5L 2s
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=5L 2s - "diatonic"= One way of distinguishing the "diatonic" scale is by considering it a [[MOSScales|moment of symmetry]] scale produced by a chain of "fifths". This will include [[12edo]]'s diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone". It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music. ==substituting step sizes== The 5L 2s MOS scale has this generalized form. L L s L L L s Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice. 2 2 1 2 2 2 1 When L=3, s=1, you have [[17edo]]: 3 3 1 3 3 3 1 When L=3, s=2, you have [[19edo]]: 3 3 2 3 3 3 2 When L=4, s=1, you have [[22edo]]: 4 4 1 4 4 4 1 When L=4, s=3, you have [[26edo]]: 4 4 3 4 4 4 3 When L=5, s=1, you have [[27edo]]: 5 5 1 5 5 5 1 When L=5, s=2, you have [[29edo]]: 5 5 2 5 5 5 2 When L=5, s=3, you have [[31edo]]: 5 5 3 5 5 5 3 When L=5, s=4, you have [[33edo]]: 5 5 4 5 5 5 4 So you have scales where L and s are nearly equal, which approach [[7edo]]: 1 1 1 1 1 1 1 And you have scales where s becomes so small it approaches zero, which would give us [[5edo]]: 1 1 0 1 1 1 0 or 1 1 1 1 1 ==a continuum of temperaments== So if 3\7 (three degrees of 7edo) is at one extreme and 2\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators. Thus, between 3\7 and 2\5 you have (3+2)\(7+5) = 5\12, five degrees of 12edo: || 3\7 || || || || 5\12 || || 2\5 || || If we carry this freshman-summing out a little further, new, larger [[edo]]s pop up in our continuum. ||||||||||||~ generator ||~ ||~ in cents ||~ tetrachord ||~ ||~ ||~ ||~ ||~ comments || ||< 3\7 ||||< ||< ||< ||= || ||= 514.286 ||= 1 1 1 || 239.2945 || 274.991 || 307.521 || 378.193 ||= || || 59\138 ||||< || || || || ||= 513.0435 ||= 20 20 19 || 238.673 || 274.370 || 308.142 || 378.8145 || || || 56\131 ||||< || || || || ||= 512.977 ||= 19 19 18 || 238.640 || 274.337 || 308.175 || 378.848 || || || 53\124 ||||< || || || || ||= 512.903 ||= 18 18 17 || 238.603 || 274.300 || 308.212 || 378.885 || || || 50\117 ||||< ||< ||< || || ||= 512.8205 ||= 17 17 16 || 238.562 || 274.259 || 308.2535 || 378.926 || || ||< 47\110 ||||< ||< ||< || || ||= 512.727 ||= 16 16 15 || 238.515 || 274.212 || 308.300 || 378.973 || || ||< 44\103 ||||< ||< ||< || || ||= 512.621 ||= 15 15 14 || 238.462 || 274.159 || 308.353 || 379.0255 || || ||< 41\96 ||||< ||< ||< || || ||= 512.500 ||= 14 14 13 || 238.402 || 274.098 || 308.414 || 379.086 || || ||< 38\89 ||||< ||< ||< || || ||= 512.360 ||= 13 13 12 || 238.331 || 274.028 || 308.484 || 379.156 || || ||< 35\82 ||||< ||< ||< || || ||= 512.195 ||= 12 12 11 || 238.249 || 273.946 || 308.566 || 379.239 || || ||< 32\75 ||||< ||< ||< || || ||= 512.000 ||= 11 11 10 || 238.152 || 273.848 || 308.664 || 379.336 || || ||< 29\68 ||||< ||< ||< || || ||= 511.765 ||= 10 10 9 || 238.034 || 273.731 || 308.781 || 379.454 || || ||< 26\61 ||||< ||< ||< || || ||= 511.475 ||= 9 9 8 || 237.889 || 273.586 || 308.926 || 379.5985 || || ||< 23\54 ||||< ||< ||< || || ||= 511.111 ||= 8 8 7 || 237.707 || 273.404 || 309.108 || 379.781 || || ||< 20\47 ||||< ||< ||< || || ||= 510.638 ||= 7 7 6 || 237.471 || 273.168 || 309.345 || 380.017 || || ||< 17\40 ||||< ||< ||< ||= || ||= 510.000 ||= 6 6 5 || 237.152 || 272.848 || 309.664 || 380.336 ||= || ||< 14\33 ||||< ||< ||< ||= || ||= 509.091 ||= 5 5 4 || 236.697 || 272.394 || 310.118 || 380.791 ||= || ||< ||||< 25\59 ||< ||< ||= || ||= 508.475 ||= 9 9 7 || 236.389 || 272.086 || 310.4265 || 381.0985 ||= || ||< 11\26 ||||< ||< ||< ||= || ||= 507.692 ||= 4 4 3 || 235.998 || 271.695 || 310.817 || 381.491 ||= || ||< ||||< 30\71 ||< ||< ||= || ||= 507.042 ||= 11 11 8 || 235.672 || 271.3695 || 311.142 || 381.846 ||= || ||< ||||< 19\45 ||< ||< ||= || ||= 506.667 ||= 7 7 5 || 235.485 || 271.182 || 311.33 || 382.003 ||= || ||< ||||< 27\64 ||< ||< ||= || ||= 506.250 ||= 10 10 7 || 