5L 5s

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There is only one significant harmonic entropy minimum with this MOS pattern: [[Archytas clan|blackwood]], in which intervals of the prime numbers 3 and 7 are all represented using steps of [[5edo]], and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5.

The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being [[Rothenberg propriety|proper]] (because 1\15 is in the middle of the range of good blackwood generators).
||||||||||~ Generator ||~ Cents ||~ Comments ||
|| 0\5 ||   ||   ||   ||   || 0 ||=   ||
||   ||   ||   ||   || 1\30 || 40 ||   ||
||   ||   ||   || 1\25 ||   || 48 ||   ||
||   ||   ||   ||   ||   || 240/(1+3*2^(1/75)) ||   ||
||   ||   || 1\20 ||   ||   || 60 ||=   ||
||   ||   ||   ||   ||   || 240/(1+3/2^(1/75)) ||   ||
||   || 1\15 ||   ||   ||   || 80 ||= Blackwood is around here
Optimum rank range (L/s=2/1) for MOS ||
||   ||   ||   || 3\40 ||   || 90 ||=   ||
||   ||   ||   ||   || 5\65 || 92.31 ||= Golden blackwood ||
||   ||   || 2\25 ||   ||   || 96 ||=   ||
|| 1\10 ||   ||   ||   ||   || 120 ||=   ||

Original HTML content:

<html><head><title>5L 5s</title></head><body>There is only one significant harmonic entropy minimum with this MOS pattern: <a class="wiki_link" href="/Archytas%20clan">blackwood</a>, in which intervals of the prime numbers 3 and 7 are all represented using steps of <a class="wiki_link" href="/5edo">5edo</a>, and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5.<br />
<br />
The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being <a class="wiki_link" href="/Rothenberg%20propriety">proper</a> (because 1\15 is in the middle of the range of good blackwood generators).<br />


<table class="wiki_table">
    <tr>
        <th colspan="5">Generator<br />
</th>
        <th>Cents<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>0\5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\30<br />
</td>
        <td>40<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\25<br />
</td>
        <td><br />
</td>
        <td>48<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>240/(1+3*2^(1/75))<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\20<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>60<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>240/(1+3/2^(1/75))<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>1\15<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>80<br />
</td>
        <td style="text-align: center;">Blackwood is around here<br />
Optimum rank range (L/s=2/1) for MOS<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3\40<br />
</td>
        <td><br />
</td>
        <td>90<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\65<br />
</td>
        <td>92.31<br />
</td>
        <td style="text-align: center;">Golden blackwood<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>2\25<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>96<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td>1\10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>120<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

</body></html>
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