2ed13/10: Difference between revisions
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Wikispaces>Andrew_Heathwaite **Imported revision 296113608 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 296175356 - Original comment: ** |
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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: | : 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> | : The original revision id was <tt>296175356</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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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. | 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]] | [[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.<br /> | 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.<br /> | ||
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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.<br /> | |||
<br /> | <br /> | ||
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Revision as of 13:02, 28 January 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-01-28 13:02:46 UTC.
- The original revision id was 296175356.
- 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:
=13:10= [[13_10|13:10]], as a frequency ratio, measures approximately 454.2 cents. It lies in the extremely xenharmonic and ambiguous territory between the perceptual category of a "third" and that of a "fourth" (see [[interseptimal]]). It appears in the [[OverToneSeries|overtone series]] between the tenth and thirteen overtones. =sqrt 13:10 as an interval= The "square root of 13:10", then, means an interval which logarithmically bisects 13:10. It's an irrational number which measures, in cents, about 227.1. =sqrt 13:10 as a scale= "The square root of 13:10" can also refer to the scale that is produced by repeatedly stacking the interval "the square root of 13:10". It is an [[EDONOI]], or "equal division of a nonoctave interval," and as such, it does not contain a perfect octave (2:1). It is also [[macrotonal]], since the smallest step, at 227.1 cents, is larger than a semitone. In terms of [[EDO|equal divisions of the octave]], it fits between [[5edo]] and [[6edo]] (about 5.284edo). Melodically, it can sound somewhat "pentatonic," but harmonically it is very different. ==harmonic content== 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]] === === ==Music== [[http://soundcloud.com/andrew_heathwaite/truffles|Truffles?]] by Andrew Heathwaite and Michael Gaiuranos ===Scala file=== ! sqrt13over10.scl The square root of 13:10 nonoctave temperament. 2 ! 227.106973952238 13/10
Original HTML content:
<html><head><title>square root of 13 over 10</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x13:10"></a><!-- ws:end:WikiTextHeadingRule:0 -->13:10</h1> <br /> <a class="wiki_link" href="/13_10">13:10</a>, as a frequency ratio, measures approximately 454.2 cents. It lies in the extremely xenharmonic and ambiguous territory between the perceptual category of a "third" and that of a "fourth" (see <a class="wiki_link" href="/interseptimal">interseptimal</a>). It appears in the <a class="wiki_link" href="/OverToneSeries">overtone series</a> between the tenth and thirteen overtones.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="sqrt 13:10 as an interval"></a><!-- ws:end:WikiTextHeadingRule:2 -->sqrt 13:10 as an interval</h1> <br /> The "square root of 13:10", then, means an interval which logarithmically bisects 13:10. It's an irrational number which measures, in cents, about 227.1.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="sqrt 13:10 as a scale"></a><!-- ws:end:WikiTextHeadingRule:4 -->sqrt 13:10 as a scale</h1> <br /> "The square root of 13:10" can also refer to the scale that is produced by repeatedly stacking the interval "the square root of 13:10". It is an <a class="wiki_link" href="/EDONOI">EDONOI</a>, or "equal division of a nonoctave interval," and as such, it does not contain a perfect octave (2:1). It is also <a class="wiki_link" href="/macrotonal">macrotonal</a>, since the smallest step, at 227.1 cents, is larger than a semitone.<br /> <br /> In terms of <a class="wiki_link" href="/EDO">equal divisions of the octave</a>, it fits between <a class="wiki_link" href="/5edo">5edo</a> and <a class="wiki_link" href="/6edo">6edo</a> (about 5.284edo). Melodically, it can sound somewhat "pentatonic," but harmonically it is very different.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="sqrt 13:10 as a scale-harmonic content"></a><!-- ws:end:WikiTextHeadingRule:6 -->harmonic content</h2> <br /> 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.<br /> <br /> 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.<br /> <br /> <!-- ws:start:WikiTextLocalImageRule:14:<img src="/file/view/sqrt13_10_harmonic_contents.jpg/111024861/sqrt13_10_harmonic_contents.jpg" alt="" title="" /> --><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" /><!-- ws:end:WikiTextLocalImageRule:14 --><br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><!-- ws:end:WikiTextHeadingRule:8 --> </h3> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="sqrt 13:10 as a scale-Music"></a><!-- ws:end:WikiTextHeadingRule:10 -->Music</h2> <br /> <a class="wiki_link_ext" href="http://soundcloud.com/andrew_heathwaite/truffles" rel="nofollow">Truffles?</a> by Andrew Heathwaite and Michael Gaiuranos<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="sqrt 13:10 as a scale-Music-Scala file"></a><!-- ws:end:WikiTextHeadingRule:12 -->Scala file</h3> <br /> ! sqrt13over10.scl<br /> The square root of 13:10 nonoctave temperament.<br /> 2<br /> !<br /> 227.106973952238<br /> 13/10</body></html>