Magic Tetrachords

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author hearneg and made on 2014-05-12 04:49:16 UTC.

The original revision id was 508246936.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

=Magic Tetrachords!= 

Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], [[Magic]] [[Tetrachord]]s!

Magic[7] has structure [[3L 4s]], with a large step of 6/5 and a small step of 25/24~28/27~36/35. The chroma is therefore 7/6, so the diminished second is 8/9 (so like, a tone in the wrong direction. Weird hey). From any mode of Magic[7] with 4/3 or 3/2 can be constructed a 1-[[MODMOS]] with 4/3 and 3/2.

Eg. Take 5|1 (see [[Modal UDP Notation]]). We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.
We obtain 3/2 as the fourth of the scale, rather than the fifth, and we want 4/3 before it. If we lower the minor fifth, 14/9, by a chroma, we obtain a diminished fifth of 4/3, where the interval in between the major fourth and the diminished fifth is a diminished second of 8/9! We now have 5|1 b5, LsLdLLs: 1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1. So we have an out of order tetrachordal scale, of chromatic genus. Why not just play the notes in order of pitch? Then, as if by 'magic', we have a tetrachordal scale! The two tetrachords are 1/1 6/5 5/4 4/3 and 1/1 16/15 9/7 4/3. Since the tetrachords are not the same our tetrachordal scale is classified as ‘mixed.’

Noting that 1|5 – sLssLsL – is the inversion of 5|1, we deduce that by raising the major fourth, 9/7, by the chroma, resulting in an augmented fourth of 3/2, we obtain the inversion of our first tetrachordal scale, 1|5 #4, sLLdLsL: 1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1, with out of order tetrachords 1/1 28/27 5/4 4/3 and 1/1 16/15 10/9 4/3.

Taking now 6|0, we have LsLsLss: 1/1 6/5 5/4 3/2 14/9 15/8 27/14 2/1.
This time lowering the minor fifth, 14/9, by a chroma, to obtain a diminished fifth of 4/3, we now have 6|0 b5, LsLdAss (where A is 7/5~45/32): 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1. Our out of order, mixed tetrachords are now of different genus, the upper, 1/1 5/4 9/7 4/3 of enharmonic genus and the lower our now familiar chromatic: 1/1 6/5 5/4 4/3.

It follows as before that from 0|6 we obtain the inverse, 0|6 #4, ssAdLsL: 1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1, with an enharmonic lower tetrachord of 1/1 28/27 16/15 4/3 and an upper chromatic tetrachord of 1/1 16/15 10/9 4/3.

Looking now at modes of Magic[7] that do not contain 4/3 or 3/2: From 4|2 and 2|4 2-MODMOS lead us to the same scales we obtained from 5|1 and 1|5 respectively, so that is of little interest.

From 3|3 however, both raising the major fourth to 3/2 and lowering the minor fifth to 4/3 leads us to 3|3 #4 b5, sLLdLLs: 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1, a symmetrical scale with chromatic tetrachords of 1/1 28/27 5/4 4/3 and 1/1 16/15 9/7 2/1, inversions of each other.

From the 1-MODMOS above, 2-MODMOS can be obtained with two diminished seconds of 8/9, and one diminished fourth of 10/9, of one diatonic and one chromatic tetrachord, moving us another step closer to the Zarlino-Ptolemy diatonic scale, and from other 1-MODMOS, other, non-diatonic tetrachordal scales can be constructed, but that is outside the scope of this article and is left as an exercise for the reader.

==Table of Results:== 
||~ UDP Notation ||~ Steps ||~ Ratios ||~ Lower Tetrachord ||~ Upper Tetrachord ||
|| 5|1 b5 || LsLdLLs || 1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1 || Chromatic: 1/1 6/5 5/4 4/3 || Chromatic: 1/1 16/15 9/7 4/3 ||
|| 1|5 #4 || sLLdLsL || 1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1 || Chromatic: 1/1 28/27 5/4 4/3 || Chromatic: 1/1 16/15 10/9 4/3 ||
|| 6|0 b5 || LsLdAss || 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1 || Chromatic: 1/1 6/5 5/4 4/3 || Enharmonic: 1/1 5/4 9/7 4/3 ||
|| 0|6 #4 || ssAdLsL || 1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1 || Enharmonic: 1/1 28/27 16/15 4/3 || Chromatic: 1/1 16/15 10/9 4/3 ||
|| 3|3 #4 b5 || sLLdLLs || 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1 || Chromatic: 1/1 28/27 5/4 4/3 || Chromatic: 1/1 16/15 9/7 2/1 ||
Any of the 7 modes of each of these scales can of course be used.
Tunings of [[19edo]], [[22edo]] and [[41edo]] are encouraged.

