960.000<br /> | <td>960.000<br /> | ||
</td> | </td> | ||
<td>240. | <td>240.00<br /> | ||
</td> | </td> | ||
<td>720.000<br /> | <td><span style="line-height: 15.6000003814697px;">720.000</span><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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</tr> | </tr> | ||
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<td><br /> | <td>21\53<br /> | ||
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<td>< | <td><span style="text-align: center;">10 1 10</span><br /> | ||
</td> | </td> | ||
<td>475.472<br /> | <td>475.472<br /> | ||
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<td>701.887<br /> | <td>701.887<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Vulture/Buzzard is around here<br /> | <td style="text-align: center;"><span style="text-align: center;">Vulture/Buzzard is around here</span><br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td>19\48<br /> | ||
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<td> | <td><span style="text-align: center;">9 1 9</span><br /> | ||
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<td>475<br /> | <td>475<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>17\43<br /> | ||
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<td style="text-align: center;">8 1 8 | <td><span style="text-align: center;">8 1 8</span><br /> | ||
</td> | </td> | ||
<td>474.419<br /> | <td>474.419<br /> | ||
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</tr> | </tr> | ||
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<td><br /> | <td>15\38<br /> | ||
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<td>< | <td><span style="text-align: center;">7 1 7</span><br /> | ||
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<td>473.684<br /> | <td>473.684<br /> | ||
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</tr> | </tr> | ||
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<td><br /> | <td>13\33<br /> | ||
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<td><br /> | <td><br /> | ||
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<td>< | <td><span style="text-align: center;">6 1 6</span><br /> | ||
</td> | </td> | ||
<td>472.727<br /> | <td>472.727<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>11\28<br /> | ||
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<td style="text-align: center;">5 1 5 | <td><span style="text-align: center;">5 1 5</span><br /> | ||
</td> | </td> | ||
<td>471.429<br /> | <td>471.429<br /> | ||
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</tr> | </tr> | ||
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<td><br /> | <td>9\23<br /> | ||
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<td>< | <td><span style="text-align: center;">4 1 4</span><br /> | ||
</td> | </td> | ||
<td>469.565<br /> | <td>469.565<br /> | ||
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<td>678.261<br /> | <td>678.261<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">L/s = 4<br /> | <td style="text-align: center;"><span style="text-align: center;">L/s = 4</span><br /> | ||
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<td>3.03 1 3.03<br /> | |||
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<td>466.769<br /> | |||
</td> | |||
<td>933.538<br /> | |||
</td> | |||
<td>200.307<br /> | |||
</td> | |||
<td>667.076<br /> | |||
</td> | |||
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = 3*2^(1/75)</span><br /> | |||
</td> | |||
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<td>7\18<br /> | <td>7\18<br /> | ||
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<td><br /> | <td><br /> | ||
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<td style="text-align: center;">3 1 3 | <td><span style="text-align: center;">3 1 3</span><br /> | ||
</td> | </td> | ||
<td>466.667<br /> | <td>466.667<br /> | ||
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<td>666.667<br /> | <td>666.667<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">L/s = 3<br /> | <td style="text-align: center;"><span style="text-align: center;">L/s = 3</span><br /> | ||
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<td>2.97 1 2.97<br /> | |||
</td> | |||
<td>466.564<br /> | |||
</td> | |||
<td>933.127<br /> | |||
</td> | |||
<td>199.691<br /> | |||
</td> | |||
<td>666.255<br /> | |||
</td> | |||
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = 3/2^(1/75)</span><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td><br /> | <td><br /> | ||
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<td><br /> | <td><br /> | ||
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<td style="text-align: center;">5 2 5 | <td><span style="text-align: center;">5 2 5</span><br /> | ||
</td> | </td> | ||
<td>464.516<br /> | <td>464.516<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td>5\13<br /> | <td>5\13<br /> | ||
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<td><br /> | <td><br /> | ||
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<td>< | <td><span style="text-align: center;">2 1 2</span><br /> | ||
</td> | </td> | ||
<td>461.538<br /> | <td>461.538<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
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<td><br /> | <td><br /> | ||
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<td style="text-align: center;">5 3 5 | <td><span style="text-align: center;">5 3 5</span><br /> | ||
</td> | </td> | ||
<td>458.824<br /> | <td>458.824<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
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| Line 480: | Line 504: | ||
<td><br /> | <td><br /> | ||
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<td style="text-align: center;">13 8 | <td><span style="text-align: center;">13 8 13</span><br /> | ||
</td> | </td> | ||
<td>458.427<br /> | <td>458.427<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
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<td style="text-align: center;">34 21 | <td><span style="text-align: center;">34 21 34</span><br /> | ||
</td> | </td> | ||
<td>458.369<br /> | <td><span style="line-height: 15.6000003814697px;">458.369</span><br /> | ||
</td> | </td> | ||
<td>916.738<br /> | <td><span style="text-align: center;">916.738</span><br /> | ||
</td> | </td> | ||
<td>175.107<br /> | <td>175.107<br /> | ||
</td> | </td> | ||
<td>633.473<br /> | <td><span style="line-height: 15.6000003814697px;">633.473</span><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
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<td>233\610<br /> | <td>233\610<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">89 55 | <td><span style="text-align: center;">89 55 89</span><br /> | ||
</td> | </td> | ||
<td>458.361<br /> | <td>458.361<br /> | ||
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<td>633.443<br /> | <td>633.443<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Golden father<br /> | <td style="text-align: center;"><span style="text-align: center;">Golden father</span><br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
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<td style="text-align: center;">55 34 | <td><span style="text-align: center;">55 34 55</span><br /> | ||
</td> | </td> | ||
<td>458.355<br /> | <td>458.355<br /> | ||
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<td>633.422<br /> | <td>633.422<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
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<td><br /> | <td><br /> | ||
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<td style="text-align: center;">21 13 | <td><span style="text-align: center;">21 13 21</span><br /> | ||
</td> | </td> | ||
<td>458.333<br /> | <td>458.333<br /> | ||
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<td>633.333<br /> | <td>633.333<br /> | ||
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<td><br /> | <td style="text-align: center;"><br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
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<td><br /> | <td><br /> | ||
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<td style="text-align: center;">8 5 8 | <td><span style="text-align: center;">8 5 8</span><br /> | ||
</td> | </td> | ||
<td>458.182<br /> | <td>458.182<br /> | ||
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</tr> | </tr> | ||
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<td>8\21<br /> | <td>8\21<br /> | ||
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<td style="text-align: center;">3 2 3 | <td><span style="text-align: center;">3 2 3</span><br /> | ||
</td> | </td> | ||
<td>457.143<br /> | <td>457.143<br /> | ||
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<td>628.571<br /> | <td>628.571<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Optimum rank range (L/s=3/2) father<br /> | <td style="text-align: center;"><span style="text-align: center;">Optimum rank range (L/s=3/2) father</span><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td>11\29<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
| Line 716: | Line 714: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td>< | <td><span style="text-align: center;">4 3 4</span><br /> | ||
</td> | </td> | ||
<td>455.172<br /> | <td>455.172<br /> | ||
| Line 748: | Line 744: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td>< | <td><span style="text-align: center;">1 1 1</span><br /> | ||
</td> | </td> | ||
<td>450.000<br /> | <td>450.000<br /> | ||