Just intonation: 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:<br>

: This revision was by author [[User:~~Andrew_Heathwaite~~|~~Andrew_Heathwaite~~]] and made on <tt>~~2009~~-04-19 15:30:01 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 19:57:40 UTC</tt>.<br>

: The original revision id was <tt>~~68459559~~</tt>.<br>

: The original revision id was <tt>142244957</tt>.<br>

: The revision comment was: <tt>~~added link to otones12-24~~</tt><br>

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

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<h4>Original Wikitext content:</h4>

<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">=Just Intonation explained=

~~Describe~~ intervals between pitches by specifying ratios (of [[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) between the frequencies of pitches, and ~~you will~~ be ~~speaking Just Intonation~~.

Just Intonation describes intervals between pitches by specifying ratios (of [[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) between the frequencies of pitches. This is sometimes distinguished from //rational intonation// by requiring that the ratios be ones of low complexity (as for example measured by [[Tenney height]]) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinatons of the just perfect fourth, 4/3, and the just minor sixth, 8/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit [[Microtempering|microtempering]] system because of certain very accurate approximations to the octave and to seven limit intervals: (8/5)^2/(4/3)^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]].

If you are used to speaking only in note names, you may need to study the relation between [[FrequencyPitchTutorial|frequency and pitch]]. I recommend Kyle Gann's //[[http://www.kylegann.com/tuning.html|Just Intonation Explained]]//. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].

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<h4>Original HTML content:</h4>

<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"><html><head><title>Just intonation</title></head><body><h1 id="toc0"><a name="Just Intonation explained"></a>Just Intonation explained</h1>

~~Describe~~ intervals between pitches by specifying ratios (of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow">rational numbers</a>) between the frequencies of pitches, and ~~you will~~ be ~~speaking Just Intonation~~.<br />

Just Intonation describes intervals between pitches by specifying ratios (of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow">rational numbers</a>) between the frequencies of pitches. This is sometimes distinguished from <em>rational intonation</em> by requiring that the ratios be ones of low complexity (as for example measured by <a class="wiki_link" href="/Tenney%20height">Tenney height</a>) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinatons of the just perfect fourth, 4/3, and the just minor sixth, 8/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit <a class="wiki_link" href="/Microtempering">microtempering</a> system because of certain very accurate approximations to the octave and to seven limit intervals: (8/5)^2/(4/3)^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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow">septimal minor third</a>.<br />

<br />

If you are used to speaking only in note names, you may need to study the relation between <a class="wiki_link" href="/FrequencyPitchTutorial">frequency and pitch</a>. I recommend Kyle Gann's <em><a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow">Just Intonation Explained</a></em>. A transparent illustration and one of just intonation's acoustic bases is the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>.<br />

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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:<br>
-: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-04-19 15:30:01 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 19:57:40 UTC</tt>.<br>
-: The original revision id was <tt>68459559</tt>.<br>
+: The original revision id was <tt>142244957</tt>.<br>
-: The revision comment was: <tt>added link to otones12-24</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>
 <h4>Original Wikitext content:</h4>
 <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">=Just Intonation explained=
-Describe intervals between pitches by specifying ratios (of [[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) between the frequencies of pitches, and you will be speaking Just Intonation.
+Just Intonation describes intervals between pitches by specifying ratios (of [[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) between the frequencies of pitches. This is sometimes distinguished from //rational intonation// by requiring that the ratios be ones of low complexity (as for example measured by [[Tenney height]]) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinatons of the just perfect fourth, 4/3, and the just minor sixth, 8/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit [[Microtempering|microtempering]] system because of certain very accurate approximations to the octave and to seven limit intervals: (8/5)^2/(4/3)^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]].
 If you are used to speaking only in note names, you may need to study the relation between [[FrequencyPitchTutorial|frequency and pitch]]. I recommend Kyle Gann's //[[http://www.kylegann.com/tuning.html|Just Intonation Explained]]//. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].
@@ Line 57: / Line 57: @@
 <h4>Original HTML content:</h4>
 <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;Just intonation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Just Intonation explained"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Just Intonation explained&lt;/h1&gt;
-  Describe intervals between pitches by specifying ratios (of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow"&gt;rational numbers&lt;/a&gt;) between the frequencies of pitches, and you will be speaking Just Intonation.&lt;br /&gt;
+  Just Intonation describes intervals between pitches by specifying ratios (of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow"&gt;rational numbers&lt;/a&gt;) between the frequencies of pitches. This is sometimes distinguished from &lt;em&gt;rational intonation&lt;/em&gt; by requiring that the ratios be ones of low complexity (as for example measured by &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinatons of the just perfect fourth, 4/3, and the just minor sixth, 8/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; system because of certain very accurate approximations to the octave and to seven limit intervals: (8/5)^2/(4/3)^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;br /&gt;
 &lt;br /&gt;
 If you are used to speaking only in note names, you may need to study the relation between &lt;a class="wiki_link" href="/FrequencyPitchTutorial"&gt;frequency and pitch&lt;/a&gt;. I recommend Kyle Gann's &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt;&lt;/em&gt;. A transparent illustration and one of just intonation's acoustic bases is the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;.&lt;br /&gt;