235.277 || 270.973 || 311.539 || 382.211 ||= || ||< 8\19 ||||< ||< ||< ||= || ||= 505.263 ||= 3 3 2 || 234.783 || 270.480 || 312.032 || 382.705 ||= Optimum rank range (L/s=3/2) diatonic || || ||||< 37\88 || || || || ||= 504.5455 ||= 14 14 9 || 234.424 || 270.121 || 312.391 || 383.0635 ||= LucyTuning || || ||< || || || || || ||= 504.356 ||= <span style="display: block; text-align: center;">pi pi 2</span> || 234.329 || 270.026 || 312.486 || 383.158 || || ||< ||||< 29\69 ||< ||< ||= || ||= 504.348 ||= 11 11 7 || 234.3255 || 270.022 || 312.490 || 383.172 ||= || ||< ||||< 21\50 ||< ||< ||= || ||= 504.000 ||= 8 8 5 || 234.152 || 269.848 || 312.664 || 383.336 ||= || || ||||< || 55\131 || || || ||= 503.817 ||= 21 21 13 || 234.060 || 269.757 || 312.755 || 383.428 || || || ||||< || || || 144\343 || ||= 503.790 ||= 55 55 34 || 234.047 || 269.743 || 312.769 || 383.441 || || || ||||< || || || || 233\555 ||= 503.784 ||= 89 89 55 || 234.0435 || 269.740 || 312.772 || 383.444 ||= Golden meantone || || ||||< || || 89\212 || || ||= 503.774 ||= 34 34 21 || 234.038 || 269.735 || 312.777 || 383.449 ||= || ||< ||||< ||< 34\81 ||< ||= || ||= 503.704 ||= 13 13 8 || 234.003 || 269.700 || 312.811 || 383.485 ||= || ||< ||||< 13\31 ||< ||< ||= || ||= 503.226 ||= 5 5 3 || 233.7645 || 269.461 || 313.051 || 383.723 ||= Meantone is in this region || ||< ||||< ||< 31\74 ||< ||= || ||= 502.703 ||= 12 12 7 || 233.503 || 269.200 || 313.312 || 383.985 ||= || || || || || || || || ||= 502.5135 ||= <span style="background-color: #ffffff;">√3 √3 1</span> || 233.408 || 269.105 || 313.407 || 384.079 || || ||< ||||< 18\43 ||< ||< ||= || ||= 502.326 ||= 7 7 4 || 233.314 || 269.011 || 313.501 || 384.183 ||= || ||< ||||< 23\55 ||< ||< ||= || ||= 501.818 ||= 9 9 5 || 233.061 || 268.7575 || 313.754 || 384.428 ||= || ||< 5\12 ||||< ||< ||< ||= || ||= 500.000 ||= 2 2 1 || 232.152 || 267.848 || 314.664 || 385.336 ||= Boundary of propriety (generators larger than this are proper) || ||< ||||< 42\101 ||< ||< || || ||= 499.010 ||= 17 17 8 || 231.6565 || 267.353 || 315.159 || 385.831 || || ||< ||||< <span style="display: block; text-align: center;">37\89</span> ||< ||< ||= ||= ||= 498.876 ||= 15 15 7 ||< 231.590 ||< 267.287 ||< 315,226 ||< 385.898 ||= || ||< ||||< <span style="display: block; text-align: center;">32\77</span> ||< ||< ||= ||= ||= 498.701 ||= 13 13 6 ||< 231.502 ||< 267.199 ||< 315.313 ||< 385.986 ||= || ||< ||||< 27\65 ||< ||< || || ||= 498.4615 ||= 11 11 5 || 231.382 || 267.079 || 315.433 || 386.105 || || ||< ||||< 22\53 ||< ||< ||= || ||= 498.113 ||= 9 9 4 || 231.208 || 266.905 || 315.609 || 386.278 ||= Pythagorean is around here || ||< ||||< 17\41 ||< ||< ||= || ||= 497.591 ||= 7 7 3 || 230.932 || 266.629 || 315.883 || 386.556 ||= || ||< ||||< 29\70 ||< ||< ||= || ||= 497.143 ||= 12 12 5 || 230.723 || 266.420 || 316.092 || 386.765 ||= || ||< ||||< 12\29 ||< ||< ||= || ||= 496.552 ||= 5 5 2 || 230.4275 || 266.124 || 316.388 || 387.061 ||= || ||< ||||< ||< 31\75 ||< ||= || ||= 496.000 ||= 13 13 5 || 230.152 || 265.848 || 316.664 || 387.336 ||= || || || || || || || 81\196 || ||= 495.918 ||= 34 34 13 || 230.111 || 265.808 || 316.705 || 387.377 || || || || || || || || || 131\317 ||= 495.899 ||= 55 55 21 || 230.101 || 265.798 || 316.714 || 387.387 || || || || || || || 50\121 || || ||= 495.868 ||= 21 21 8 || 230.0855 || 265.782 || 316.73 || 387.402 || || ||< ||||< 19\46 ||< ||< ||= || ||= 495.652 ||= 8 8 3 || 229.978 || 265.6745 || 316.837 || 387.511 ||= || ||< ||||< ||< ||< ||= || ||= 495.393 ||= <span style="display: block; text-align: center;">e e 1</span> || 229.848 || 265.545 || 316.967 || 387.639 ||= <span style="display: block; text-align: center;">L/s = e</span> || || ||||< 26\63 || || || || ||= <span style="display: block; text-align: center;">495.238</span> ||= 11 11 4 || 229.771 || 265.4675 || 317.045 || 387.717 ||= <span style="display: block; text-align: center;"> </span> || ||< 7\17 ||||< ||< ||< ||= || ||= 494.118 ||= 3 3 1 || 229.210 || 264.907 || 317.596 || 388.286 ||= L/s = 3 || || ||||< || || || || ||= 493.553 ||= pi pi 1 || 228.928 || 264.625 || 317.887 || 388.56 ||= <span style="display: block; text-align: center;">L/s = pi</span> || ||< ||||< 23\56 ||< ||< ||= || ||= 492.857 ||= 10 10 3 || 228.580 || 264.277 || 318.235 || 388.908 ||= || ||< ||||< 16\39 ||< ||< ||= || ||= 492.308 ||= 7 7 2 || 228.305 || 264.002 || 318.51 || 389.182 ||= || ||< ||||< 25\61 ||< ||< ||= || ||= 491.803 ||= 11 11 3 || 228.053 || 263.750 || 318.761 || 389.436 ||= || ||< 9\22 ||||< ||< ||< ||= || ||= 490.909 ||= 4 4 1 || 227.606 || 263.303 || 319.209 || 389.882 ||= (No-5's) superpyth is in this region L/s = 4 || ||< ||||< 20\49 ||< ||< ||= || ||= 489.796 ||= 9 9 2 || 227.050 || 262.746 || 319.766 || 390.438 ||= || ||< 11\27 ||||< ||< ||< ||= || ||= 488.889 ||= 5 5 1 || 226.596 || 262.293 || 320.219 || 390.892 ||= || ||< 13\32 ||||< ||< ||< ||= || ||= 487.500 ||= 6 6 1 || 225.9015 || 261.598 || 320.914 || 391.596 ||= || ||< 15\37 ||||< ||< ||< || || ||= 486.4865 ||= 7 7 1 || 225.395 || 261.092 || 321.4205 || 392.093 || || ||< 17\42 ||||< ||< ||< ||= ||= ||= 485.714 ||= 8 8 1 ||< 225.009 ||< 260.7055 ||< 321.807 ||< 392.479 ||= || ||< 19\47 ||||< ||< ||< || || ||= 485.106 ||= 9 9 1 || 224.705 || 260.402 || 322.111 || 392.783 || || ||< 21\52 ||||< ||< ||< || || ||= 484.615 ||= 10 10 1 || 224.459 || 260.156 || 322.356 || 393.0285 || || ||< 23\57 ||||< ||< ||< || || ||= 484.2105 ||= 11 11 1 || 224.257 || 259.954 || 322.5585 || 393.231 || || ||< 25\62 ||||< ||< ||< || || ||= 483.871 ||= 12 12 1 || 224.087 || 259.784 || 322.728 || 393.401 || || ||< 27\67 ||||< ||< ||< || || ||= 483.582 ||= 13 13 1 || 223.943 || 259.6395 || 322.873 || 393.545 || || ||< 29\72 ||||< ||< ||< || || ||= 483.333 ||= 14 14 1 || 223.818 || 259.515 || 322.997 || 393.6695 || || ||< 31\77 ||||< ||< ||< || || ||= 483.117 ||= 15 15 1 || 223.710 || 259.407 || 323.105 || 393.778 || || ||< 33\82 ||||< ||< ||< || || ||= 482.927 ||= 16 16 1 || 223.615 || 259.312 || 323.200 || 393.873 || || || 35\87 ||||< ||< ||< || || ||= 482.759 ||= 17 17 1 || 223.531 || 259.228 || 323.2845 || 393.957 || || || 37\92 ||||< ||< ||< || || ||= 482.609 ||= 18 18 1 || 223.456 || 259.153 || 323.539 || 394.032 || || || 39\97 ||||< || || || || ||= 482.474 ||= 19 19 1 || 223.389 || 259.0855 || 323.427 || 394.099 || || || 41\102 ||||< || || || || ||= 482.353 ||= 20 20 1 || 223.328 || 259.025 || 323.487 || 394.160 || || || 43\107 ||||< || || || || ||= 482.243 ||= 21 21 1 || 223.273 || 258.970 || 323.542 || 394.215 || || || 45\112 ||||< || || || || ||= 482.143 ||= 22 22 1 || 223.223 || 258.920 || 323.592 || 394.265 || || || 47\117 ||||< || || || || ||= 482.051 ||= 23 23 1 || 223.177 || 258.874 || 323.638 || 394.311 || || || 49\122 ||||< || || || || ||= 481.967 ||= 24 24 1 || 223.135 || 258.832 || 322.680 || 394.353 || || ||< 2\5 ||||< ||< ||< ||= || ||= 480.000 ||= 1 1 0 || 222.152 || 257.848 || 324.664 || 395.336 ||= || Temperaments above 5\12 on this chart are called "negative temperaments" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 8\19) and 1/4-comma (close to 13\31). As these tunings approach 3\7, the majors become flatter and the minors become sharper. Temperaments below 5\12 on this chart are called "positive temperaments" and they include Pythagorean tuning itself (well approximated by 22\53) as well as superpyth temperaments such as 7\17 and 9\22. As these tunings approach 2\5, the majors become sharper and the minors become flatter. Around 9\22, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4. [[image:5L2s.jpg]] 5L 2s contains the pentatonic MOS [[2L 3s]] and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either [[7L 5s]] or [[5L 7s]].
