5L 2s: Difference between revisions

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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 ||~ scale in steps of an [[edo]] ||~   ||~   ||~   ||~   ||~ comments ||
||= 3\7 ||=   ||=   ||=   ||=   ||=   ||= 514.286 ||= 1 1 1 1 1 1 1 || 239.2945 || 274.991 || 307.521 || 378.193 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 17\40 ||= 510.000 ||= 6 6 5 6 6 6 5 || 237.152 || 272.848 || 309.664 || 380.336 ||=   ||
||=   ||=   ||=   ||=   ||= 14\33 ||=   ||= 509.091 ||= 5 5 4 5 5 5 4 || 236.697 || 272.394 || 310.118 || 380.791 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 25\59 ||= 508.475 ||= 9 9 7 9 9 9 7 || 236.389 || 272.086 || 310.4265 || 381.0985 ||=   ||
||=   ||=   ||=   ||= 11\26 ||=   ||=   ||= 507.692 ||= 4 4 3 4 4 4 3 || 235.998 || 271.695 || 310.817 || 381.491 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 30\71 ||= 507.042 ||= 11 11 8 11 11 11 8 || 235.672 || 271.3695 || 311.142 || 381.846 ||=   ||
||=   ||=   ||=   ||=   ||= 19\45 ||=   ||= 506.667 ||= 7 7 5 7 7 7 5 || 235.485 || 271.182 || 311.33 || 382.003 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 27\64 ||= 506.250 ||= 10 10 7 10 10 10 7 || 235.277 || 270.973 || 311.539 || 382.211 ||=   ||
||=   ||=   ||= 8\19 ||=   ||=   ||=   ||= 505.263 ||= 3 3 2 3 3 3 2 ||   ||   || 312.032 || 382.705 ||= Optimum rank range (L/s=3/2) diatonic ||
||=   ||=   ||=   ||=   ||=   ||= 29\69 ||= 504.348 ||= 11 11 7 11 11 11 7 ||   ||   || 312.490 || 383.172 ||=   ||
||=   ||=   ||=   ||=   ||= 21\50 ||=   ||= 504.000 ||= 8 8 5 8 8 8 5 ||   ||   || 312.664 || 383.336 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 34\81 ||= 503.704 ||= 13 13 8 13 13 13 8 ||   ||   || 312.811 || 383.485 ||= Golden meantone ||
||=   ||=   ||=   ||= 13\31 ||=   ||=   ||= 503.226 ||= 5 5 3 5 5 5 3 ||   ||   || 313.051 || 383.723 ||= Meantone is in this region ||
||=   ||=   ||=   ||=   ||=   ||= 31\74 ||= 502.703 ||= 12 12 7 12 12 12 7 ||   ||   || 313.312 || 383.985 ||=   ||
||=   ||=   ||=   ||=   ||= 18\43 ||=   ||= 502.326 ||= 7 7 4 7 7 7 4 ||   ||   || 313.501 || 384.183 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 23\55 ||= 501.818 ||= 9 9 5 9 9 9 5 ||   ||   || 313.754 || 384.428 ||=   ||
||=   ||= 5\12 ||=   ||=   ||=   ||=   ||= 500.000 ||= 2 2 1 2 2 2 1 ||   ||   || 314.664 || 385.336 ||= Boundary of propriety (generators
larger than this are proper) ||
||=   ||=   ||=   ||=   ||=   ||= 22\53 ||= 498.113 ||= 9 9 4 9 9 9 4 ||   ||   || 315.609 || 386.278 ||= Pythagorean is around here ||
||=   ||=   ||=   ||=   ||= 17\41 ||=   ||= 497.591 ||= 7 7 3 7 7 7 3 ||   ||   || 315.883 || 386.556 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 29\70 ||= 497.143 ||= 12 12 5 12 12 12 5 ||   ||   || 316.092 || 386.765 ||=   ||
||=   ||=   ||=   ||= 12\29 ||=   ||=   ||= 496.552 ||= 5 5 2 5 5 5 2 ||   ||   || 316.388 || 387.061 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 31\75 ||= 496.000 ||= 13 13 5 13 13 13 5 ||   ||   || 316.664 || 387.336 ||=   ||
||=   ||=   ||=   ||=   ||= 19\46 ||=   ||= 495.652 ||= 8 8 3 8 8 8 3 ||   ||   || 316.837 || 387.511 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 26\63 ||= 495.238 ||= 11 11 4 11 11 11 4 ||   ||   || 317.045 || 387.717 ||=   ||
||=   ||=   ||= 7\17 ||=   ||=   ||=   ||= 494.118 ||= 3 3 1 3 3 3 1 ||   ||   || 317.596 || 388.286 ||= L/s = 3 ||
||=   ||=   ||=   ||=   ||=   ||= 23\56 ||= 492.857 ||= 10 10 3 10 10 10 3 ||   ||   || 318.235 || 388.908 ||=   ||
||=   ||=   ||=   ||=   ||= 16\39 ||=   ||= 492.308 ||= 7 7 2 7 7 7 2 ||   ||   || 318.51 || 389.182 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 25\61 ||= 491.803 ||= 11 11 3 11 11 11 3 ||   ||   || 318.761 || 389.436 ||=   ||
