5L 2s: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~Andrew_Heathwaite~~|~~Andrew_Heathwaite~~]] and made on <tt>~~2010~~-10-~~26 17~~:04:46 UTC</tt>.

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-04-25 13:21:41 UTC</tt>.

: The original revision id was <tt>~~173863945~~</tt>.

: The original revision id was <tt>222738126</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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If we carry this freshman-summing out a little further, new, larger [[edo]]s pop up in our continuum.

||||||||||||~ generator ||~ in cents ||~ scale in steps of an [[edo]] ||

||||||||||||~ generator ||~ in cents ||~ scale in steps of an [[edo]] ||= ||

||= 3\7 ||= ||= ||= ||= ||= ||= 514.286 ||= 1 1 1 1 1 1 1 ||

||= 3\7 ||= ||= ||= ||= ||= ||= 514.286 ||= 1 1 1 1 1 1 1 ||= ||

||= ||= ||= ||= ||= ||= 17\40 ||= 510.000 ||= 6 6 5 6 6 6 5 ||

||= ||= ||= ||= ||= ||= 17\40 ||= 510.000 ||= 6 6 5 6 6 6 5 ||= ||

||= ||= ||= ||= ||= 14\33 ||= ||= 509.091 ||= 5 5 4 5 5 5 4 ||

||= ||= ||= ||= ||= 14\33 ||= ||= 509.091 ||= 5 5 4 5 5 5 4 ||= ||

||= ||= ||= ||= ||= ||= 25\59 ||= 508.475 ||= 9 9 7 9 9 9 7 ||

||= ||= ||= ||= ||= ||= 25\59 ||= 508.475 ||= 9 9 7 9 9 9 7 ||= ||

||= ||= ||= ||= 11\26 ||= ||= ||= 507.692 ||= 4 4 3 4 4 4 3 ||

||= ||= ||= ||= 11\26 ||= ||= ||= 507.692 ||= 4 4 3 4 4 4 3 ||= ||

||= ||= ||= ||= ||= ||= 30\71 ||= 507.042 ||= 11 11 8 11 11 11 8 ||

||= ||= ||= ||= ||= ||= 30\71 ||= 507.042 ||= 11 11 8 11 11 11 8 ||= ||

||= ||= ||= ||= ||= 19\45 ||= ||= 506.667 ||= 7 7 5 7 7 7 5 ||

||= ||= ||= ||= ||= 19\45 ||= ||= 506.667 ||= 7 7 5 7 7 7 5 ||= ||

||= ||= ||= ||= ||= ||= 27\64 ||= 506.250 ||= 10 10 7 10 10 10 7 ||

||= ||= ||= ||= ||= ||= 27\64 ||= 506.250 ||= 10 10 7 10 10 10 7 ||= ||

||= ||= ||= 8\19 ||= ||= ||= ||= 506.263 ||= 3 3 2 3 3 3 2 ||

||= ||= ||= 8\19 ||= ||= ||= ||= 506.263 ||= 3 3 2 3 3 3 2 ||= ||

||= ||= ||= ||= ||= ||= 29\69 ||= 504.348 ||= 11 11 7 11 11 11 7 ||

||= ||= ||= ||= ||= ||= 29\69 ||= 504.348 ||= 11 11 7 11 11 11 7 ||= ||

||= ||= ||= ||= ||= 21\50 ||= ||= 504.000 ||= 8 8 5 8 8 8 5 ||

||= ||= ||= ||= ||= 21\50 ||= ||= 504.000 ||= 8 8 5 8 8 8 5 ||= ||

||= ||= ||= ||= ||= ||= 34\81 ||= 503.704 ||= 13 13 8 13 13 13 8 ||

||= ||= ||= ||= ||= ||= 34\81 ||= 503.704 ||= 13 13 8 13 13 13 8 ||= ||

||= ||= ||= ||= 13\31 ||= ||= ||= 503.226 ||= 5 5 3 5 5 5 3 ||

||= ||= ||= ||= 13\31 ||= ||= ||= 503.226 ||= 5 5 3 5 5 5 3 ||= Meantone is in this region ||

