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Non radical intervals with musical significance: Difference between revisions

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Wikispaces>Sarzadoce
**Imported revision 586762787 - Original comment: **
Wikispaces>Sarzadoce
**Imported revision 586762957 - 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:Sarzadoce|Sarzadoce]] and made on <tt>2016-07-10 19:55:26 UTC</tt>.<br>
: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2016-07-10 20:01:53 UTC</tt>.<br>
: The original revision id was <tt>586762787</tt>.<br>
: The original revision id was <tt>586762957</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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<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">Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.
<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">Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.


There are many non-radical intervals which have musical significance. By **non-radical** is meant a number that cannot be written in the form
There are many non-radical intervals which have musical significance. By **non-radical** is meant a number that cannot be written in the form [[math]] a^{1/b} [[math]], where a and b are integers. What follows is a list of musically significant non-radical intervals.
&lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span class="math" style="display: inline-block; width: 1.94em;"&gt;&lt;span style="clip: rect(1.04em,1000em,2.38em,-0.54em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.17em; width: 1.6em;"&gt;&lt;span class="mrow"&gt;&lt;span class="msubsup"&gt;&lt;span style="display: inline-block; height: 0px; position: relative; width: 1.61em;"&gt;&lt;span style="clip: rect(1.86em,1000em,2.7em,-0.54em); left: 0em; position: absolute; top: -2.5em;"&gt;&lt;span class="mi" style="font-family: MathJax_Math;"&gt;//a//&lt;/span&gt;&lt;/span&gt;&lt;span style="left: 0.51em; position: absolute; top: -2.72em;"&gt;&lt;span class="texatom"&gt;&lt;span class="mrow"&gt;&lt;span class="mn" style="font-family: MathJax_Main; font-size: 70.7%;"&gt;1&lt;/span&gt;&lt;span class="texatom"&gt;&lt;span class="mrow"&gt;&lt;span class="mo" style="font-family: MathJax_Main; font-size: 70.7%;"&gt;/&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mi" style="font-family: MathJax_Math; font-size: 70.7%;"&gt;//b//&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
, where a and b are integers. What follows is a list of musically significant non-radical intervals.


|| **Ratio** || **Cents** || **Name** || **Musical Significance** ||
|| **Ratio** || **Cents** || **Name** || **Musical Significance** ||
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[[math]] || 833.09 || [[math]]
[[math]] || 833.09 || [[math]]
\text{Phi } (\phi)
\text{Phi } (\phi)
[[math]] || "Linear phi," the unique interval whose continued fraction convergents ||
[[math]] || "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [[https://en.wikipedia.org/wiki/Diophantine_approximation#General_upper_bound|Dirichlet's Approximation Theorem]]. ||
|| [[math]]
|| [[math]]
e \approx 2.7183
e \approx 2.7183
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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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Non radical intervals with musical significance&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.&lt;br /&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Non radical intervals with musical significance&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are many non-radical intervals which have musical significance. By &lt;strong&gt;non-radical&lt;/strong&gt; is meant a number that cannot be written in the form&lt;br /&gt;
There are many non-radical intervals which have musical significance. By &lt;strong&gt;non-radical&lt;/strong&gt; is meant a number that cannot be written in the form &lt;a class="wiki_link" href="/math"&gt;math&lt;/a&gt; a^{1/b} &lt;a class="wiki_link" href="/math"&gt;math&lt;/a&gt;, where a and b are integers. What follows is a list of musically significant non-radical intervals.&lt;br /&gt;
&lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span style="display: inline-block; width: 1.94em;" class="math"&gt;&lt;span style="clip: rect(1.04em,1000em,2.38em,-0.54em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.17em; width: 1.6em;"&gt;&lt;span class="mrow"&gt;&lt;span class="msubsup"&gt;&lt;span style="display: inline-block; height: 0px; position: relative; width: 1.61em;"&gt;&lt;span style="clip: rect(1.86em,1000em,2.7em,-0.54em); left: 0em; position: absolute; top: -2.5em;"&gt;&lt;span style="font-family: MathJax_Math;" class="mi"&gt;&lt;em&gt;a&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="left: 0.51em; position: absolute; top: -2.72em;"&gt;&lt;span class="texatom"&gt;&lt;span class="mrow"&gt;&lt;span style="font-family: MathJax_Main; font-size: 70.7%;" class="mn"&gt;1&lt;/span&gt;&lt;span class="texatom"&gt;&lt;span class="mrow"&gt;&lt;span style="font-family: MathJax_Main; font-size: 70.7%;" class="mo"&gt;/&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: MathJax_Math; font-size: 70.7%;" class="mi"&gt;&lt;em&gt;b&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
, where a and b are integers. What follows is a list of musically significant non-radical intervals.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;


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  --&gt;&lt;script type="math/tex"&gt;\text{Phi } (\phi)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
  --&gt;&lt;script type="math/tex"&gt;\text{Phi } (\phi)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&amp;quot;Linear phi,&amp;quot; the unique interval whose continued fraction convergents&lt;br /&gt;
         &lt;td&gt;&amp;quot;Linear phi,&amp;quot; the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Diophantine_approximation#General_upper_bound" rel="nofollow"&gt;Dirichlet's Approximation Theorem&lt;/a&gt;.&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Retrieved from "https://en.xen.wiki/w/Non_radical_intervals_with_musical_significance"