||=   ||=   ||=   ||= 9\22 ||=   ||=   ||= 490.909 ||= 4 4 1 4 4 4 1 ||   ||   || 319.209 || 389.882 ||= (No-5's) superpyth is in this region
L/s = 4 ||
||=   ||=   ||=   ||=   ||=   ||= 20\49 ||= 489.796 ||= 9 9 2 9 9 9 2 ||   ||   || 319.766 || 390.438 ||=   ||
||=   ||=   ||=   ||=   ||= 11\27 ||=   ||= 488.889 ||= 5 5 1 5 5 5 1 ||   ||   || 320.219 || 390.892 ||=   ||
||=   ||=   ||=   ||=   ||=   ||= 13\32 ||= 487.500 ||= 6 6 1 6 6 6 1 ||   ||   || 320.914 || 391.596 ||=   ||
||= 2\5 ||=   ||=   ||=   ||=   ||=   ||= 480.000 ||= 1 1 0 1 1 1 0 ||   ||   || 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:&lt;h1&gt; --><h1 id="toc0"><a name="x5L 2s - &quot;diatonic&quot;"></a><!-- ws:end:WikiTextHeadingRule:0 -->5L 2s - &quot;diatonic&quot;</h1>
 <br />
One way of distinguishing the &quot;diatonic&quot; scale is by considering it a <a class="wiki_link" href="/MOSScales">moment of symmetry</a> scale produced by a chain of &quot;fifths&quot;. 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 &quot;tone&quot;.<br />
<br />
It may be misleading to call 5L 2s &quot;diatonic,&quot; 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:&lt;h2&gt; --><h2 id="toc1"><a name="x5L 2s - &quot;diatonic&quot;-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:&lt;h2&gt; --><h2 id="toc2"><a name="x5L 2s - &quot;diatonic&quot;-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 &quot;freshman sums&quot; 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 />
<br />


<table class="wiki_table">
    <tr>
        <th colspan="6">generator<br />
</th>
        <th>in cents<br />
</th>
        <th>scale in steps of an <a class="wiki_link" href="/edo">edo</a><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th>comments<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">3\7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">514.286<br />
</td>
        <td style="text-align: center;">1 1 1 1 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 style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">17\40<br />
</td>
        <td style="text-align: center;">510.000<br />
</td>
        <td style="text-align: center;">6 6 5 6 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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">14\33<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">509.091<br />
</td>
        <td style="text-align: center;">5 5 4 5 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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">25\59<br />
</td>
        <td style="text-align: center;">508.475<br />
</td>
        <td style="text-align: center;">9 9 7 9 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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">11\26<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">507.692<br />
</td>
        <td style="text-align: center;">4 4 3 4 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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">30\71<br />
</td>
        <td style="text-align: center;">507.042<br />
</td>
        <td style="text-align: center;">11 11 8 11 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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">19\45<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">506.667<br />
</td>
        <td style="text-align: center;">7 7 5 7 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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">27\64<br />
</td>
        <td style="text-align: center;">506.250<br />
</td>
        <td style="text-align: center;">10 10 7 10 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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">8\19<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">505.263<br />
