5L 2s: Difference between revisions

Revision as of 16:31, 4 November 2009

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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, while excluding just intonation scales that use more than one size of "tone".

==substituting step sizes== 

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

==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.

|| 3\7 ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 17\40 ||
||   ||   ||   ||   || 14\33 ||   ||
||   ||   ||   ||   ||   || 25\59 ||
||   ||   ||   || 11\26 ||   ||   ||
||   ||   ||   ||   ||   || 30\71 ||
||   ||   ||   ||   || 19\45 ||   ||
||   ||   ||   ||   ||   || 27\63 ||
||   ||   || 8\19 ||   ||   ||   ||
||   ||   ||   ||   ||   || 29\69 ||
||   ||   ||   ||   || 21\50 ||   ||
||   ||   ||   ||   ||   || 34\81 ||
||   ||   ||   || 13\31 ||   ||   ||
||   ||   ||   ||   ||   || 31\74 ||
||   ||   ||   ||   || 18\43 ||   ||
||   ||   ||   ||   ||   || 23\55 ||
||   || 5\12 ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 22\53 ||
||   ||   ||   ||   || 17\41 ||   ||
||   ||   ||   ||   ||   || 29\60 ||
||   ||   ||   || 12\29 ||   ||   ||
||   ||   ||   ||   ||   || 31\75 ||
||   ||   ||   ||   || 19\46 ||   ||
||   ||   ||   ||   ||   || 26\63 ||
||   ||   || 7\17 ||   ||   ||   ||
||   ||   ||   ||   ||   || 23\56 ||
||   ||   ||   ||   || 16\39 ||   ||
||   ||   ||   ||   ||   || 25\61 ||
||   ||   ||   || 9\22 ||   ||   ||
||   ||   ||   ||   ||   || 28\49 ||
||   ||   ||   ||   || 11\27 ||   ||
||   ||   ||   ||   ||   || 19\32 ||
|| 2\5 ||   ||   ||   ||   ||   ||

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.

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, while excluding just intonation scales that use more than one size of &quot;tone&quot;.<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 />
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 />
<!-- 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>
        <td>3\7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>17\40<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>14\33<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>25\59<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11\26<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>30\71<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>19\45<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>27\63<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>8\19<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>29\69<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>21\50<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>34\81<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\31<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>31\74<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>18\43<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>23\55<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5\12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>22\53<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>17\41<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>29\60<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>12\29<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>31\75<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>19\46<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>26\63<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>23\56<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>16\39<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>25\61<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9\22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>28\49<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11\27<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>19\32<br />
</td>
    </tr>
    <tr>
        <td>2\5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><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.</body></html>

5L 2s: Difference between revisions

Revision as of 16:31, 4 November 2009

IMPORTED REVISION FROM WIKISPACES

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