Original HTML content:
<html><head><title>5L 2s</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x5L 2s - "diatonic""></a><!-- ws:end:WikiTextHeadingRule:0 -->5L 2s - "diatonic"</h1>
<br />
One way of distinguishing the "diatonic" scale is by considering it a <a class="wiki_link" href="/MOSScales">moment of symmetry</a> scale produced by a chain of "fifths". This will include <a class="wiki_link" href="/12edo">12edo</a>'s diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone".<br />
<br />
It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x5L 2s - "diatonic"-substituting step sizes"></a><!-- ws:end:WikiTextHeadingRule:2 -->substituting step sizes</h2>
<br />
The 5L 2s MOS scale has this generalized form.<br />
L L s L L L s<br />
<br />
Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.<br />
2 2 1 2 2 2 1<br />
<br />
When L=3, s=1, you have <a class="wiki_link" href="/17edo">17edo</a>:<br />
3 3 1 3 3 3 1<br />
<br />
When L=3, s=2, you have <a class="wiki_link" href="/19edo">19edo</a>:<br />
3 3 2 3 3 3 2<br />
<br />
When L=4, s=1, you have <a class="wiki_link" href="/22edo">22edo</a>:<br />
4 4 1 4 4 4 1<br />
<br />
When L=4, s=3, you have <a class="wiki_link" href="/26edo">26edo</a>:<br />
4 4 3 4 4 4 3<br />
<br />
When L=5, s=1, you have <a class="wiki_link" href="/27edo">27edo</a>:<br />
5 5 1 5 5 5 1<br />
<br />
When L=5, s=2, you have <a class="wiki_link" href="/29edo">29edo</a>:<br />
5 5 2 5 5 5 2<br />
<br />
When L=5, s=3, you have <a class="wiki_link" href="/31edo">31edo</a>:<br />
5 5 3 5 5 5 3<br />
<br />
When L=5, s=4, you have <a class="wiki_link" href="/33edo">33edo</a>:<br />
5 5 4 5 5 5 4<br />
<br />
So you have scales where L and s are nearly equal, which approach <a class="wiki_link" href="/7edo">7edo</a>:<br />
1 1 1 1 1 1 1<br />
<br />
And you have scales where s becomes so small it approaches zero, which would give us <a class="wiki_link" href="/5edo">5edo</a>:<br />
1 1 0 1 1 1 0 or 1 1 1 1 1<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x5L 2s - "diatonic"-a continuum of temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->a continuum of temperaments</h2>
<br />
So if 3\7 (three degrees of 7edo) is at one extreme and 2\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators. Thus, between 3\7 and 2\5 you have (3+2)\(7+5) = 5\12, five degrees of 12edo:<br />
<br />
<table class="wiki_table">
<tr>
<td>3\7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>5\12<br />
</td>
</tr>
<tr>
<td>2\5<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
If we carry this freshman-summing out a little further, new, larger <a class="wiki_link" href="/edo">edo</a>s pop up in our continuum.<br />
<table class="wiki_table">
<tr>
<th colspan="6">generator<br />
</th>
<th><br />
</th>
<th>in cents<br />
</th>
<th>tetrachord<br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th>comments<br />
</th>
</tr>
<tr>
<td style="text-align: left;">3\7<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">514.286<br />
</td>
<td style="text-align: center;">1 1 1<br />
</td>
<td>239.2945<br />
</td>
<td>274.991<br />
</td>
<td>307.521<br />
</td>
<td>378.193<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>59\138<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">513.0435<br />
</td>
<td style="text-align: center;">20 20 19<br />
</td>
<td>238.673<br />
</td>
<td>274.370<br />
</td>
<td>308.142<br />
</td>
<td>378.8145<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>56\131<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.977<br />
</td>
<td style="text-align: center;">19 19 18<br />
</td>
<td>238.640<br />
</td>
<td>274.337<br />
</td>
<td>308.175<br />
</td>
<td>378.848<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>53\124<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.903<br />
</td>
<td style="text-align: center;">18 18 17<br />
</td>
<td>238.603<br />
</td>
<td>274.300<br />
</td>
<td>308.212<br />
</td>
<td>378.885<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>50\117<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.8205<br />
</td>
<td style="text-align: center;">17 17 16<br />
</td>
<td>238.562<br />
</td>
<td>274.259<br />
</td>
<td>308.2535<br />
</td>
<td>378.926<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">47\110<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.727<br />
</td>
<td style="text-align: center;">16 16 15<br />
</td>
<td>238.515<br />
</td>
<td>274.212<br />
</td>
<td>308.300<br />
</td>
<td>378.973<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">44\103<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.621<br />
</td>
<td style="text-align: center;">15 15 14<br />
</td>
<td>238.462<br />
</td>
<td>274.159<br />
</td>
<td>308.353<br />
</td>
<td>379.0255<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">41\96<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.500<br />
</td>
<td style="text-align: center;">14 14 13<br />
</td>
<td>238.402<br />
</td>
<td>274.098<br />
</td>
<td>308.414<br />
</td>
<td>379.086<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">38\89<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.360<br />
</td>
<td style="text-align: center;">13 13 12<br />
</td>
<td>238.331<br />
</td>
<td>274.028<br />
</td>
<td>308.484<br />
</td>
<td>379.156<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">35\82<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.195<br />
