Just intonation: Difference between revisions

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=Just Intonation explained=  

The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit [[Microtempering|microtempering]] system because of very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the [[http://en.wikipedia.org/wiki/Septimal_minor_third|septimal minor third]].&lt;/ref&gt;

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The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit &amp;lt;a class=&amp;quot;wiki_link&amp;quot; href=&amp;quot;/Microtempering&amp;quot;&amp;gt;microtempering&amp;lt;/a&amp;gt; system because of very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the &amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://en.wikipedia.org/wiki/Septimal_minor_third&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;septimal minor third&amp;lt;/a&amp;gt;.&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-1" class="reference"&gt;&lt;a href="#cite_note-1"&gt;[1]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:8 --&gt;&lt;br /&gt;

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  &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Just_intonation" rel="nofollow"&gt;Wikipedia article on just intonation&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://nowitzky.hostwebs.com/justint/" rel="nofollow"&gt;Just Intonation&lt;/a&gt; by Mark Nowitzky &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeAm2lPL" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt; by Kyle Gann &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe2iC7Nq" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/Octave.html" rel="nofollow"&gt;Anatomy of an Octave&lt;/a&gt; by Kyle Gann &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe30LCev" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.dbdoty.com/Words/What-is-Just-Intonation.html" rel="nofollow"&gt;What is Just Intonation?&lt;/a&gt; by David B. Doty &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe3MeWVq" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/faq/#whatisJI" rel="nofollow"&gt;What is &amp;quot;just intonation&amp;quot;?&lt;/a&gt; by Carl Lumma &lt;a class="wiki_link_ext" href="http://www.webcitation.org/65NwFAKLh" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.dbdoty.com/Words/werntz.html" rel="nofollow"&gt;A Response to Julia Werntz&lt;/a&gt; by David B. Doty &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe38KWx4" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/gws/commaseq.htm" rel="nofollow"&gt;Comma Sequences&lt;/a&gt; by Gene Ward Smith &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe4rPLZ0" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

&lt;li id="cite_note-1"&gt;&lt;a href="#cite_ref-1"&gt;^&lt;/a&gt; Just Intonation is sometimes distinguished from &lt;em&gt;rational intonation,&lt;/em&gt; by requiring that the ratios be lower than some arbitrary complexity (as for example measured by &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;, &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt;, etc.) but there is no clear dividing line. The matter is partially a question of intent.&lt;br /&gt;

The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; system because of very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow"&gt;septimal minor third&lt;/a&gt;.&lt;/li&gt;

&lt;li id="cite_note-2"&gt;&lt;a href="#cite_ref-2"&gt;^&lt;/a&gt; [2] All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are &amp;quot;stretched&amp;quot; according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic &amp;quot;noise&amp;quot;. A breathily played flute has a large addition of inharmonic material, a &amp;quot;jinashi&amp;quot; shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.&lt;/li&gt;