</td>
        <td style="text-align: center;">3 3 2 3 3 3 2<br />
</td>
        <td><br />
</td>
        <td><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 style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">29\69<br />
</td>
        <td style="text-align: center;">504.348<br />
</td>
        <td style="text-align: center;">11 11 7 11 11 11 7<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">21\50<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">504.000<br />
</td>
        <td style="text-align: center;">8 8 5 8 8 8 5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>312.664<br />
</td>
        <td>383.336<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">34\81<br />
</td>
        <td style="text-align: center;">503.704<br />
</td>
        <td style="text-align: center;">13 13 8 13 13 13 8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>312.811<br />
</td>
        <td>383.485<br />
</td>
        <td style="text-align: center;">Golden meantone<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">13\31<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">503.226<br />
</td>
        <td style="text-align: center;">5 5 3 5 5 5 3<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">31\74<br />
</td>
        <td style="text-align: center;">502.703<br />
</td>
        <td style="text-align: center;">12 12 7 12 12 12 7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>313.312<br />
</td>
        <td>383.985<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">18\43<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">502.326<br />
</td>
        <td style="text-align: center;">7 7 4 7 7 7 4<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">23\55<br />
</td>
        <td style="text-align: center;">501.818<br />
</td>
        <td style="text-align: center;">9 9 5 9 9 9 5<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;">5\12<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">500.000<br />
</td>
        <td style="text-align: center;">2 2 1 2 2 2 1<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>314.664<br />
</td>
        <td>385.336<br />
</td>
        <td style="text-align: center;">Boundary of propriety (generators<br />
larger than this are proper)<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">22\53<br />
</td>
        <td style="text-align: center;">498.113<br />
</td>
        <td style="text-align: center;">9 9 4 9 9 9 4<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">17\41<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">497.591<br />
</td>
        <td style="text-align: center;">7 7 3 7 7 7 3<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">29\70<br />
</td>
        <td style="text-align: center;">497.143<br />
</td>
        <td style="text-align: center;">12 12 5 12 12 12 5<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">12\29<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">496.552<br />
</td>
        <td style="text-align: center;">5 5 2 5 5 5 2<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">31\75<br />
</td>
        <td style="text-align: center;">496.000<br />
</td>
        <td style="text-align: center;">13 13 5 13 13 13 5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>316.664<br />
</td>
        <td>387.336<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">19\46<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">495.652<br />
</td>
        <td style="text-align: center;">8 8 3 8 8 8 3<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">26\63<br />
</td>
        <td style="text-align: center;">495.238<br />