</td>
<td style="text-align: center;">12 12 11<br />
</td>
<td>238.249<br />
</td>
<td>273.946<br />
</td>
<td>308.566<br />
</td>
<td>379.239<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">32\75<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">512.000<br />
</td>
<td style="text-align: center;">11 11 10<br />
</td>
<td>238.152<br />
</td>
<td>273.848<br />
</td>
<td>308.664<br />
</td>
<td>379.336<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">29\68<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">511.765<br />
</td>
<td style="text-align: center;">10 10 9<br />
</td>
<td>238.034<br />
</td>
<td>273.731<br />
</td>
<td>308.781<br />
</td>
<td>379.454<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">26\61<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">511.475<br />
</td>
<td style="text-align: center;">9 9 8<br />
</td>
<td>237.889<br />
</td>
<td>273.586<br />
</td>
<td>308.926<br />
</td>
<td>379.5985<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">23\54<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">511.111<br />
</td>
<td style="text-align: center;">8 8 7<br />
</td>
<td>237.707<br />
</td>
<td>273.404<br />
</td>
<td>309.108<br />
</td>
<td>379.781<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">20\47<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">510.638<br />
</td>
<td style="text-align: center;">7 7 6<br />
</td>
<td>237.471<br />
</td>
<td>273.168<br />
</td>
<td>309.345<br />
</td>
<td>380.017<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">17\40<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">510.000<br />
</td>
<td style="text-align: center;">6 6 5<br />
</td>
<td>237.152<br />
</td>
<td>272.848<br />
</td>
<td>309.664<br />
</td>
<td>380.336<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">14\33<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">509.091<br />
</td>
<td style="text-align: center;">5 5 4<br />
</td>
<td>236.697<br />
</td>
<td>272.394<br />
</td>
<td>310.118<br />
</td>
<td>380.791<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">25\59<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">508.475<br />
</td>
<td style="text-align: center;">9 9 7<br />
</td>
<td>236.389<br />
</td>
<td>272.086<br />
</td>
<td>310.4265<br />
</td>
<td>381.0985<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">11\26<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">507.692<br />
</td>
<td style="text-align: center;">4 4 3<br />
</td>
<td>235.998<br />
</td>
<td>271.695<br />
</td>
<td>310.817<br />
</td>
<td>381.491<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">30\71<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">507.042<br />
</td>
<td style="text-align: center;">11 11 8<br />
</td>
<td>235.672<br />
</td>
<td>271.3695<br />
</td>
<td>311.142<br />
</td>
<td>381.846<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">19\45<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">506.667<br />
</td>
<td style="text-align: center;">7 7 5<br />
</td>
<td>235.485<br />
</td>
<td>271.182<br />
</td>
<td>311.33<br />
</td>
<td>382.003<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">27\64<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">506.250<br />
</td>
<td style="text-align: center;">10 10 7<br />
</td>
<td>235.277<br />
</td>
<td>270.973<br />
</td>
<td>311.539<br />
</td>
<td>382.211<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">8\19<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">505.263<br />
</td>
<td style="text-align: center;">3 3 2<br />
</td>
<td>234.783<br />
</td>
<td>270.480<br />
</td>
<td>312.032<br />
</td>
<td>382.705<br />
</td>
<td style="text-align: center;">Optimum rank range (L/s=3/2) diatonic<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td colspan="2" style="text-align: left;">37\88<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">504.5455<br />
</td>
<td style="text-align: center;">14 14 9<br />
</td>
<td>234.424<br />
</td>
<td>270.121<br />
</td>
<td>312.391<br />
</td>
<td>383.0635<br />
</td>
<td style="text-align: center;">LucyTuning<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">504.356<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">pi pi 2</span><br />
</td>
<td>234.329<br />
</td>
<td>270.026<br />
</td>
<td>312.486<br />
</td>
<td>383.158<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">29\69<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">504.348<br />
</td>
<td style="text-align: center;">11 11 7<br />
</td>
<td>234.3255<br />
</td>
<td>270.022<br />
</td>
<td>312.490<br />
</td>
<td>383.172<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">21\50<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">504.000<br />
</td>
<td style="text-align: center;">8 8 5<br />
</td>
<td>234.152<br />
</td>
<td>269.848<br />
</td>
<td>312.664<br />
</td>
<td>383.336<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td>55\131<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">503.817<br />
</td>
<td style="text-align: center;">21 21 13<br />
</td>
<td>234.060<br />
</td>
<td>269.757<br />
</td>
<td>312.755<br />
</td>
<td>383.428<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>144\343<br />
</td>
<td><br />
</td>
<td style="text-align: center;">503.790<br />
</td>
<td style="text-align: center;">55 55 34<br />
</td>
<td>234.047<br />
</td>
<td>269.743<br />