||= ||= ||= ||= ||= ||= 31\74 ||= 502.703 ||= 12 12 7 12 12 12 7 ||

||= ||= ||= ||= ||= ||= 31\74 ||= 502.703 ||= 12 12 7 12 12 12 7 ||= ||

||= ||= ||= ||= ||= 18\43 ||= ||= 502.326 ||= 7 7 4 7 7 7 4 ||

||= ||= ||= ||= ||= 18\43 ||= ||= 502.326 ||= 7 7 4 7 7 7 4 ||= ||

||= ||= ||= ||= ||= ||= 23\55 ||= 501.818 ||= 9 9 5 9 9 9 5 ||

||= ||= ||= ||= ||= ||= 23\55 ||= 501.818 ||= 9 9 5 9 9 9 5 ||= ||

||= ||= 5\12 ||= ||= ||= ||= ||= 500.000 ||= 2 2 1 2 2 2 1 ||

||= ||= 5\12 ||= ||= ||= ||= ||= 500.000 ||= 2 2 1 2 2 2 1 ||= Boundary of propriety (generators

||= ||= ||= ||= ||= ||= 22\53 ||= 498.113 ||= 9 9 4 9 9 9 4 ||

larger than this are proper) ||

||= ||= ||= ||= ||= 17\41 ||= ||= 497.591 ||= 7 7 3 7 7 7 3 ||

||= ||= ||= ||= ||= ||= 22\53 ||= 498.113 ||= 9 9 4 9 9 9 4 ||= Pythagorean is around here ||