</td>
        <td style="text-align: center;">11 11 4 11 11 11 4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>317.045<br />
</td>
        <td>387.717<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">7\17<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">494.118<br />
</td>
        <td style="text-align: center;">3 3 1 3 3 3 1<br />
</td>
        <td><br />
</td>
        <td><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 style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">23\56<br />
</td>
        <td style="text-align: center;">492.857<br />
</td>
        <td style="text-align: center;">10 10 3 10 10 10 3<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">16\39<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">492.308<br />
</td>
        <td style="text-align: center;">7 7 2 7 7 7 2<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">25\61<br />
</td>
        <td style="text-align: center;">491.803<br />
</td>
        <td style="text-align: center;">11 11 3 11 11 11 3<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">9\22<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">490.909<br />
</td>
        <td style="text-align: center;">4 4 1 4 4 4 1<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">20\49<br />
</td>
        <td style="text-align: center;">489.796<br />
</td>
        <td style="text-align: center;">9 9 2 9 9 9 2<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">11\27<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">488.889<br />
</td>
        <td style="text-align: center;">5 5 1 5 5 5 1<br />
</td>
        <td><br />
</td>
        <td><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: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">13\32<br />
</td>
        <td style="text-align: center;">487.500<br />
</td>
        <td style="text-align: center;">6 6 1 6 6 6 1<br />
</td>
        <td><br />
</td>
        <td><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: center;">2\5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">480.000<br />
</td>
        <td style="text-align: center;">1 1 0 1 1 1 0<br />
</td>
        <td><br />
</td>
        <td><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 &quot;negative temperaments&quot; (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 &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.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:970:&lt;img src=&quot;/file/view/5L2s.jpg/103741463/5L2s.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/5L2s.jpg/103741463/5L2s.jpg" alt="5L2s.jpg" title="5L2s.jpg" /><!-- ws:end:WikiTextLocalImageRule:970 --><br />
<br />
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>

@@ 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:JosephRuhf|JosephRuhf]] and made on <tt>2014-05-13 16:43:45 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2014-06-04 14:39:51 UTC</tt>.<br>
-: The original revision id was <tt>508685194</tt>.<br>
+: The original revision id was <tt>512838698</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]] ||~   ||~   ||~ comments ||
+||||||||||||~ generator ||~ in cents ||~ scale in steps of an [[edo]] ||~   ||~   ||~   ||~   ||~ comments ||
-||= 3\7 ||=   ||=   ||=   ||=   ||=   ||= 514.286 ||= 1 1 1 1 1 1 1 || 307.521 || 378.193 ||=   ||
+||= 3\7 ||=   ||=   ||=   ||=   ||=   ||= 514.286 ||= 1 1 1 1 1 1 1 || 239.2945 || 274.991 || 307.521 || 378.193 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 17\40 ||= 510.000 ||= 6 6 5 6 6 6 5 || 309.664 || 380.336 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 17\40 ||= 510.000 ||= 6 6 5 6 6 6 5 || 237.152 || 272.848 || 309.664 || 380.336 ||=   ||