</td>
<td>312.769<br />
</td>
<td>383.441<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>233\555<br />
</td>
<td style="text-align: center;">503.784<br />
</td>
<td style="text-align: center;">89 89 55<br />
</td>
<td>234.0435<br />
</td>
<td>269.740<br />
</td>
<td>312.772<br />
</td>
<td>383.444<br />
</td>
<td style="text-align: center;">Golden meantone<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td>89\212<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">503.774<br />
</td>
<td style="text-align: center;">34 34 21<br />
</td>
<td>234.038<br />
</td>
<td>269.735<br />
</td>
<td>312.777<br />
</td>
<td>383.449<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;">34\81<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">503.704<br />
</td>
<td style="text-align: center;">13 13 8<br />
</td>
<td>234.003<br />
</td>
<td>269.700<br />
</td>
<td>312.811<br />
</td>
<td>383.485<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">13\31<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">503.226<br />
</td>
<td style="text-align: center;">5 5 3<br />
</td>
<td>233.7645<br />
</td>
<td>269.461<br />
</td>
<td>313.051<br />
</td>
<td>383.723<br />
</td>
<td style="text-align: center;">Meantone is in this region<br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;">31\74<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">502.703<br />
</td>
<td style="text-align: center;">12 12 7<br />
</td>
<td>233.503<br />
</td>
<td>269.200<br />
</td>
<td>313.312<br />
</td>
<td>383.985<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 style="text-align: center;">502.5135<br />
</td>
<td style="text-align: center;"><span style="background-color: #ffffff;">√3 √3 1</span><br />
</td>
<td>233.408<br />
</td>
<td>269.105<br />
</td>
<td>313.407<br />
</td>
<td>384.079<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">18\43<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">502.326<br />
</td>
<td style="text-align: center;">7 7 4<br />
</td>
<td>233.314<br />
</td>
<td>269.011<br />
</td>
<td>313.501<br />
</td>
<td>384.183<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">23\55<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">501.818<br />
</td>
<td style="text-align: center;">9 9 5<br />
</td>
<td>233.061<br />
</td>
<td>268.7575<br />
</td>
<td>313.754<br />
</td>
<td>384.428<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">5\12<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">500.000<br />
</td>
<td style="text-align: center;">2 2 1<br />
</td>
<td>232.152<br />
</td>
<td>267.848<br />
</td>
<td>314.664<br />
</td>
<td>385.336<br />
</td>
<td style="text-align: center;">Boundary of propriety<br />
(generators larger than this are proper)<br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">42\101<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">499.010<br />
</td>
<td style="text-align: center;">17 17 8<br />
</td>
<td>231.6565<br />
</td>
<td>267.353<br />
</td>
<td>315.159<br />
</td>
<td>385.831<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;"><span style="display: block; text-align: center;">37\89</span><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">498.876<br />
</td>
<td style="text-align: center;">15 15 7<br />
</td>
<td style="text-align: left;">231.590<br />
</td>
<td style="text-align: left;">267.287<br />
</td>
<td style="text-align: left;">315,226<br />
</td>
<td style="text-align: left;">385.898<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;"><span style="display: block; text-align: center;">32\77</span><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">498.701<br />
</td>
<td style="text-align: center;">13 13 6<br />
</td>
<td style="text-align: left;">231.502<br />
</td>
<td style="text-align: left;">267.199<br />
</td>
<td style="text-align: left;">315.313<br />
</td>
<td style="text-align: left;">385.986<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">27\65<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">498.4615<br />
</td>
<td style="text-align: center;">11 11 5<br />
</td>
<td>231.382<br />
</td>
<td>267.079<br />
</td>
<td>315.433<br />
</td>
<td>386.105<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">22\53<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">498.113<br />
</td>
<td style="text-align: center;">9 9 4<br />
</td>
<td>231.208<br />
</td>
<td>266.905<br />
</td>
<td>315.609<br />
</td>
<td>386.278<br />
</td>
<td style="text-align: center;">Pythagorean is around here<br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">17\41<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">497.591<br />
</td>
<td style="text-align: center;">7 7 3<br />
</td>
<td>230.932<br />
</td>
<td>266.629<br />
</td>
<td>315.883<br />
</td>
<td>386.556<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">29\70<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">497.143<br />
</td>
<td style="text-align: center;">12 12 5<br />
</td>
<td>230.723<br />
</td>
<td>266.420<br />
</td>
<td>316.092<br />
</td>
<td>386.765<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">12\29<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">496.552<br />
</td>
<td style="text-align: center;">5 5 2<br />
</td>
<td>230.4275<br />
</td>
<td>266.124<br />
</td>
<td>316.388<br />
</td>
<td>387.061<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;">31\75<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">496.000<br />
