4L 3s
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Original Wikitext content:
4L 3s refers to the structure of moment of symmetry scales with generators ranging from 1\4edo (one degree of [[4edo]], 300¢) to 2\7edo (two degrees of [[7edo]], or approx. 342.857¢). The spectrum looks like this: ||||||||||||||~ Generator ||~ Tetrachord ||~ g in cents ||~ 2g ||~ 3g ||~ 4g ||~ Comments || || 1\4 || || || || || || || 1 0 1 || 300.000 || 600.000 || 900.000 || 0.000 ||= || || || || || || || || 8\31 || 7 1 7 || 309.677 || 619.355 || 929.023 || 38.71 ||= Myna is around here || || || || || || || 7\27 || || 6 1 6 || 311.111 || 622.222 || 933.333 || 44.444 ||= || || || || || || 6\23 || || || 5 1 5 || 313.043 || 626.087 || 939.13 || 52.174 ||= || || || || || 5\19 || || || || 4 1 4 || 315.789 || 631.579 || 947.368 || 63.158 ||= L/s = 4 || || || || || || 9\34 || || || 7 2 7 || 317.647 || 634.294 || 951.941 || 70.588 ||= Hanson/Keemun is around here || || || || || || || || || pi 1 pi || 319.272 || 638.545 || 957.817 || 77.089 ||= <span style="display: block; text-align: center;">L/s = pi</span> || || || || 4\15 || || || || || 3 1 3 || 320.000 || 640.000 || 960.000 || 80.000 ||= L/s = 3 || || || || || || || || || e 1 e || 321.6245 || 641.249 || 964.874 || 86.498 || <span style="display: block; text-align: center;">L/s = e</span> || || || || || || 11\41 || || || 8 3 8 || 321.951 || 643.902 || 965.854 || 87.805 || || || || || || || || || 29\108 || 21 8 21 || 322.222 || 644.444 || 966.667 || 88.889 || || || || || || || || 18\67 || || 13 5 13 || 322.388 || 644.776 || 967.364 || 89.522 || || || || || || 7\26 || || || || 5 2 5 || 323.077 || 646.154 || 969.231 || 92.308 ||= Orgone is around here || || || 3\11 || || || || || || 2 1 2 || 327.273 || 654.545 || 981.818 || 109.091 ||= Boundary of propriety (generators larger than this are proper) || || || || || || || || || <span style="background-color: #ffffff;">√3 1 √3</span> || 330.217 || 660.434 || 990.651 || 120.868 || || || || || || 8\29 || || || || 5 3 5 || 331.034 || 662.069 || 993.013 || 124.138 ||= || || || || || || || 21\76 || || 13 8 13 || 331.579 || 663.158 || 994.739 || 126.316 ||= || || || || || || || || 34\123 || 21 13 21 || 331.707 || 663.415 || 995.122 || 126.829 ||= Unnamed golden temperament || || || || || || 13\47 || || || 8 5 8 || 331.915 || 663.83 || 995.745 || 127.66 ||= || || || || || || || || || pi 2 pi || 332.3165 || 664.633 || 996.9495 || 129.266 || || || || || 5\18 || || || || || 3 2 3 || 333.333 || 666.667 || 1000.000 || 133.333 ||= Optimum rank range (L/s=3/2) || || || || || 7\25 || || || || 4 3 4 || 336.000 || 672.000 || 1008.000 || 144.000 ||= || || || || || || 9\32 || || || 5 4 5 || 337.5 || 675 || 1012.5 || 150 ||= Sixix || || || || || || || 11\39 || || 6 5 6 || 338.462 || 676.923 || 1015.385 || 153.846 ||= Sixix || || || || || || || || 13\46 || 7 6 7 || 339.130 || 678.261 || 1017.391 || 156.522 ||= (17/14)^3=9/5 || || 2\7 || || || || || || || 1 1 1 || 342.857 || 685.714 || 1028.571 || 171.429 ||= || There are two notable harmonic entropy minima: [[Kleismic family|hanson/keemun]], in which the generator is 6/5 and 6 of them make a 3/1, and [[Starling temperaments|myna]], in which the generator is also 6/5 but now **10** of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).
