Slendric: Difference between revisions

Revision as of 19:47, 19 August 2011

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This revision was by author keenanpepper and made on 2011-08-19 19:47:40 UTC.

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Slendric, a member of the [[Gamelismic clan]], has 8/7 as a generator, and three of them make a 3/2. Thus the gamelisma, 1029/1024, is tempered out. Since 1029/1024 is a relatively small comma (8.4 cents), and the error is distributed over several intervals, slendric is quite an accurate temperament (approximating many intervals within 1 or 2 cents).

The disadvantage, if you want to think of it that way, is that approximations to the 5th harmonic do not occur until you go a large number of generators away from the unison. In other words, the 5th harmonic must have a large [[complexity]]. Possible extensions of slendric to the full 7 limit include [[mothra]] (tempering out 81/80) and [[rodan]] (even more complex).

This article concerns the basic 2.3.7 subgroup temperament, slendric itself.

==Interval chains== 
|| 296.81 || 530.50 || 764.19 || 997.88 || 31.56 || 265.25 || 498.94 || 732.63 || 966.31 || 0. || 233.69 || 467.37 || 701.06 || 934.75 || 1168.44 || 202.12 || 435.81 || 669.50 || 903.19 ||
|| 32/27 ||   || 14/9 || 16/9 ||   || 7/6 || 4/3 || 32/21 || 7/4 || 1/1 || 8/7 || 21/16 || 3/2 || 7/4 ||   || 9/8 || 9/7 ||   || 27/16 ||
==MOSes== 
===5-note and 6-note (both proper)=== 
There is a 5-note MOS, Lssss, in which L is 7/6 and s is 8/7; and a 6-note MOS, LLLLLs, in which L is 8/7 and s is the characteristic small interval of slendric representing both 64/63 and 49/48.

Both of these scales are somewhat lacking in harmonic resources relative to similar-sized scales of other temperaments. Even within the 2.3.7 subgroup, [[superpyth]] and [[semaphore]] have pentatonic scales with more consonant intervals and chords; or if more accuracy is desired a 2.3.7 JI scale could be used.

Slendric really shines when used with larger scales than these. The 5-note MOS, however, has a special role in organizing the intervals of slendric because it is so close to [[5edo]] (see below).

===11-note (LsLsLsLsLss, improper)=== 
The 11-note MOS has 9/8 "whole tones" in alternation with ~32 cent "sixth tones", with the exception of one pair of adjacent "sixth tones".

|| Small ("minor") interval || 31.56 || 63.13 || 265.25 || 296.81 || 498.94 || 530.50 || 732.63 || 764.19 || 966.31 || 997.88 ||
|| JI intervals represented ||   ||   || 7/6 || 32/27 || 4/3 ||   || 32/21 || 14/9 || 7/4 || 16/9 ||
|| Large ("major") interval || 202.12 || 233.69 || 435.81 || 467.37 || 669.50 || 701.06 || 903.19 || 934.75 || 1136.87 || 1168.44 ||
|| JI intervals represented || 9/8 || 8/7 || 9/7 || 21/16 ||   || 3/2 || 27/16 || 12/7 ||   ||   ||
==Alternate way of organizing intervals== 
Instead of organizing the intervals according to larger and larger MOSes (none of which are proper until at least 26 notes), the intervals of slendric can be organized according to how many steps of [[5edo]], or equivalently the 5-note MOS, they correspond to. The "major" interval of a class is the one that's just larger than the corresponding 5edo interval, and the "minor" interval is just smaller.
|| Steps of 5edo || 1 || 2 || 3 || 4 ||
|| "Augmented" interval || 296.81 || 530.50 || 764.19 || 997.88 ||
|| JI intervals represented || 32/27 ||   || 14/9 || 16/9 ||
|| "Major" interval || 265.25 || 498.94 || 732.63 || 966.31 ||
|| JI intervals represented || 7/6 || 4/3 || 32/21 || 7/4 ||
|| "Minor" interval || 233.69 || 467.37 || 701.06 || 934.75 ||
|| JI intervals represented || 8/7 || 21/16 || 3/2 || 12/7 ||
|| "Diminished" interval || 202.12 || 435.81 || 669.50 || 903.19 ||
|| JI intervals represented || 9/8 || 9/7 ||   || 27/16 ||