</td>
<td style="text-align: center;">13 13 5<br />
</td>
<td>230.152<br />
</td>
<td>265.848<br />
</td>
<td>316.664<br />
</td>
<td>387.336<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>81\196<br />
</td>
<td><br />
</td>
<td style="text-align: center;">495.918<br />
</td>
<td style="text-align: center;">34 34 13<br />
</td>
<td>230.111<br />
</td>
<td>265.808<br />
</td>
<td>316.705<br />
</td>
<td>387.377<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>131\317<br />
</td>
<td style="text-align: center;">495.899<br />
</td>
<td style="text-align: center;">55 55 21<br />
</td>
<td>230.101<br />
</td>
<td>265.798<br />
</td>
<td>316.714<br />
</td>
<td>387.387<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>50\121<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">495.868<br />
</td>
<td style="text-align: center;">21 21 8<br />
</td>
<td>230.0855<br />
</td>
<td>265.782<br />
</td>
<td>316.73<br />
</td>
<td>387.402<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">19\46<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">495.652<br />
</td>
<td style="text-align: center;">8 8 3<br />
</td>
<td>229.978<br />
</td>
<td>265.6745<br />
</td>
<td>316.837<br />
</td>
<td>387.511<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">495.393<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">e e 1</span><br />
</td>
<td>229.848<br />
</td>
<td>265.545<br />
</td>
<td>316.967<br />
</td>
<td>387.639<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = e</span><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td colspan="2" style="text-align: left;">26\63<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">495.238</span><br />
</td>
<td style="text-align: center;">11 11 4<br />
</td>
<td>229.771<br />
</td>
<td>265.4675<br />
</td>
<td>317.045<br />
</td>
<td>387.717<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;"><br />
</span><br />
</td>
</tr>
<tr>
<td style="text-align: left;">7\17<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">494.118<br />
</td>
<td style="text-align: center;">3 3 1<br />
</td>
<td>229.210<br />
</td>
<td>264.907<br />
</td>
<td>317.596<br />
</td>
<td>388.286<br />
</td>
<td style="text-align: center;">L/s = 3<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">493.553<br />
</td>
<td style="text-align: center;">pi pi 1<br />
</td>
<td>228.928<br />
</td>
<td>264.625<br />
</td>
<td>317.887<br />
</td>
<td>388.56<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = pi</span><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">23\56<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">492.857<br />
</td>
<td style="text-align: center;">10 10 3<br />
</td>
<td>228.580<br />
</td>
<td>264.277<br />
</td>
<td>318.235<br />
</td>
<td>388.908<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">16\39<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">492.308<br />
</td>
<td style="text-align: center;">7 7 2<br />
</td>
<td>228.305<br />
</td>
<td>264.002<br />
</td>
<td>318.51<br />
</td>
<td>389.182<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">25\61<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">491.803<br />
</td>
<td style="text-align: center;">11 11 3<br />
</td>
<td>228.053<br />
</td>
<td>263.750<br />
</td>
<td>318.761<br />
</td>
<td>389.436<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">9\22<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">490.909<br />
</td>
<td style="text-align: center;">4 4 1<br />
</td>
<td>227.606<br />
</td>
<td>263.303<br />
</td>
<td>319.209<br />
</td>
<td>389.882<br />
</td>
<td style="text-align: center;">(No-5's) superpyth is in this region<br />
L/s = 4<br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td colspan="2" style="text-align: left;">20\49<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">489.796<br />
</td>
<td style="text-align: center;">9 9 2<br />
</td>
<td>227.050<br />
</td>
<td>262.746<br />
</td>
<td>319.766<br />
</td>
<td>390.438<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">11\27<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">488.889<br />
</td>
<td style="text-align: center;">5 5 1<br />
</td>
<td>226.596<br />
</td>
<td>262.293<br />
</td>
<td>320.219<br />
</td>
<td>390.892<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">13\32<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">487.500<br />
</td>
<td style="text-align: center;">6 6 1<br />
</td>
<td>225.9015<br />
</td>
<td>261.598<br />
</td>
<td>320.914<br />
</td>
<td>391.596<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">15\37<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">486.4865<br />
</td>
<td style="text-align: center;">7 7 1<br />
</td>
<td>225.395<br />
</td>
<td>261.092<br />
</td>
<td>321.4205<br />
</td>
<td>392.093<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">17\42<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">485.714<br />
</td>
<td style="text-align: center;">8 8 1<br />
</td>
<td style="text-align: left;">225.009<br />
</td>
<td style="text-align: left;">260.7055<br />
</td>
<td style="text-align: left;">321.807<br />
</td>
<td style="text-align: left;">392.479<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">19\47<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">485.106<br />
</td>
<td style="text-align: center;">9 9 1<br />
</td>
<td>224.705<br />
