5L 2s
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Original Wikitext content:
=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, while excluding just intonation scales that use more than one size of "tone". This produces a generalized diatonic scale with the 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 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. || 3\7 || || || || || || || || || || || || || || || || || || 14\33 || || || || || || || || || || || || || 11\26 || || || || || || || || || || || || || || || 19\35 || || || || || || || || || || || || 8\19 || || || || || || || || || || || || || || || || 21\50 || || || || || || || || || || || || || 13\31 || || || || || || || || || || || || || || || 18\43 || || || || || || || || || || || 5\12 || || || || || || || || || || || || || || || || || 17\41 || || || || || || || || || || || || || 12\29 || || || || || || || || || || || || || || || 19\46 || || || || || || || || || || || || 7\17 || || || || || || || || || || || || || || || || 16\39 || || || || || || || || || || || || || 9\22 || || || || || || || || || || || || || || || 11\27 || || || || || || || || || || 2\5 || || || || || ||
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, while excluding just intonation scales that use more than one size of "tone".<br />
<br />
This produces a generalized diatonic scale with the 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 />
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 />
<br />
<table class="wiki_table">
<tr>
<td>3\7<br />
</td>
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<td>14\33<br />
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<td>11\26<br />
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<td>19\35<br />
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<td>8\19<br />
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<td>21\50<br />
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<td>13\31<br />
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<td>18\43<br />
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<td>5\12<br />
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<td>17\41<br />
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<td>12\29<br />
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<td>19\46<br />
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<td>7\17<br />
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<td>16\39<br />
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<td>9\22<br />
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<td>11\27<br />
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<td>2\5<br />
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</table>
</body></html>