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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | =Magic Tetrachords!= |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-05-09 16:18:41 UTC</tt>.<br>
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| : The original revision id was <tt>507845610</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Magic Tetrachords!=
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| Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], [[Magic]] [[Tetrachord]]s! | | Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], [[Magic|Magic]] [[tetrachord|Tetrachord]]s! |
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| 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. | | Magic[7] has structure [[3L_4s|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|MODMOS]] with 4/3 and 3/2. |
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| | Eg. Take 5|1 (see [[Modal_UDP_Notation|Modal UDP Notation]]). We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1. |
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| Eg. Take 5|1. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.
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| 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.’ | | 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.’ |
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| Taking now 6|0, we have LsLsLss: 1/1 6/5 5/4 3/2 14/9 15/8 27/14 2/1. | | Taking now 6|0, we have LsLsLss: 1/1 6/5 5/4 3/2 14/9 15/8 27/14 2/1. |
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| 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. | | 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. |
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| 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. | | 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. |
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| ==Table of Results:== | | ==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 || | | {| class="wikitable" |
| || 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 || | | ! | [[Modal_UDP_Notation|UDP Notation]] |
| || 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 || | | ! | Steps |
| || 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 || | | ! | Ratios |
| | ! | Lower [[tetrachord|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 |
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| | | | 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 |
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| | | | 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 |
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| | | | 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 |
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| | | | 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. | | Any of the 7 modes of each of these scales can of course be used. |
| Tunings of [[19edo]], [[22edo]] and [[41edo]] are encouraged.</pre></div>
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| <h4>Original HTML content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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>
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| 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 />
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| <br />
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| 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 />
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| Eg. Take 5|1. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.<br />
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| 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 />
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| 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 />
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| Taking now 6|0, we have LsLsLss: 1/1 6/5 5/4 3/2 14/9 15/8 27/14 2/1.<br />
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| 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 />
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| 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 />
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| 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 />
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| 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 />
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| <br />
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| 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 />
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| <!-- 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>
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| <table class="wiki_table">
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| <tr>
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| <th>UDP Notation<br />
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| </th>
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| <th>Steps<br />
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| </th>
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| <th>Ratios<br />
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| </th>
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| <th>Lower Tetrachord<br />
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| </th>
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| <th>Upper Tetrachord<br />
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| </th>
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| </tr>
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| <tr>
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| <td>5|1 b5<br />
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| </td>
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| <td>LsLdLLs<br />
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| </td>
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| <td>1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1<br />
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| </td>
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| <td>Chromatic: 1/1 6/5 5/4 4/3<br />
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| </td>
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| <td>Chromatic: 1/1 16/15 9/7 4/3<br />
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| </td>
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| </tr>
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| <tr>
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| <td>1|5 #4<br />
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| </td>
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| <td>sLLdLsL<br />
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| </td>
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| <td>1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1<br />
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| </td>
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| <td>Chromatic: 1/1 28/27 5/4 4/3<br />
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| </td>
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| <td>Chromatic: 1/1 16/15 10/9 4/3<br />
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| </td>
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| </tr>
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| <tr>
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| <td>6|0 b5<br />
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| </td>
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| <td>LsLdAss<br />
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| </td>
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| <td>1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1<br />
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| </td>
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| <td>Chromatic: 1/1 6/5 5/4 4/3<br />
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| </td>
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| <td>Enharmonic: 1/1 5/4 9/7 4/3<br />
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| </td>
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| </tr>
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| <tr>
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| <td>0|6 #4<br />
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| </td>
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| <td>ssAdLsL<br />
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| </td>
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| <td>1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1<br />
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| </td>
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| <td>Enharmonic: 1/1 28/27 16/15 4/3<br />
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| </td>
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| <td>Chromatic: 1/1 16/15 10/9 4/3<br />
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| </td>
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| </tr>
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| <tr>
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| <td>3|3 #4 b5<br />
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| </td>
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| <td>sLLdLLs<br />
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| </td>
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| <td>1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1<br />
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| </td>
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| <td>Chromatic: 1/1 28/27 5/4 4/3<br />
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| </td>
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| <td>Chromatic: 1/1 16/15 9/7 2/1<br />
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| </td>
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| </tr>
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| </table>
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| Any of the 7 modes of each of these scales can of course be used.<br />
| | Tunings of [[19edo|19edo]], [[22edo|22edo]] and [[41edo|41edo]] are encouraged. |
| 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></pre></div> | | [[Category:magic]] |
| | [[Category:MODMOS]] |
| | [[Category:Tetrachords]] |