Original HTML content:

<html><head><title>Magic Tetrachords</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Magic Tetrachords!"></a><!-- ws:end:WikiTextHeadingRule:0 -->Magic Tetrachords!</h1>
 <br />
Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], <a class="wiki_link" href="/Magic">Magic</a> <a class="wiki_link" href="/Tetrachord">Tetrachord</a>s!<br />
<br />
Magic[7] has structure <a class="wiki_link" href="/3L%204s">3L 4s</a>, with a large step of 6/5 and a small step of 25/24~28/27~36/35. The chroma is therefore 7/6, so the diminished second is 8/9 (so like, a tone in the wrong direction. Weird hey). From any mode of Magic[7] with 4/3 or 3/2 can be constructed a 1-<a class="wiki_link" href="/MODMOS">MODMOS</a> with 4/3 and 3/2.<br />
<br />
Eg. Take 5|1 (see <a class="wiki_link" href="/Modal%20UDP%20Notation">Modal UDP Notation</a>). We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.<br />
We obtain 3/2 as the fourth of the scale, rather than the fifth, and we want 4/3 before it. If we lower the minor fifth, 14/9, by a chroma, we obtain a diminished fifth of 4/3, where the interval in between the major fourth and the diminished fifth is a diminished second of 8/9! We now have 5|1 b5, LsLdLLs: 1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1. So we have an out of order tetrachordal scale, of chromatic genus. Why not just play the notes in order of pitch? Then, as if by 'magic', we have a tetrachordal scale! The two tetrachords are 1/1 6/5 5/4 4/3 and 1/1 16/15 9/7 4/3. Since the tetrachords are not the same our tetrachordal scale is classified as ‘mixed.’<br />
<br />
Noting that 1|5 – sLssLsL – is the inversion of 5|1, we deduce that by raising the major fourth, 9/7, by the chroma, resulting in an augmented fourth of 3/2, we obtain the inversion of our first tetrachordal scale, 1|5 #4, sLLdLsL: 1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1, with out of order tetrachords 1/1 28/27 5/4 4/3 and 1/1 16/15 10/9 4/3.<br />
<br />
Taking now 6|0, we have LsLsLss: 1/1 6/5 5/4 3/2 14/9 15/8 27/14 2/1.<br />
This time lowering the minor fifth, 14/9, by a chroma, to obtain a diminished fifth of 4/3, we now have 6|0 b5, LsLdAss (where A is 7/5~45/32): 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1. Our out of order, mixed tetrachords are now of different genus, the upper, 1/1 5/4 9/7 4/3 of enharmonic genus and the lower our now familiar chromatic: 1/1 6/5 5/4 4/3.<br />
<br />
It follows as before that from 0|6 we obtain the inverse, 0|6 #4, ssAdLsL: 1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1, with an enharmonic lower tetrachord of 1/1 28/27 16/15 4/3 and an upper chromatic tetrachord of 1/1 16/15 10/9 4/3.<br />
<br />
Looking now at modes of Magic[7] that do not contain 4/3 or 3/2: From 4|2 and 2|4 2-MODMOS lead us to the same scales we obtained from 5|1 and 1|5 respectively, so that is of little interest.<br />
<br />
From 3|3 however, both raising the major fourth to 3/2 and lowering the minor fifth to 4/3 leads us to 3|3 #4 b5, sLLdLLs: 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1, a symmetrical scale with chromatic tetrachords of 1/1 28/27 5/4 4/3 and 1/1 16/15 9/7 2/1, inversions of each other.<br />
<br />
From the 1-MODMOS above, 2-MODMOS can be obtained with two diminished seconds of 8/9, and one diminished fourth of 10/9, of one diatonic and one chromatic tetrachord, moving us another step closer to the Zarlino-Ptolemy diatonic scale, and from other 1-MODMOS, other, non-diatonic tetrachordal scales can be constructed, but that is outside the scope of this article and is left as an exercise for the reader.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Magic Tetrachords!-Table of Results:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Table of Results:</h2>
 

<table class="wiki_table">
    <tr>
        <th>UDP Notation<br />
</th>
        <th>Steps<br />
</th>
        <th>Ratios<br />
</th>
        <th>Lower Tetrachord<br />
</th>
        <th>Upper Tetrachord<br />
</th>
    </tr>
    <tr>
        <td>5|1 b5<br />
</td>
        <td>LsLdLLs<br />
</td>
        <td>1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1<br />
</td>
        <td>Chromatic: 1/1 6/5 5/4 4/3<br />
</td>
        <td>Chromatic: 1/1 16/15 9/7 4/3<br />
</td>
    </tr>
    <tr>
        <td>1|5 #4<br />
</td>
        <td>sLLdLsL<br />
</td>
        <td>1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1<br />
</td>
        <td>Chromatic: 1/1 28/27 5/4 4/3<br />
</td>
        <td>Chromatic: 1/1 16/15 10/9 4/3<br />
</td>
    </tr>
    <tr>
        <td>6|0 b5<br />
</td>
        <td>LsLdAss<br />
</td>
        <td>1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1<br />
</td>
        <td>Chromatic: 1/1 6/5 5/4 4/3<br />
</td>
        <td>Enharmonic: 1/1 5/4 9/7 4/3<br />
</td>
    </tr>
    <tr>
        <td>0|6 #4<br />
</td>
        <td>ssAdLsL<br />
</td>
        <td>1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1<br />
</td>
        <td>Enharmonic: 1/1 28/27 16/15 4/3<br />
</td>
        <td>Chromatic: 1/1 16/15 10/9 4/3<br />
</td>
    </tr>
    <tr>
        <td>3|3 #4 b5<br />
</td>
        <td>sLLdLLs<br />
</td>
        <td>1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1<br />
</td>
        <td>Chromatic: 1/1 28/27 5/4 4/3<br />
</td>
        <td>Chromatic: 1/1 16/15 9/7 2/1<br />
</td>
    </tr>
</table>

Any of the 7 modes of each of these scales can of course be used.<br />
Tunings of <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/22edo">22edo</a> and <a class="wiki_link" href="/41edo">41edo</a> are encouraged.</body></html>