21/16: Difference between revisions

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-14 21:22:12 UTC</tt>.<br>

: The original revision id was <tt>254175870</tt>.<br>

<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">21/16, the septimal sub-fourth, is an interval of the [[7-limit|7 prime-limit]] measuring approximately 470.8¢. It can be thought of as the 21st overtone, octave reduced. Since 21 is 3*7, 21 can be treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3_2|3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7_4|7/4]] in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion [[4_3|4/3]] and 21/8 (an octave above 21/16) sounds more consonant than 21/16.

21/16 is [[21_20|21/20]] away from 5/4. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8_7|8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11_8|11/8]] is also nearby, so that 21/16 can step up by the small semitone of 22/21 (about 80.5¢) to 11/8. These are all movements that assume a stable fundamental, of course, and other movements are possible.

<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;21_16&lt;/title&gt;&lt;/head&gt;&lt;body&gt;21/16, the septimal sub-fourth, is an interval of the &lt;a class="wiki_link" href="/7-limit"&gt;7 prime-limit&lt;/a&gt; measuring approximately 470.8¢. It can be thought of as the 21st overtone, octave reduced. Since 21 is 3*7, 21 can be treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt; between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths (&lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion &lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt; and 21/8 (an octave above 21/16) sounds more consonant than 21/16.&lt;br /&gt;

21/16 is &lt;a class="wiki_link" href="/21_20"&gt;21/20&lt;/a&gt; away from 5/4. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by &lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;, the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; system, &lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt; is also nearby, so that 21/16 can step up by the small semitone of 22/21 (about 80.5¢) to 11/8. These are all movements that assume a stable fundamental, of course, and other movements are possible.&lt;br /&gt;