||= ||= ||= ||= ||= ||= 29\70 ||= 497.143 ||= 12 12 5 12 12 12 5 ||

||= ||= ||= ||= ||= 17\41 ||= ||= 497.591 ||= 7 7 3 7 7 7 3 ||= ||

||= ||= ||= ||= 12\29 ||= ||= ||= 496.552 ||= 5 5 2 5 5 5 2 ||

||= ||= ||= ||= ||= ||= 29\70 ||= 497.143 ||= 12 12 5 12 12 12 5 ||= ||

||= ||= ||= ||= ||= ||= 31\75 ||= 496.000 ||= 13 13 5 13 13 13 5 ||

||= ||= ||= ||= 12\29 ||= ||= ||= 496.552 ||= 5 5 2 5 5 5 2 ||= ||

||= ||= ||= ||= ||= 19\46 ||= ||= 495.652 ||= 8 8 3 8 8 8 3 ||

||= ||= ||= ||= ||= ||= 31\75 ||= 496.000 ||= 13 13 5 13 13 13 5 ||= ||

||= ||= ||= ||= ||= ||= 26\63 ||= 495.238 ||= 11 11 4 11 11 11 4 ||

||= ||= ||= ||= ||= 19\46 ||= ||= 495.652 ||= 8 8 3 8 8 8 3 ||= ||

||= ||= ||= 7\17 ||= ||= ||= ||= 494.118 ||= 3 3 1 3 3 3 1 ||

||= ||= ||= ||= ||= ||= 26\63 ||= 495.238 ||= 11 11 4 11 11 11 4 ||= ||

||= ||= ||= ||= ||= ||= 23\56 ||= 492.857 ||= 10 10 3 10 10 10 3 ||

||= ||= ||= 7\17 ||= ||= ||= ||= 494.118 ||= 3 3 1 3 3 3 1 ||= ||

||= ||= ||= ||= ||= 16\39 ||= ||= 492.308 ||= 7 7 2 7 7 7 2 ||

||= ||= ||= ||= ||= ||= 23\56 ||= 492.857 ||= 10 10 3 10 10 10 3 ||= ||

||= ||= ||= ||= ||= ||= 25\61 ||= 491.803 ||= 11 11 3 11 11 11 3 ||

||= ||= ||= ||= ||= 16\39 ||= ||= 492.308 ||= 7 7 2 7 7 7 2 ||= ||

||= ||= ||= ||= 9\22 ||= ||= ||= 490.909 ||= 4 4 1 4 4 4 1 ||

||= ||= ||= ||= ||= ||= 25\61 ||= 491.803 ||= 11 11 3 11 11 11 3 ||= ||

||= ||= ||= ||= ||= ||= 20\49 ||= 489.796 ||= 9 9 2 9 9 9 2 ||

||= ||= ||= ||= 9\22 ||= ||= ||= 490.909 ||= 4 4 1 4 4 4 1 ||= (No-5's) superpyth is in this region ||

||= ||= ||= ||= ||= 11\27 ||= ||= 488.889 ||= 5 5 1 5 5 5 1 ||

||= ||= ||= ||= ||= ||= 20\49 ||= 489.796 ||= 9 9 2 9 9 9 2 ||= ||

||= ||= ||= ||= ||= ||= 13\32 ||= 487.500 ||= 6 6 1 6 6 6 1 ||

||= ||= ||= ||= ||= 11\27 ||= ||= 488.889 ||= 5 5 1 5 5 5 1 ||= ||

||= 2\5 ||= ||= ||= ||= ||= ||= 480.000 ||= 1 1 0 1 1 1 0 ||

||= ||= ||= ||= ||= ||= 13\32 ||= 487.500 ||= 6 6 1 6 6 6 1 ||= ||

||= 2\5 ||= ||= ||= ||= ||= ||= 480.000 ||= 1 1 0 1 1 1 0 ||= ||

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.

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<th>scale in steps of an <a class="wiki_link" href="/edo">edo</a>

</th>

</td>

</tr>

<tr>

Line 204:

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</td>

<td style="text-align: center;">1 1 1 1 1 1 1

</td>

</td>

</tr>

Line 222:

Line 227:

</td>

<td style="text-align: center;">6 6 5 6 6 6 5

</td>

</td>

</tr>

Line 240:

Line 247:

</td>

<td style="text-align: center;">5 5 4 5 5 5 4

</td>

</td>

</tr>

Line 258:

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</td>

<td style="text-align: center;">9 9 7 9 9 9 7

</td>

</td>

</tr>

Line 276:

Line 287:

</td>

<td style="text-align: center;">4 4 3 4 4 4 3

</td>

</td>

</tr>

Line 294:

Line 307:

</td>

<td style="text-align: center;">11 11 8 11 11 11 8

</td>

</td>

</tr>

Line 312:

Line 327:

</td>

<td style="text-align: center;">7 7 5 7 7 7 5

</td>

</td>

</tr>

Line 330:

Line 347:

</td>

<td style="text-align: center;">10 10 7 10 10 10 7

</td>

</td>

</tr>

Line 348:

Line 367:

</td>

<td style="text-align: center;">3 3 2 3 3 3 2

</td>

</td>

</tr>

Line 366:

Line 387:

</td>

<td style="text-align: center;">11 11 7 11 11 11 7

</td>

</td>

</tr>

Line 384:

Line 407:

</td>

<td style="text-align: center;">8 8 5 8 8 8 5

</td>

</td>

</tr>

Line 402:

Line 427:

</td>

<td style="text-align: center;">13 13 8 13 13 13 8

</td>

</td>

</tr>

Line 420:

Line 447:

</td>

<td style="text-align: center;">5 5 3 5 5 5 3

</td>

<td style="text-align: center;">Meantone is in this region

</td>

</tr>

Line 438:

Line 467:

</td>

<td style="text-align: center;">12 12 7 12 12 12 7

</td>

</td>

</tr>

Line 456:

Line 487:

</td>

<td style="text-align: center;">7 7 4 7 7 7 4

</td>

</td>

</tr>

Line 474:

Line 507:

</td>

<td style="text-align: center;">9 9 5 9 9 9 5

</td>

</td>

</tr>

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Line 527:

</td>

<td style="text-align: center;">2 2 1 2 2 2 1

</td>

<td style="text-align: center;">Boundary of propriety (generators

larger than this are proper)

</td>

</tr>

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Line 548:

</td>

<td style="text-align: center;">9 9 4 9 9 9 4

</td>

<td style="text-align: center;">Pythagorean is around here

</td>

</tr>

Line 528:

Line 568:

</td>

<td style="text-align: center;">7 7 3 7 7 7 3

</td>

</td>

</tr>

Line 546:

Line 588:

</td>

<td style="text-align: center;">12 12 5 12 12 12 5

</td>

</td>

</tr>

Line 564:

Line 608:

</td>

<td style="text-align: center;">5 5 2 5 5 5 2

</td>

</td>

</tr>

Line 582:

Line 628:

</td>

<td style="text-align: center;">13 13 5 13 13 13 5

</td>

</td>

</tr>

Line 600:

Line 648:

</td>

<td style="text-align: center;">8 8 3 8 8 8 3

</td>

</td>

</tr>

Line 618:

Line 668:

</td>

<td style="text-align: center;">11 11 4 11 11 11 4

</td>

</td>

</tr>

Line 636:

Line 688:

</td>

<td style="text-align: center;">3 3 1 3 3 3 1

</td>

</td>

</tr>

Line 654:

Line 708:

</td>

<td style="text-align: center;">10 10 3 10 10 10 3

</td>

</td>

</tr>

Line 672:

Line 728:

</td>

<td style="text-align: center;">7 7 2 7 7 7 2

</td>

</td>

</tr>

Line 690:

Line 748:

</td>

<td style="text-align: center;">11 11 3 11 11 11 3

</td>

</td>

</tr>

Line 708:

Line 768:

</td>

<td style="text-align: center;">4 4 1 4 4 4 1

</td>

<td style="text-align: center;">(No-5's) superpyth is in this region

</td>

</tr>

Line 726:

Line 788:

</td>

<td style="text-align: center;">9 9 2 9 9 9 2

</td>

</td>

</tr>

Line 744:

Line 808:

</td>

<td style="text-align: center;">5 5 1 5 5 5 1

</td>

</td>

</tr>

Line 762:

Line 828:

</td>

<td style="text-align: center;">6 6 1 6 6 6 1

</td>

</td>

</tr>

Line 780:

Line 848:

</td>

<td style="text-align: center;">1 1 0 1 1 1 0

</td>

</td>

</tr>

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Temperaments below 5/12 on this chart are called &quot;positive temperaments&quot; 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.

<img src="/file/view/5L2s.jpg/103741463/5L2s.jpg" alt="5L2s.jpg" title="5L2s.jpg" />

<img src="/file/view/5L2s.jpg/103741463/5L2s.jpg" alt="5L2s.jpg" title="5L2s.jpg" />

