Magic Tetrachords - Revision history

Fredg999 category edits: Moving from Category:Tetrachord to Category:Tetrachords using Cat-a-lot

2022-06-08T17:05:36Z

Moving from Category:Tetrachord to Category:Tetrachords using Cat-a-lot

← Older revision		Revision as of 17:05, 8 June 2022
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	[[Category:magic]]		[[Category:magic]]
	[[Category:MODMOS]]		[[Category:MODMOS]]
	[[Category:~~tetrachord~~]]		[[Category:Tetrachords]]

Keenan Pepper: Clean up categories

2021-04-24T19:38:41Z

Clean up categories

← Older revision		Revision as of 19:38, 24 April 2021
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	Tunings of [[19edo\|19edo]], [[22edo\|22edo]] and [[41edo\|41edo]] are encouraged.		Tunings of [[19edo\|19edo]], [[22edo\|22edo]] and [[41edo\|41edo]] are encouraged.
	[[Category:magic]]		[[Category:magic]]
	[[Category:~~modmos]]~~		[[Category:MODMOS]]
	~~[[Category:mos~~]]
	[[Category:tetrachord]]		[[Category:tetrachord]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

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Wikispaces>hearneg: Imported revision 508247528 - Original comment:

2014-05-12T04:54:27Z

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← Older revision		Revision as of 04:54, 12 May 2014
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	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:hearneg\|hearneg]] and made on <tt>2014-05-12 04:49:16 UTC</tt>.<br>		: This revision was by author [[User:hearneg\|hearneg]] and made on <tt>2014-05-12 04:54:27 UTC</tt>.<br>
	: The original revision id was <tt>~~508246936~~</tt>.<br>		: The original revision id was <tt>508247528</tt>.<br>
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	==Table of Results:==		==Table of Results:==
	\|\|~ UDP Notation \|\|~ Steps \|\|~ Ratios \|\|~ Lower Tetrachord \|\|~ Upper Tetrachord \|\|		\|\|~ [[Modal UDP Notation\|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 \|\|		\|\| 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 \|\|		\|\| 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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	<table class="wiki_table">		<table class="wiki_table">
	<tr>		<tr>
	<th>UDP Notation<br />		<th><a class="wiki_link" href="/Modal%20UDP%20Notation">UDP Notation</a><br />
	</th>		</th>
	<th>Steps<br />		<th>Steps<br />
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	<th>Ratios<br />		<th>Ratios<br />
	</th>		</th>
	<th>Lower Tetrachord<br />		<th>Lower <a class="wiki_link" href="/Tetrachord">Tetrachord</a><br />
	</th>		</th>
	<th>Upper Tetrachord<br />		<th>Upper Tetrachord<br />

Wikispaces>hearneg: Imported revision 508246936 - Original comment:

2014-05-12T04:49:16Z

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Wikispaces>hearneg: Imported revision 507845610 - Original comment:

2014-05-09T16:18:41Z

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Wikispaces>hearneg: Imported revision 507843428 - Original comment:

2014-05-09T16:05:09Z

**Imported revision 507843428 - Original comment: **

← Older revision		Revision as of 16:05, 9 May 2014
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:hearneg\|hearneg]] and made on <tt>2014-05-09 15:49:28 UTC</tt>.<br>		: This revision was by author [[User:hearneg\|hearneg]] and made on <tt>2014-05-09 16:05:09 UTC</tt>.<br>
	: The original revision id was <tt>~~507840372~~</tt>.<br>		: The original revision id was <tt>507843428</tt>.<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.		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.		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.		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.
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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.		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, 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.

	==Table of Results:==		==Table of Results:==
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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 \|\|		\|\| 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 \|\|		\|\| 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 \|\|</pre></div>		\|\| 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.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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>		<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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	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 />		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 />		<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 />		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 />		<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 />		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 />		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 />		<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, but that is outside the scope of this article, and is left as an exercise for the reader.<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 />		<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>		<!-- 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>		</table>

	</body></html></pre></div>		Any of the 7 modes of each of these scales can of course be used.<br />
			Tunings of 19edo, 22edo and 41edo are encouraged.</body></html></pre></div>

Wikispaces>hearneg: Imported revision 507840372 - Original comment:

2014-05-09T15:49:28Z

**Imported revision 507840372 - Original comment: **

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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-05-09 15:49:28 UTC</tt>.<br>
: The original revision id was <tt>507840372</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>
<h4>Original Wikitext content:</h4>
<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!=

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

Magic[7] has structure 3L4s, 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) and the augmented seventh 9/4. 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. 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, 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 ||</pre></div>
<h4>Original HTML content:</h4>
<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><h1 id="toc0"><a name="Magic Tetrachords!"></a>Magic Tetrachords!</h1>
<br />
Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], Magic Tetrachords!<br />
<br />
Magic[7] has structure 3L4s, 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) and the augmented seventh 9/4. From any mode of Magic[7] with 4/3 or 3/2 can be constructed a 1-MODMOS with 4/3 and 3/2.<br />
<br />
Eg. Take 5|1. 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, but that is outside the scope of this article, and is left as an exercise for the reader.<br />
<br />
<h2 id="toc1"><a name="Magic Tetrachords!-Table of Results:"></a>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>

</body></html></pre></div>

← Older revision		Revision as of 04:49, 12 May 2014
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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>		: This revision was by author [[User:hearneg\|hearneg]] and made on <tt>2014-05-12 04:49:16 UTC</tt>.<br>
	: The original revision id was <tt>~~507845610~~</tt>.<br>		: The original revision id was <tt>508246936</tt>.<br>
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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]], 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. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.		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.’		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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	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 />		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 />		<br />
	Eg. Take 5\|1. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.<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 />		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 />		<br />

← Older revision		Revision as of 16:18, 9 May 2014
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:hearneg\|hearneg]] and made on <tt>2014-05-09 16:05:09 UTC</tt>.<br>		: This revision was by author [[User:hearneg\|hearneg]] and made on <tt>2014-05-09 16:18:41 UTC</tt>.<br>
	: The original revision id was <tt>~~507843428~~</tt>.<br>		: The original revision id was <tt>507845610</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>		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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	<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!=		<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!=

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

	Magic[7] has structure ~~3L4s~~, 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) ~~and the augmented seventh 9/4~~. 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]], 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. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.		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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	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 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.		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:==		==Table of Results:==
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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 \|\|		\|\| 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>		Tunings of [[19edo]], [[22edo]] and [[41edo]] are encouraged.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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>		<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>
	<br />		<br />
	Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], Magic ~~Tetrachords~~!<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 />		<br />
	Magic[7] has structure ~~3L4s~~, 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) ~~and the augmented seventh 9/4~~. From any mode of Magic[7] with 4/3 or 3/2 can be constructed a 1-MODMOS with 4/3 and 3/2.<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 />		<br />
	Eg. Take 5\|1. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.<br />		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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	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 />		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 />		<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 />		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 />		<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>		<!-- 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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	Any of the 7 modes of each of these scales can of course be used.<br />		Any of the 7 modes of each of these scales can of course be used.<br />
	Tunings of 19edo, 22edo and 41edo are encouraged.</body></html></pre></div>		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>

Magic Tetrachords - Revision history

Fredg999 category edits: Moving from Category:Tetrachord to Category:Tetrachords using Cat-a-lot

Keenan Pepper: Clean up categories

Wikispaces>FREEZE at 00:00, 17 July 2018

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