-||=   ||=   ||=   ||=   ||= 14\33 ||=   ||= 509.091 ||= 5 5 4 5 5 5 4 || 310.118 || 380.791 ||=   ||
+||=   ||=   ||=   ||=   ||= 14\33 ||=   ||= 509.091 ||= 5 5 4 5 5 5 4 || 236.697 || 272.394 || 310.118 || 380.791 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 25\59 ||= 508.475 ||= 9 9 7 9 9 9 7 || 310.4265 || 381.0485 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 25\59 ||= 508.475 ||= 9 9 7 9 9 9 7 || 236.389 || 272.086 || 310.4265 || 381.0985 ||=   ||
-||=   ||=   ||=   ||= 11\26 ||=   ||=   ||= 507.692 ||= 4 4 3 4 4 4 3 || 310.817 || 381.491 ||=   ||
+||=   ||=   ||=   ||= 11\26 ||=   ||=   ||= 507.692 ||= 4 4 3 4 4 4 3 || 235.998 || 271.695 || 310.817 || 381.491 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 30\71 ||= 507.042 ||= 11 11 8 11 11 11 8 || 311.142 || 381.846 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 30\71 ||= 507.042 ||= 11 11 8 11 11 11 8 || 235.672 || 271.3695 || 311.142 || 381.846 ||=   ||
-||=   ||=   ||=   ||=   ||= 19\45 ||=   ||= 506.667 ||= 7 7 5 7 7 7 5 || 311.33 || 382.003 ||=   ||
+||=   ||=   ||=   ||=   ||= 19\45 ||=   ||= 506.667 ||= 7 7 5 7 7 7 5 || 235.485 || 271.182 || 311.33 || 382.003 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 27\64 ||= 506.250 ||= 10 10 7 10 10 10 7 || 311.539 || 382.211 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 27\64 ||= 506.250 ||= 10 10 7 10 10 10 7 || 235.277 || 270.973 || 311.539 || 382.211 ||=   ||
-||=   ||=   ||= 8\19 ||=   ||=   ||=   ||= 505.263 ||= 3 3 2 3 3 3 2 || 312.032 || 382.705 ||= Optimum rank range (L/s=3/2) diatonic ||
+||=   ||=   ||= 8\19 ||=   ||=   ||=   ||= 505.263 ||= 3 3 2 3 3 3 2 ||   ||   || 312.032 || 382.705 ||= Optimum rank range (L/s=3/2) diatonic ||
-||=   ||=   ||=   ||=   ||=   ||= 29\69 ||= 504.348 ||= 11 11 7 11 11 11 7 || 312.490 || 383.172 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 29\69 ||= 504.348 ||= 11 11 7 11 11 11 7 ||   ||   || 312.490 || 383.172 ||=   ||
-||=   ||=   ||=   ||=   ||= 21\50 ||=   ||= 504.000 ||= 8 8 5 8 8 8 5 || 312.664 || 383.336 ||=   ||
+||=   ||=   ||=   ||=   ||= 21\50 ||=   ||= 504.000 ||= 8 8 5 8 8 8 5 ||   ||   || 312.664 || 383.336 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 34\81 ||= 503.704 ||= 13 13 8 13 13 13 8 || 312.811 || 383.485 ||= Golden meantone ||
+||=   ||=   ||=   ||=   ||=   ||= 34\81 ||= 503.704 ||= 13 13 8 13 13 13 8 ||   ||   || 312.811 || 383.485 ||= Golden meantone ||
-||=   ||=   ||=   ||= 13\31 ||=   ||=   ||= 503.226 ||= 5 5 3 5 5 5 3 || 313.051 || 383.723 ||= Meantone is in this region ||
+||=   ||=   ||=   ||= 13\31 ||=   ||=   ||= 503.226 ||= 5 5 3 5 5 5 3 ||   ||   || 313.051 || 383.723 ||= Meantone is in this region ||
-||=   ||=   ||=   ||=   ||=   ||= 31\74 ||= 502.703 ||= 12 12 7 12 12 12 7 || 313.312 || 383.985 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 31\74 ||= 502.703 ||= 12 12 7 12 12 12 7 ||   ||   || 313.312 || 383.985 ||=   ||
-||=   ||=   ||=   ||=   ||= 18\43 ||=   ||= 502.326 ||= 7 7 4 7 7 7 4 || 313.501 || 384.183 ||=   ||
+||=   ||=   ||=   ||=   ||= 18\43 ||=   ||= 502.326 ||= 7 7 4 7 7 7 4 ||   ||   || 313.501 || 384.183 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 23\55 ||= 501.818 ||= 9 9 5 9 9 9 5 || 313.754 || 384.428 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 23\55 ||= 501.818 ||= 9 9 5 9 9 9 5 ||   ||   || 313.754 || 384.428 ||=   ||
-||=   ||= 5\12 ||=   ||=   ||=   ||=   ||= 500.000 ||= 2 2 1 2 2 2 1 || 314.664 || 385.336 ||= Boundary of propriety (generators
+||=   ||= 5\12 ||=   ||=   ||=   ||=   ||= 500.000 ||= 2 2 1 2 2 2 1 ||   ||   || 314.664 || 385.336 ||= Boundary of propriety (generators
 larger than this are proper) ||
