5L 3s
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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 || || || || || || || || <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 ||= || || 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.
Original HTML content:
<html><head><title>5L 3s</title></head><body>5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of <a class="wiki_link" href="/5edo">5edo</a> = 480¢) to 3\8 (three degrees of <a class="wiki_link" href="/8edo">8edo</a> = 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:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th><br />
</th>
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<th><br />
</th>
<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 />
</th>
<th><span style="display: block; text-align: center;">Comments</span><br />
</th>
</tr>
<tr>
<td>2\5<br />
</td>
<td><br />
</td>
<td><br />
</td>
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</td>
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<td><span style="text-align: center;">1 0 1</span><br />
</td>
<td>480.000<br />
</td>
<td>960.000<br />
</td>
<td>240.00<br />
</td>
<td><span style="line-height: 15.6000003814697px;">720.000</span><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>21\53<br />
</td>
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</td>
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</td>
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</td>
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</td>
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</td>
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</td>
<td><span style="text-align: center;">10 1 10</span><br />
</td>
<td>475.472<br />
</td>
<td>950.943<br />
</td>
<td>226.415<br />
</td>
<td>701.887<br />
</td>
<td style="text-align: center;"><span style="text-align: center;">Vulture/Buzzard is around here</span><br />
</td>
</tr>
<tr>
<td>19\48<br />
</td>
<td><br />
</td>
<td><br />
</td>
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</td>
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</td>
<td><br />
</td>
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</td>
<td><br />
</td>
<td><span style="text-align: center;">9 1 9</span><br />
</td>
<td>475<br />
</td>
<td>950<br />
</td>
<td>225<br />
</td>
<td>700<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>17\43<br />
</td>
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</td>
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</td>
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</td>
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</td>
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</td>
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</td>
<td><br />
</td>
<td><span style="text-align: center;">8 1 8</span><br />
</td>
<td>474.419<br />
</td>
<td>948.837<br />
</td>
<td>223.256<br />
</td>
<td>697.674<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>15\38<br />
</td>
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</td>
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</td>
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</td>
<td><br />
</td>
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</td>
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</td>
<td><span style="text-align: center;">7 1 7</span><br />
</td>
<td>473.684<br />
</td>
<td>947.368<br />
</td>
<td>221.053<br />
</td>
<td>694.737<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>13\33<br />
</td>
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</td>
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</td>
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</td>
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</td>
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</td>
<td><span style="text-align: center;">6 1 6</span><br />
</td>
<td>472.727<br />
</td>
<td>945.455<br />
</td>
<td>218.181<br />
</td>
<td>690.909<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>11\28<br />
</td>
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</td>
<td><br />
</td>
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</td>
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</td>
<td><br />
</td>
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</td>
<td><br />
</td>
<td><span style="text-align: center;">5 1 5</span><br />
</td>
<td>471.429<br />
</td>
<td>942.857<br />
</td>
<td>214.286<br />
</td>
<td>685.714<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>9\23<br />
</td>
<td><br />
</td>
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</td>
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</td>
<td><br />
</td>
<td><br />
</td>
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</td>
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</td>
<td><span style="text-align: center;">4 1 4</span><br />
</td>
<td>469.565<br />
</td>
<td>939.130<br />
</td>
<td>208.696<br />
</td>
<td>678.261<br />
</td>
<td style="text-align: center;"><span style="text-align: center;">L/s = 4</span><br />
</td>
</tr>
<tr>
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</td>
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</td>
<td><br />
</td>
<td><br />
</td>
<td>3.03 1 3.03<br />
</td>
<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>
</tr>
<tr>
<td>7\18<br />
</td>
<td><br />
</td>
<td><br />
</td>
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</td>
<td><br />
</td>
<td><br />
</td>
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</td>
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</td>
<td><span style="text-align: center;">3 1 3</span><br />
</td>
<td>466.667<br />
</td>
<td>933.333<br />
</td>
<td>200.000<br />
</td>
<td>666.667<br />
</td>
<td style="text-align: center;"><span style="text-align: center;">L/s = 3</span><br />
</td>
</tr>
<tr>
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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>
<td>12\31<br />
</td>
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</td>
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<td><span style="text-align: center;">5 2 5</span><br />
</td>
<td>464.516<br />
</td>
<td>929.032<br />
</td>
<td>193.549<br />
</td>
<td>658.065<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>5\13<br />
</td>
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<td><span style="text-align: center;">2 1 2</span><br />
</td>
<td>461.538<br />
</td>
<td>923.077<br />
</td>
<td>184.615<br />
</td>
<td>646.154<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>13\34<br />
</td>
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</td>
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<td><span style="text-align: center;">5 3 5</span><br />
</td>
<td>458.824<br />
</td>
<td>917.647<br />
</td>
<td>176.471<br />
</td>
<td>635.294<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>34\89<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><span style="text-align: center;">13 8 13</span><br />
</td>
<td>458.427<br />
</td>
<td>916.854<br />
</td>
<td>175.281<br />
</td>
<td>633.708<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
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</td>
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</td>
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</td>
<td>89\233<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><span style="text-align: center;">34 21 34</span><br />
</td>
<td><span style="line-height: 15.6000003814697px;">458.369</span><br />
</td>
<td><span style="text-align: center;">916.738</span><br />
</td>
<td>175.107<br />
</td>
<td><span style="line-height: 15.6000003814697px;">633.473</span><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
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</td>
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</td>
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</td>
<td><br />
</td>
<td>233\610<br />
</td>
<td><span style="text-align: center;">89 55 89</span><br />
</td>
<td>458.361<br />
</td>
<td>916.721<br />
</td>
<td>175.082<br />
</td>
<td>633.443<br />
</td>
<td style="text-align: center;"><span style="text-align: center;">Golden father</span><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
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</td>
<td><br />
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<td>144\377<br />
</td>
<td><br />
</td>
<td><span style="text-align: center;">55 34 55</span><br />
</td>
<td>458.355<br />
</td>
<td>916.711<br />
</td>
<td>175.066<br />
</td>
<td>633.422<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>55\144<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><span style="text-align: center;">21 13 21</span><br />
</td>
<td>458.333<br />
</td>
<td>916.666<br />
</td>
<td>175<br />
</td>
<td>633.333<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>21\55<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><span style="text-align: center;">8 5 8</span><br />
</td>
<td>458.182<br />
</td>
<td>916.364<br />
</td>
<td>174.545<br />
</td>
<td>632.727<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>8\21<br />
</td>
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</td>
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</td>
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</td>
<td><br />
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<td><span style="text-align: center;">3 2 3</span><br />
</td>
<td>457.143<br />
</td>
<td>914.286<br />
</td>
<td>171.429<br />
</td>
<td>628.571<br />
</td>
<td style="text-align: center;"><span style="text-align: center;">Optimum rank range (L/s=3/2) father</span><br />
</td>
</tr>
<tr>
<td>11\29<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><span style="text-align: center;">4 3 4</span><br />
</td>
<td>455.172<br />
</td>
<td>910.345<br />
</td>
<td>165.517<br />
</td>
<td>620.690<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>3\8<br />
</td>
<td><br />
</td>
<td><br />
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<td><br />
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<td><span style="text-align: center;">1 1 1</span><br />
</td>
<td>450.000<br />
</td>
<td>900.000<br />
</td>
<td>150.000<br />
</td>
<td>600.000<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
The only notable harmonic entropy minimum is Vulture/<a class="wiki_link" href="/Hemifamity%20temperaments">Buzzard</a>, 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.</body></html>