</td>
<td>260.402<br />
</td>
<td>322.111<br />
</td>
<td>392.783<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">21\52<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">484.615<br />
</td>
<td style="text-align: center;">10 10 1<br />
</td>
<td>224.459<br />
</td>
<td>260.156<br />
</td>
<td>322.356<br />
</td>
<td>393.0285<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">23\57<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">484.2105<br />
</td>
<td style="text-align: center;">11 11 1<br />
</td>
<td>224.257<br />
</td>
<td>259.954<br />
</td>
<td>322.5585<br />
</td>
<td>393.231<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">25\62<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">483.871<br />
</td>
<td style="text-align: center;">12 12 1<br />
</td>
<td>224.087<br />
</td>
<td>259.784<br />
</td>
<td>322.728<br />
</td>
<td>393.401<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">27\67<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">483.582<br />
</td>
<td style="text-align: center;">13 13 1<br />
</td>
<td>223.943<br />
</td>
<td>259.6395<br />
</td>
<td>322.873<br />
</td>
<td>393.545<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">29\72<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">483.333<br />
</td>
<td style="text-align: center;">14 14 1<br />
</td>
<td>223.818<br />
</td>
<td>259.515<br />
</td>
<td>322.997<br />
</td>
<td>393.6695<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">31\77<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">483.117<br />
</td>
<td style="text-align: center;">15 15 1<br />
</td>
<td>223.710<br />
</td>
<td>259.407<br />
</td>
<td>323.105<br />
</td>
<td>393.778<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">33\82<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.927<br />
</td>
<td style="text-align: center;">16 16 1<br />
</td>
<td>223.615<br />
</td>
<td>259.312<br />
</td>
<td>323.200<br />
</td>
<td>393.873<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>35\87<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.759<br />
</td>
<td style="text-align: center;">17 17 1<br />
</td>
<td>223.531<br />
</td>
<td>259.228<br />
</td>
<td>323.2845<br />
</td>
<td>393.957<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>37\92<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.609<br />
</td>
<td style="text-align: center;">18 18 1<br />
</td>
<td>223.456<br />
</td>
<td>259.153<br />
</td>
<td>323.539<br />
</td>
<td>394.032<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>39\97<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.474<br />
</td>
<td style="text-align: center;">19 19 1<br />
</td>
<td>223.389<br />
</td>
<td>259.0855<br />
</td>
<td>323.427<br />
</td>
<td>394.099<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>41\102<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.353<br />
</td>
<td style="text-align: center;">20 20 1<br />
</td>
<td>223.328<br />
</td>
<td>259.025<br />
</td>
<td>323.487<br />
</td>
<td>394.160<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>43\107<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.243<br />
</td>
<td style="text-align: center;">21 21 1<br />
</td>
<td>223.273<br />
</td>
<td>258.970<br />
</td>
<td>323.542<br />
</td>
<td>394.215<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>45\112<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.143<br />
</td>
<td style="text-align: center;">22 22 1<br />
</td>
<td>223.223<br />
</td>
<td>258.920<br />
</td>
<td>323.592<br />
</td>
<td>394.265<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>47\117<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">482.051<br />
</td>
<td style="text-align: center;">23 23 1<br />
</td>
<td>223.177<br />
</td>
<td>258.874<br />
</td>
<td>323.638<br />
</td>
<td>394.311<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>49\122<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">481.967<br />
</td>
<td style="text-align: center;">24 24 1<br />
</td>
<td>223.135<br />
</td>
<td>258.832<br />
</td>
<td>322.680<br />
</td>
<td>394.353<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;">2\5<br />
</td>
<td colspan="2" style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">480.000<br />
</td>
<td style="text-align: center;">1 1 0<br />
</td>
<td>222.152<br />
</td>
<td>257.848<br />
</td>
<td>324.664<br />
</td>
<td>395.336<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
Temperaments above 5\12 on this chart are called "negative temperaments" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 8\19) and 1/4-comma (close to 13\31). As these tunings approach 3\7, the majors become flatter and the minors become sharper.<br />
<br />
Temperaments below 5\12 on this chart are called "positive temperaments" and they include Pythagorean tuning itself (well approximated by 22\53) as well as superpyth temperaments such as 7\17 and 9\22. As these tunings approach 2\5, the majors become sharper and the minors become flatter. Around 9\22, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.<br />
<br />
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5L 2s contains the pentatonic MOS <a class="wiki_link" href="/2L%203s">2L 3s</a> and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either <a class="wiki_link" href="/7L%205s">7L 5s</a> or <a class="wiki_link" href="/5L%207s">5L 7s</a>.</body></html>