Original HTML content:

<html><head><title>Slendric</title></head><body>Slendric, a member of the <a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a>, has 8/7 as a generator, and three of them make a 3/2. Thus the gamelisma, 1029/1024, is tempered out. Since 1029/1024 is a relatively small comma (8.4 cents), and the error is distributed over several intervals, slendric is quite an accurate temperament (approximating many intervals within 1 or 2 cents).<br />
<br />
The disadvantage, if you want to think of it that way, is that approximations to the 5th harmonic do not occur until you go a large number of generators away from the unison. In other words, the 5th harmonic must have a large <a class="wiki_link" href="/complexity">complexity</a>. Possible extensions of slendric to the full 7 limit include <a class="wiki_link" href="/mothra">mothra</a> (tempering out 81/80) and <a class="wiki_link" href="/rodan">rodan</a> (even more complex).<br />
<br />
This article concerns the basic 2.3.7 subgroup temperament, slendric itself.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Interval chains"></a><!-- ws:end:WikiTextHeadingRule:0 -->Interval chains</h2>
 

<table class="wiki_table">
    <tr>
        <td>296.81<br />
</td>
        <td>530.50<br />
</td>
        <td>764.19<br />
</td>
        <td>997.88<br />
</td>
        <td>31.56<br />
</td>
        <td>265.25<br />
</td>
        <td>498.94<br />
</td>
        <td>732.63<br />
</td>
        <td>966.31<br />
</td>
        <td>0.<br />
</td>
        <td>233.69<br />
</td>
        <td>467.37<br />
</td>
        <td>701.06<br />
</td>
        <td>934.75<br />
</td>
        <td>1168.44<br />
</td>
        <td>202.12<br />
</td>
        <td>435.81<br />
</td>
        <td>669.50<br />
</td>
        <td>903.19<br />
</td>
    </tr>
    <tr>
        <td>32/27<br />
</td>
        <td><br />
</td>
        <td>14/9<br />
</td>
        <td>16/9<br />
</td>
        <td><br />
</td>
        <td>7/6<br />
</td>
        <td>4/3<br />
</td>
        <td>32/21<br />
</td>
        <td>7/4<br />
</td>
        <td>1/1<br />
</td>
        <td>8/7<br />
</td>
        <td>21/16<br />
</td>
        <td>3/2<br />
</td>
        <td>7/4<br />
</td>
        <td><br />
</td>
        <td>9/8<br />
</td>
        <td>9/7<br />
</td>
        <td><br />
</td>
        <td>27/16<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-MOSes"></a><!-- ws:end:WikiTextHeadingRule:2 -->MOSes</h2>
 <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-MOSes-5-note and 6-note (both proper)"></a><!-- ws:end:WikiTextHeadingRule:4 -->5-note and 6-note (both proper)</h3>
 There is a 5-note MOS, Lssss, in which L is 7/6 and s is 8/7; and a 6-note MOS, LLLLLs, in which L is 8/7 and s is the characteristic small interval of slendric representing both 64/63 and 49/48.<br />
<br />
Both of these scales are somewhat lacking in harmonic resources relative to similar-sized scales of other temperaments. Even within the 2.3.7 subgroup, <a class="wiki_link" href="/superpyth">superpyth</a> and <a class="wiki_link" href="/semaphore">semaphore</a> have pentatonic scales with more consonant intervals and chords; or if more accuracy is desired a 2.3.7 JI scale could be used.<br />
<br />
Slendric really shines when used with larger scales than these. The 5-note MOS, however, has a special role in organizing the intervals of slendric because it is so close to <a class="wiki_link" href="/5edo">5edo</a> (see below).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-MOSes-11-note (LsLsLsLsLss, improper)"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-note (LsLsLsLsLss, improper)</h3>
 The 11-note MOS has 9/8 &quot;whole tones&quot; in alternation with ~32 cent &quot;sixth tones&quot;, with the exception of one pair of adjacent &quot;sixth tones&quot;.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Small (&quot;minor&quot;) interval<br />
</td>
        <td>31.56<br />
</td>
        <td>63.13<br />
</td>
        <td>265.25<br />
</td>
        <td>296.81<br />
</td>
        <td>498.94<br />
</td>
        <td>530.50<br />
</td>
        <td>732.63<br />
</td>
        <td>764.19<br />
</td>
        <td>966.31<br />
</td>
        <td>997.88<br />
</td>
    </tr>
    <tr>
        <td>JI intervals represented<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7/6<br />
</td>
        <td>32/27<br />
</td>
        <td>4/3<br />
</td>
        <td><br />
</td>
        <td>32/21<br />
</td>
        <td>14/9<br />
</td>
        <td>7/4<br />
</td>
        <td>16/9<br />
</td>
    </tr>
    <tr>
        <td>Large (&quot;major&quot;) interval<br />
</td>
        <td>202.12<br />
</td>
        <td>233.69<br />
</td>
        <td>435.81<br />
</td>
        <td>467.37<br />
</td>
        <td>669.50<br />
</td>
        <td>701.06<br />
</td>
        <td>903.19<br />
</td>
        <td>934.75<br />
</td>
        <td>1136.87<br />
</td>
        <td>1168.44<br />
</td>
    </tr>
    <tr>
        <td>JI intervals represented<br />
</td>
        <td>9/8<br />
</td>
        <td>8/7<br />
</td>
        <td>9/7<br />
</td>
        <td>21/16<br />
</td>
        <td><br />
</td>
        <td>3/2<br />
</td>
        <td>27/16<br />
</td>
        <td>12/7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Alternate way of organizing intervals"></a><!-- ws:end:WikiTextHeadingRule:8 -->Alternate way of organizing intervals</h2>
 Instead of organizing the intervals according to larger and larger MOSes (none of which are proper until at least 26 notes), the intervals of slendric can be organized according to how many steps of <a class="wiki_link" href="/5edo">5edo</a>, or equivalently the 5-note MOS, they correspond to. The &quot;major&quot; interval of a class is the one that's just larger than the corresponding 5edo interval, and the &quot;minor&quot; interval is just smaller.<br />