-||=   ||=   ||=   ||=   ||=   ||= 22\53 ||= 498.113 ||= 9 9 4 9 9 9 4 || 315.609 || 386.278 ||= Pythagorean is around here ||
+||=   ||=   ||=   ||=   ||=   ||= 22\53 ||= 498.113 ||= 9 9 4 9 9 9 4 ||   ||   || 315.609 || 386.278 ||= Pythagorean is around here ||
-||=   ||=   ||=   ||=   ||= 17\41 ||=   ||= 497.591 ||= 7 7 3 7 7 7 3 || 315.883 || 386.556 ||=   ||
+||=   ||=   ||=   ||=   ||= 17\41 ||=   ||= 497.591 ||= 7 7 3 7 7 7 3 ||   ||   || 315.883 || 386.556 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 29\70 ||= 497.143 ||= 12 12 5 12 12 12 5 || 316.092 || 386.765 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 29\70 ||= 497.143 ||= 12 12 5 12 12 12 5 ||   ||   || 316.092 || 386.765 ||=   ||
-||=   ||=   ||=   ||= 12\29 ||=   ||=   ||= 496.552 ||= 5 5 2 5 5 5 2 || 316.388 || 387.061 ||=   ||
+||=   ||=   ||=   ||= 12\29 ||=   ||=   ||= 496.552 ||= 5 5 2 5 5 5 2 ||   ||   || 316.388 || 387.061 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 31\75 ||= 496.000 ||= 13 13 5 13 13 13 5 || 316.664 || 387.336 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 31\75 ||= 496.000 ||= 13 13 5 13 13 13 5 ||   ||   || 316.664 || 387.336 ||=   ||
-||=   ||=   ||=   ||=   ||= 19\46 ||=   ||= 495.652 ||= 8 8 3 8 8 8 3 || 316.837 || 387.511 ||=   ||
+||=   ||=   ||=   ||=   ||= 19\46 ||=   ||= 495.652 ||= 8 8 3 8 8 8 3 ||   ||   || 316.837 || 387.511 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 26\63 ||= 495.238 ||= 11 11 4 11 11 11 4 || 317.045 || 387.717 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 26\63 ||= 495.238 ||= 11 11 4 11 11 11 4 ||   ||   || 317.045 || 387.717 ||=   ||
-||=   ||=   ||= 7\17 ||=   ||=   ||=   ||= 494.118 ||= 3 3 1 3 3 3 1 || 317.596 || 388.286 ||= L/s = 3 ||
+||=   ||=   ||= 7\17 ||=   ||=   ||=   ||= 494.118 ||= 3 3 1 3 3 3 1 ||   ||   || 317.596 || 388.286 ||= L/s = 3 ||
-||=   ||=   ||=   ||=   ||=   ||= 23\56 ||= 492.857 ||= 10 10 3 10 10 10 3 || 318.235 || 388.908 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 23\56 ||= 492.857 ||= 10 10 3 10 10 10 3 ||   ||   || 318.235 || 388.908 ||=   ||
-||=   ||=   ||=   ||=   ||= 16\39 ||=   ||= 492.308 ||= 7 7 2 7 7 7 2 || 318.51 || 389.182 ||=   ||
+||=   ||=   ||=   ||=   ||= 16\39 ||=   ||= 492.308 ||= 7 7 2 7 7 7 2 ||   ||   || 318.51 || 389.182 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 25\61 ||= 491.803 ||= 11 11 3 11 11 11 3 || 318.761 || 389.436 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 25\61 ||= 491.803 ||= 11 11 3 11 11 11 3 ||   ||   || 318.761 || 389.436 ||=   ||
-||=   ||=   ||=   ||= 9\22 ||=   ||=   ||= 490.909 ||= 4 4 1 4 4 4 1 || 319.209 || 389.882 ||= (No-5's) superpyth is in this region
+||=   ||=   ||=   ||= 9\22 ||=   ||=   ||= 490.909 ||= 4 4 1 4 4 4 1 ||   ||   || 319.209 || 389.882 ||= (No-5's) superpyth is in this region
 L/s = 4 ||
-||=   ||=   ||=   ||=   ||=   ||= 20\49 ||= 489.796 ||= 9 9 2 9 9 9 2 || 319.766 || 390.438 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 20\49 ||= 489.796 ||= 9 9 2 9 9 9 2 ||   ||   || 319.766 || 390.438 ||=   ||
-||=   ||=   ||=   ||=   ||= 11\27 ||=   ||= 488.889 ||= 5 5 1 5 5 5 1 || 320.219 || 390.892 ||=   ||
+||=   ||=   ||=   ||=   ||= 11\27 ||=   ||= 488.889 ||= 5 5 1 5 5 5 1 ||   ||   || 320.219 || 390.892 ||=   ||
-||=   ||=   ||=   ||=   ||=   ||= 13\32 ||= 487.500 ||= 6 6 1 6 6 6 1 || 320.914 || 391.596 ||=   ||
+||=   ||=   ||=   ||=   ||=   ||= 13\32 ||= 487.500 ||= 6 6 1 6 6 6 1 ||   ||   || 320.914 || 391.596 ||=   ||
-||= 2\5 ||=   ||=   ||=   ||=   ||=   ||= 480.000 ||= 1 1 0 1 1 1 0 || 324.664 || 395.336 ||=   ||
+||= 2\5 ||=   ||=   ||=   ||=   ||=   ||= 480.000 ||= 1 1 0 1 1 1 0 ||   ||   || 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.