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></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-10-26 17:04:46 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-04-25 13:21:41 UTC</tt>.<br>
-: The original revision id was <tt>173863945</tt>.<br>
+: The original revision id was <tt>222738126</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>
@@ Line 60: / Line 60: @@
 If we carry this freshman-summing out a little further, new, larger [[edo]]s pop up in our continuum.
-||||||||||||~ generator ||~ in cents ||~ scale in steps of an [[edo]] ||
+||||||||||||~ generator ||~ in cents ||~ scale in steps of an [[edo]] ||=   ||
-||= 3\7 ||=   ||=   ||=   ||=   ||=   ||= 514.286 ||= 1 1 1 1 1 1 1 ||
+||= 3\7 ||=   ||=   ||=   ||=   ||=   ||= 514.286 ||= 1 1 1 1 1 1 1 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 17\40 ||= 510.000 ||= 6 6 5 6 6 6 5 ||
+||=   ||=   ||=   ||=   ||=   ||= 17\40 ||= 510.000 ||= 6 6 5 6 6 6 5 ||=   ||
-||=   ||=   ||=   ||=   ||= 14\33 ||=   ||= 509.091 ||= 5 5 4 5 5 5 4 ||
+||=   ||=   ||=   ||=   ||= 14\33 ||=   ||= 509.091 ||= 5 5 4 5 5 5 4 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 25\59 ||= 508.475 ||= 9 9 7 9 9 9 7 ||
+||=   ||=   ||=   ||=   ||=   ||= 25\59 ||= 508.475 ||= 9 9 7 9 9 9 7 ||=   ||
-||=   ||=   ||=   ||= 11\26 ||=   ||=   ||= 507.692 ||= 4 4 3 4 4 4 3 ||
+||=   ||=   ||=   ||= 11\26 ||=   ||=   ||= 507.692 ||= 4 4 3 4 4 4 3 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 30\71 ||= 507.042 ||= 11 11 8 11 11 11 8 ||
+||=   ||=   ||=   ||=   ||=   ||= 30\71 ||= 507.042 ||= 11 11 8 11 11 11 8 ||=   ||
-||=   ||=   ||=   ||=   ||= 19\45 ||=   ||= 506.667 ||= 7 7 5 7 7 7 5 ||
+||=   ||=   ||=   ||=   ||= 19\45 ||=   ||= 506.667 ||= 7 7 5 7 7 7 5 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 27\64 ||= 506.250 ||= 10 10 7 10 10 10 7 ||
+||=   ||=   ||=   ||=   ||=   ||= 27\64 ||= 506.250 ||= 10 10 7 10 10 10 7 ||=   ||
-||=   ||=   ||= 8\19 ||=   ||=   ||=   ||= 506.263 ||= 3 3 2 3 3 3 2 ||
+||=   ||=   ||= 8\19 ||=   ||=   ||=   ||= 506.263 ||= 3 3 2 3 3 3 2 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 29\69 ||= 504.348 ||= 11 11 7 11 11 11 7 ||
+||=   ||=   ||=   ||=   ||=   ||= 29\69 ||= 504.348 ||= 11 11 7 11 11 11 7 ||=   ||
-||=   ||=   ||=   ||=   ||= 21\50 ||=   ||= 504.000 ||= 8 8 5 8 8 8 5 ||
+||=   ||=   ||=   ||=   ||= 21\50 ||=   ||= 504.000 ||= 8 8 5 8 8 8 5 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 34\81 ||= 503.704 ||= 13 13 8 13 13 13 8 ||
+||=   ||=   ||=   ||=   ||=   ||= 34\81 ||= 503.704 ||= 13 13 8 13 13 13 8 ||=   ||
-||=   ||=   ||=   ||= 13\31 ||=   ||=   ||= 503.226 ||= 5 5 3 5 5 5 3 ||
+||=   ||=   ||=   ||= 13\31 ||=   ||=   ||= 503.226 ||= 5 5 3 5 5 5 3 ||= Meantone is in this region ||
-||=   ||=   ||=   ||=   ||=   ||= 31\74 ||= 502.703 ||= 12 12 7 12 12 12 7 ||
+||=   ||=   ||=   ||=   ||=   ||= 31\74 ||= 502.703 ||= 12 12 7 12 12 12 7 ||=   ||
-||=   ||=   ||=   ||=   ||= 18\43 ||=   ||= 502.326 ||= 7 7 4 7 7 7 4 ||
+||=   ||=   ||=   ||=   ||= 18\43 ||=   ||= 502.326 ||= 7 7 4 7 7 7 4 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 23\55 ||= 501.818 ||= 9 9 5 9 9 9 5 ||
+||=   ||=   ||=   ||=   ||=   ||= 23\55 ||= 501.818 ||= 9 9 5 9 9 9 5 ||=   ||
-||=   ||= 5\12 ||=   ||=   ||=   ||=   ||= 500.000 ||= 2 2 1 2 2 2 1 ||
+||=   ||= 5\12 ||=   ||=   ||=   ||=   ||= 500.000 ||= 2 2 1 2 2 2 1 ||= Boundary of propriety (generators
-||=   ||=   ||=   ||=   ||=   ||= 22\53 ||= 498.113 ||= 9 9 4 9 9 9 4 ||
+larger than this are proper) ||
-||=   ||=   ||=   ||=   ||= 17\41 ||=   ||= 497.591 ||= 7 7 3 7 7 7 3 ||
+||=   ||=   ||=   ||=   ||=   ||= 22\53 ||= 498.113 ||= 9 9 4 9 9 9 4 ||= Pythagorean is around here ||
-||=   ||=   ||=   ||=   ||=   ||= 29\70 ||= 497.143 ||= 12 12 5 12 12 12 5 ||
+||=   ||=   ||=   ||=   ||= 17\41 ||=   ||= 497.591 ||= 7 7 3 7 7 7 3 ||=   ||
-||=   ||=   ||=   ||= 12\29 ||=   ||=   ||= 496.552 ||= 5 5 2 5 5 5 2 ||
+||=   ||=   ||=   ||=   ||=   ||= 29\70 ||= 497.143 ||= 12 12 5 12 12 12 5 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 31\75 ||= 496.000 ||= 13 13 5 13 13 13 5 ||