<table class="wiki_table">
    <tr>
        <td>Steps of 5edo<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>4<br />
</td>
    </tr>
    <tr>
        <td>&quot;Augmented&quot; interval<br />
</td>
        <td>296.81<br />
</td>
        <td>530.50<br />
</td>
        <td>764.19<br />
</td>
        <td>997.88<br />
</td>
    </tr>
    <tr>
        <td>JI intervals represented<br />
</td>
        <td>32/27<br />
</td>
        <td><br />
</td>
        <td>14/9<br />
</td>
        <td>16/9<br />
</td>
    </tr>
    <tr>
        <td>&quot;Major&quot; interval<br />
</td>
        <td>265.25<br />
</td>
        <td>498.94<br />
</td>
        <td>732.63<br />
</td>
        <td>966.31<br />
</td>
    </tr>
    <tr>
        <td>JI intervals represented<br />
</td>
        <td>7/6<br />
</td>
        <td>4/3<br />
</td>
        <td>32/21<br />
</td>
        <td>7/4<br />
</td>
    </tr>
    <tr>
        <td>&quot;Minor&quot; interval<br />
</td>
        <td>233.69<br />
</td>
        <td>467.37<br />
</td>
        <td>701.06<br />
</td>
        <td>934.75<br />
</td>
    </tr>
    <tr>
        <td>JI intervals represented<br />
</td>
        <td>8/7<br />
</td>
        <td>21/16<br />
</td>
        <td>3/2<br />
</td>
        <td>12/7<br />
</td>
    </tr>
    <tr>
        <td>&quot;Diminished&quot; interval<br />
</td>
        <td>202.12<br />
</td>
        <td>435.81<br />
</td>
        <td>669.50<br />
</td>
        <td>903.19<br />
</td>
    </tr>
    <tr>
        <td>JI intervals represented<br />
</td>
        <td>9/8<br />
</td>
        <td>9/7<br />
</td>
        <td><br />
</td>
        <td>27/16<br />
</td>
    </tr>
</table>

</body></html>

Slendric: Difference between revisions

Revision as of 19:47, 19 August 2011

IMPORTED REVISION FROM WIKISPACES

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