Original HTML content:
<html><head><title>4L 3s</title></head><body>4L 3s refers to the structure of moment of symmetry scales with generators ranging from 1\4edo (one degree of <a class="wiki_link" href="/4edo">4edo</a>, 300¢) to 2\7edo (two degrees of <a class="wiki_link" href="/7edo">7edo</a>, or approx. 342.857¢). The spectrum looks like this:<br />
<table class="wiki_table">
<tr>
<th colspan="7">Generator<br />
</th>
<th>Tetrachord<br />
</th>
<th>g in cents<br />
</th>
<th>2g<br />
</th>
<th>3g<br />
</th>
<th>4g<br />
</th>
<th>Comments<br />
</th>
</tr>
<tr>
<td>1\4<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>1 0 1<br />
</td>
<td>300.000<br />
</td>
<td>600.000<br />
</td>
<td>900.000<br />
</td>
<td>0.000<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>8\31<br />
</td>
<td>7 1 7<br />
</td>
<td>309.677<br />
</td>
<td>619.355<br />
</td>
<td>929.023<br />
</td>
<td>38.71<br />
</td>
<td style="text-align: center;">Myna is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\27<br />
</td>
<td><br />
</td>
<td>6 1 6<br />
</td>
<td>311.111<br />
</td>
<td>622.222<br />
</td>
<td>933.333<br />
</td>
<td>44.444<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>6\23<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5 1 5<br />
</td>
<td>313.043<br />
</td>
<td>626.087<br />
</td>
<td>939.13<br />
</td>
<td>52.174<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5\19<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>4 1 4<br />
</td>
<td>315.789<br />
</td>
<td>631.579<br />
</td>
<td>947.368<br />
</td>
<td>63.158<br />
</td>
<td style="text-align: center;">L/s = 4<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>9\34<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7 2 7<br />
</td>
<td>317.647<br />
</td>
<td>634.294<br />
</td>
<td>951.941<br />
</td>
<td>70.588<br />
</td>
<td style="text-align: center;">Hanson/Keemun is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>pi 1 pi<br />
</td>
<td>319.272<br />
</td>
<td>638.545<br />
</td>
<td>957.817<br />
</td>
<td>77.089<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = pi</span><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>4\15<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>3 1 3<br />
</td>
<td>320.000<br />
</td>
<td>640.000<br />
</td>
<td>960.000<br />
</td>
<td>80.000<br />
</td>
<td style="text-align: center;">L/s = 3<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>e 1 e<br />
</td>
<td>321.6245<br />
</td>
<td>641.249<br />
</td>
<td>964.874<br />
</td>
<td>86.498<br />
</td>
<td><span style="display: block; text-align: center;">L/s = e</span><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>11\41<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>8 3 8<br />
</td>
<td>321.951<br />
</td>
<td>643.902<br />
</td>
<td>965.854<br />
</td>
<td>87.805<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>29\108<br />
</td>
<td>21 8 21<br />
</td>
<td>322.222<br />
</td>
<td>644.444<br />
</td>
<td>966.667<br />
</td>
<td>88.889<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>18\67<br />
</td>
<td><br />
</td>
<td>13 5 13<br />
</td>
<td>322.388<br />
</td>
<td>644.776<br />
</td>
<td>967.364<br />
</td>
<td>89.522<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\26<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5 2 5<br />
</td>
<td>323.077<br />
</td>
<td>646.154<br />
</td>
<td>969.231<br />
</td>
<td>92.308<br />
</td>
<td style="text-align: center;">Orgone is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>3\11<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>2 1 2<br />
</td>
<td>327.273<br />
</td>
<td>654.545<br />
</td>
<td>981.818<br />
</td>
<td>109.091<br />
</td>
<td style="text-align: center;">Boundary of propriety (generators<br />
larger than this are proper)<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><span style="background-color: #ffffff;">√3 1 √3</span><br />
</td>
<td>330.217<br />
</td>
<td>660.434<br />
</td>
<td>990.651<br />
</td>
<td>120.868<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>8\29<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5 3 5<br />
</td>
<td>331.034<br />
</td>
<td>662.069<br />
</td>
<td>993.013<br />
</td>
<td>124.138<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>21\76<br />
</td>
<td><br />
</td>
<td>13 8 13<br />
</td>
<td>331.579<br />
</td>
<td>663.158<br />
</td>
<td>994.739<br />
</td>
<td>126.316<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>34\123<br />
</td>
<td>21 13 21<br />
</td>
<td>331.707<br />
</td>
<td>663.415<br />
</td>
<td>995.122<br />
</td>
<td>126.829<br />
</td>
<td style="text-align: center;">Unnamed golden temperament<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>13\47<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>8 5 8<br />
</td>
<td>331.915<br />
</td>
<td>663.83<br />
</td>
<td>995.745<br />
</td>
<td>127.66<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>pi 2 pi<br />
</td>
<td>332.3165<br />
</td>
<td>664.633<br />
</td>
<td>996.9495<br />
</td>
<td>129.266<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>5\18<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>3 2 3<br />
</td>
<td>333.333<br />
</td>
<td>666.667<br />
</td>
<td>1000.000<br />
</td>
<td>133.333<br />
</td>
<td style="text-align: center;">Optimum rank range (L/s=3/2)<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\25<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>4 3 4<br />
</td>
<td>336.000<br />
</td>
<td>672.000<br />
</td>
<td>1008.000<br />
</td>
<td>144.000<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>9\32<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5 4 5<br />
</td>
<td>337.5<br />
</td>
<td>675<br />
</td>
<td>1012.5<br />
</td>
<td>150<br />
</td>
<td style="text-align: center;">Sixix<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>11\39<br />
</td>
<td><br />
</td>
<td>6 5 6<br />
</td>
<td>338.462<br />
</td>
<td>676.923<br />
</td>
<td>1015.385<br />
</td>
<td>153.846<br />
</td>
<td style="text-align: center;">Sixix<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>13\46<br />
</td>
<td>7 6 7<br />
</td>
<td>339.130<br />
</td>
<td>678.261<br />
</td>
<td>1017.391<br />
</td>
<td>156.522<br />
</td>
<td style="text-align: center;">(17/14)^3=9/5<br />
</td>
</tr>
<tr>
<td>2\7<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>1 1 1<br />
</td>
<td>342.857<br />
</td>
<td>685.714<br />
</td>
<td>1028.571<br />
</td>
<td>171.429<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
There are two notable harmonic entropy minima: <a class="wiki_link" href="/Kleismic%20family">hanson/keemun</a>, in which the generator is 6/5 and 6 of them make a 3/1, and <a class="wiki_link" href="/Starling%20temperaments">myna</a>, in which the generator is also 6/5 but now <strong>10</strong> of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).</body></html>