+||=   ||=   ||=   ||= 12\29 ||=   ||=   ||= 496.552 ||= 5 5 2 5 5 5 2 ||=   ||
-||=   ||=   ||=   ||=   ||= 19\46 ||=   ||= 495.652 ||= 8 8 3 8 8 8 3 ||
+||=   ||=   ||=   ||=   ||=   ||= 31\75 ||= 496.000 ||= 13 13 5 13 13 13 5 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 26\63 ||= 495.238 ||= 11 11 4 11 11 11 4 ||
+||=   ||=   ||=   ||=   ||= 19\46 ||=   ||= 495.652 ||= 8 8 3 8 8 8 3 ||=   ||
-||=   ||=   ||= 7\17 ||=   ||=   ||=   ||= 494.118 ||= 3 3 1 3 3 3 1 ||
+||=   ||=   ||=   ||=   ||=   ||= 26\63 ||= 495.238 ||= 11 11 4 11 11 11 4 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 23\56 ||= 492.857 ||= 10 10 3 10 10 10 3 ||
+||=   ||=   ||= 7\17 ||=   ||=   ||=   ||= 494.118 ||= 3 3 1 3 3 3 1 ||=   ||
-||=   ||=   ||=   ||=   ||= 16\39 ||=   ||= 492.308 ||= 7 7 2 7 7 7 2 ||
+||=   ||=   ||=   ||=   ||=   ||= 23\56 ||= 492.857 ||= 10 10 3 10 10 10 3 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 25\61 ||= 491.803 ||= 11 11 3 11 11 11 3 ||
+||=   ||=   ||=   ||=   ||= 16\39 ||=   ||= 492.308 ||= 7 7 2 7 7 7 2 ||=   ||
-||=   ||=   ||=   ||= 9\22 ||=   ||=   ||= 490.909 ||= 4 4 1 4 4 4 1 ||
+||=   ||=   ||=   ||=   ||=   ||= 25\61 ||= 491.803 ||= 11 11 3 11 11 11 3 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 20\49 ||= 489.796 ||= 9 9 2 9 9 9 2 ||
+||=   ||=   ||=   ||= 9\22 ||=   ||=   ||= 490.909 ||= 4 4 1 4 4 4 1 ||= (No-5's) superpyth is in this region ||
-||=   ||=   ||=   ||=   ||= 11\27 ||=   ||= 488.889 ||= 5 5 1 5 5 5 1 ||
+||=   ||=   ||=   ||=   ||=   ||= 20\49 ||= 489.796 ||= 9 9 2 9 9 9 2 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 13\32 ||= 487.500 ||= 6 6 1 6 6 6 1 ||
+||=   ||=   ||=   ||=   ||= 11\27 ||=   ||= 488.889 ||= 5 5 1 5 5 5 1 ||=   ||
-||= 2\5 ||=   ||=   ||=   ||=   ||=   ||= 480.000 ||= 1 1 0 1 1 1 0 ||
+||=   ||=   ||=   ||=   ||=   ||= 13\32 ||= 487.500 ||= 6 6 1 6 6 6 1 ||=   ||
+||= 2\5 ||=   ||=   ||=   ||=   ||=   ||= 480.000 ||= 1 1 0 1 1 1 0 ||=   ||
 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.
@@ Line 187: / Line 188: @@
          &lt;th&gt;scale in steps of an &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;&lt;br /&gt;
 &lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
@@ Line 204: / Line 207: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;1 1 1 1 1 1 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 222: / Line 227: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;6 6 5 6 6 6 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 240: / Line 247: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5 5 4 5 5 5 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 258: / Line 267: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;9 9 7 9 9 9 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 276: / Line 287: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;4 4 3 4 4 4 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 294: / Line 307: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11 11 8 11 11 11 8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 312: / Line 327: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7 7 5 7 7 7 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 330: / Line 347: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;10 10 7 10 10 10 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 348: / Line 367: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;3 3 2 3 3 3 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 366: / Line 387: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11 11 7 11 11 11 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 384: / Line 407: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;8 8 5 8 8 8 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 402: / Line 427: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;13 13 8 13 13 13 8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 420: / Line 447: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5 5 3 5 5 5 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Meantone is in this region&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 438: / Line 467: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;12 12 7 12 12 12 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 456: / Line 487: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7 7 4 7 7 7 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 474: / Line 507: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;9 9 5 9 9 9 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 492: / Line 527: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;2 2 1 2 2 2 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Boundary of propriety (generators&lt;br /&gt;
