2ed13/10: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~Andrew_Heathwaite~~|~~Andrew_Heathwaite~~]] and made on <tt>2012-01-28 02:~~48:09~~ UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-28 13:02:46 UTC</tt>.

: The original revision id was <tt>~~296113608~~</tt>.

: The original revision id was <tt>296175356</tt>.

: The revision comment was: <tt></tt>

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

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Sqrt 13:10 (the scale) contains no octaves, and also no close approximation of the third harmonic (the perfect fifth). However, it comes very close to certain just intervals involving the numbers 5, 7, 11 and 13: in particular: 8/7, 13/10, 22/13, 11/5, 5/2, 20/7, 13/4, 26/7, 11/2, and 44/7. These near-just intervals can be combined so as to make available a set of 20 harmonic and subharmonic chords.

From (8/7)^2 ~ 13/10, we can conclude 640/637 is tempered out; from 22/13 ~ (13/10)^2 that 2200/2197 is tempered out; and from 5/2 ~ (8/7)*(11/5) 176/175 is tempered out. The other approximations follow from these, suggesting either the rank three temperament tempering them out, or the rank two no-threes 2.5.7.11.13 subgroup temperament. That has a mapping [<1 1 3 1 2|, <0 7 -1 13 9|], and could be called the no-threes 16&21 temperament. It can be extended to the full 13-limit as 21&37.

[[image:sqrt13_10_harmonic_contents.jpg]]

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Sqrt 13:10 (the scale) contains no octaves, and also no close approximation of the third harmonic (the perfect fifth). However, it comes very close to certain just intervals involving the numbers 5, 7, 11 and 13: in particular: 8/7, 13/10, 22/13, 11/5, 5/2, 20/7, 13/4, 26/7, 11/2, and 44/7. These near-just intervals can be combined so as to make available a set of 20 harmonic and subharmonic chords.

From (8/7)^2 ~ 13/10, we can conclude 640/637 is tempered out; from 22/13 ~ (13/10)^2 that 2200/2197 is tempered out; and from 5/2 ~ (8/7)*(11/5) 176/175 is tempered out. The other approximations follow from these, suggesting either the rank three temperament tempering them out, or the rank two no-threes 2.5.7.11.13 subgroup temperament. That has a mapping [&lt;1 1 3 1 2|, &lt;0 7 -1 13 9|], and could be called the no-threes 16&amp;21 temperament. It can be extended to the full 13-limit as 21&amp;37.

<img src="/file/view/sqrt13_10_harmonic_contents.jpg/111024861/sqrt13_10_harmonic_contents.jpg" alt="sqrt13_10_harmonic_contents.jpg" title="sqrt13_10_harmonic_contents.jpg" />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-01-28 02:48:09 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-28 13:02:46 UTC</tt>.<br>
-: The original revision id was <tt>296113608</tt>.<br>
+: The original revision id was <tt>296175356</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 23: / Line 23: @@
 Sqrt 13:10 (the scale) contains no octaves, and also no close approximation of the third harmonic (the perfect fifth). However, it comes very close to certain just intervals involving the numbers 5, 7, 11 and 13: in particular: 8/7, 13/10, 22/13, 11/5, 5/2, 20/7, 13/4, 26/7, 11/2, and 44/7. These near-just intervals can be combined so as to make available a set of 20 harmonic and subharmonic chords.
+From (8/7)^2 ~ 13/10, we can conclude 640/637 is tempered out; from 22/13 ~ (13/10)^2 that 2200/2197 is tempered out; and from 5/2 ~ (8/7)*(11/5) 176/175 is tempered out. The other approximations follow from these, suggesting either the rank three temperament tempering them out, or the rank two no-threes 2.5.7.11.13 subgroup temperament. That has a mapping [&lt;1 1 3 1 2|, &lt;0 7 -1 13 9|], and could be called the no-threes 16&amp;21 temperament. It can be extended to the full 13-limit as 21&amp;37.
 [[image:sqrt13_10_harmonic_contents.jpg]]
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   &lt;br /&gt;
 Sqrt 13:10 (the scale) contains no octaves, and also no close approximation of the third harmonic (the perfect fifth). However, it comes very close to certain just intervals involving the numbers 5, 7, 11 and 13: in particular: 8/7, 13/10, 22/13, 11/5, 5/2, 20/7, 13/4, 26/7, 11/2, and 44/7. These near-just intervals can be combined so as to make available a set of 20 harmonic and subharmonic chords.&lt;br /&gt;
+&lt;br /&gt;
+From (8/7)^2 ~ 13/10, we can conclude 640/637 is tempered out; from 22/13 ~ (13/10)^2 that 2200/2197 is tempered out; and from 5/2 ~ (8/7)*(11/5) 176/175 is tempered out. The other approximations follow from these, suggesting either the rank three temperament tempering them out, or the rank two no-threes 2.5.7.11.13 subgroup temperament. That has a mapping [&amp;lt;1 1 3 1 2|, &amp;lt;0 7 -1 13 9|], and could be called the no-threes 16&amp;amp;21 temperament. It can be extended to the full 13-limit as 21&amp;amp;37.&lt;br /&gt;
 &lt;br /&gt;
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