@@ Line 188: / Line 188: @@
 &lt;/th&gt;
          &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;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;&lt;br /&gt;
@@ Line 212: / Line 216: @@
 &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&gt;239.2945&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;274.991&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;307.521&lt;br /&gt;
@@ Line 236: / Line 244: @@
 &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&gt;237.152&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;272.848&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;309.664&lt;br /&gt;
@@ Line 260: / Line 272: @@
 &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&gt;236.697&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;272.394&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;310.118&lt;br /&gt;
@@ Line 284: / Line 300: @@
 &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&gt;236.389&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;272.086&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;310.4265&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;381.0485&lt;br /&gt;
+         &lt;td&gt;381.0985&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 308: / Line 328: @@
 &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&gt;235.998&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;271.695&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;310.817&lt;br /&gt;
@@ Line 332: / Line 356: @@
 &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&gt;235.672&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;271.3695&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;311.142&lt;br /&gt;
@@ Line 356: / Line 384: @@
 &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&gt;235.485&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;271.182&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;311.33&lt;br /&gt;
@@ Line 380: / Line 412: @@
 &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&gt;235.277&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;270.973&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;311.539&lt;br /&gt;
@@ Line 404: / Line 440: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;312.032&lt;br /&gt;
@@ Line 428: / Line 468: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;312.490&lt;br /&gt;
@@ Line 452: / Line 496: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;312.664&lt;br /&gt;
@@ Line 476: / Line 524: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;312.811&lt;br /&gt;
@@ Line 500: / Line 552: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;313.051&lt;br /&gt;
@@ Line 524: / Line 580: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;313.312&lt;br /&gt;
@@ Line 548: / Line 608: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;313.501&lt;br /&gt;
@@ Line 572: / Line 636: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;313.754&lt;br /&gt;
@@ Line 596: / Line 664: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;314.664&lt;br /&gt;
@@ Line 621: / Line 693: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;315.609&lt;br /&gt;
@@ Line 645: / Line 721: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;315.883&lt;br /&gt;
@@ Line 669: / Line 749: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;316.092&lt;br /&gt;
@@ Line 693: / Line 777: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;316.388&lt;br /&gt;
@@ Line 717: / Line 805: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;316.664&lt;br /&gt;
@@ Line 741: / Line 833: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;316.837&lt;br /&gt;
@@ Line 765: / Line 861: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;317.045&lt;br /&gt;
@@ Line 789: / Line 889: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;317.596&lt;br /&gt;
@@ Line 813: / Line 917: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;318.235&lt;br /&gt;
@@ Line 837: / Line 945: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;318.51&lt;br /&gt;
@@ Line 861: / Line 973: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;318.761&lt;br /&gt;
@@ Line 885: / Line 1,001: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;319.209&lt;br /&gt;
@@ Line 910: / Line 1,030: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;319.766&lt;br /&gt;
@@ Line 934: / Line 1,058: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;320.219&lt;br /&gt;
@@ Line 958: / Line 1,086: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;320.914&lt;br /&gt;
@@ Line 982: / Line 1,114: @@
 &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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;324.664&lt;br /&gt;
@@ Line 997: / Line 1,133: @@
 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:834:&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:834 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:970:&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:970 --&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>

5L 2s: Difference between revisions

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