+larger than this are proper)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 510: / Line 548: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;9 9 4 9 9 9 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Pythagorean is around here&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 528: / Line 568: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7 7 3 7 7 7 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 546: / Line 588: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;12 12 5 12 12 12 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 564: / Line 608: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5 5 2 5 5 5 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 582: / Line 628: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;13 13 5 13 13 13 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 600: / Line 648: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;8 8 3 8 8 8 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 618: / Line 668: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11 11 4 11 11 11 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 636: / Line 688: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;3 3 1 3 3 3 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 654: / Line 708: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;10 10 3 10 10 10 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 672: / Line 728: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7 7 2 7 7 7 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 690: / Line 748: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11 11 3 11 11 11 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 708: / Line 768: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;4 4 1 4 4 4 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(No-5's) superpyth is in this region&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 726: / Line 788: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;9 9 2 9 9 9 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 744: / Line 808: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5 5 1 5 5 5 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 762: / Line 828: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;6 6 1 6 6 6 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 780: / Line 848: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;1 1 0 1 1 1 0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 789: / Line 859: @@
 Temperaments below 5/12 on this chart are called &amp;quot;positive temperaments&amp;quot; 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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;lt;img src=&amp;quot;/file/view/5L2s.jpg/103741463/5L2s.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/5L2s.jpg/103741463/5L2s.jpg" alt="5L2s.jpg" title="5L2s.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:630 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:698:&amp;lt;img src=&amp;quot;/file/view/5L2s.jpg/103741463/5L2s.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/5L2s.jpg/103741463/5L2s.jpg" alt="5L2s.jpg" title="5L2s.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:698 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 L 2s contains the pentatonic MOS &lt;a class="wiki_link" href="/2L%203s"&gt;2L 3s&lt;/a&gt; and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either &lt;a class="wiki_link" href="/7L%205s"&gt;7L 5s&lt;/a&gt; or &lt;a class="wiki_link" href="/5L%207s"&gt;5L 7s&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>