Just intonation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~Andrew_Heathwaite~~|~~Andrew_Heathwaite~~]] and made on <tt>2010-09-~~23 13~~:13:05 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2010-11-01 07:28:18 UTC</tt>.<br>

: The original revision id was <tt>~~164916685~~</tt>.<br>

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//6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).//

~~=Just Intonation Propaganda?=~~

Some people feel that there is a certain amount of ideological and propagandistic baggage associated to just intonation, and indeed some people do become infuriated when it is pointed out that the line between JI and non-JI is fuzzy or that [[Microtempering|microtempering]] sounds like JI. There is more than pig-headedness or quasi-religious sentiment involved in such responses, however, as the low prime limit rational numbers have a structure to them which can be grasped and employed by the composer. JI proponents are often reacting against promoters of equal temperament, especially 12-et, who often seem to feel that concern about close approximation to just intervals gets in the way of music.

=Variations on 'Just'=

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See also: [[Gallery of Just Intervals]]</pre></div>

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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"><html><head><title>Just intonation</title></head><body><a href="#Just Intonation explained">Just Intonation explained</a> | <a href="#Just Intonation used">Just Intonation used</a> | <a href="#toc3"> </a> | <a href="#Just Intonation Propaganda?">Just Intonation Propaganda?</a> | <a href="#Variations on 'Just'">Variations on 'Just'</a> | <a href="#Scalesmith's gallery of Just Intonation scales">Scalesmith's gallery of Just Intonation scales</a>

<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><a href="#Just Intonation explained">Just Intonation explained</a> | <a href="#Just Intonation used">Just Intonation used</a> | <a href="#toc3"> </a> | <a href="#Variations on 'Just'">Variations on 'Just'</a> | <a href="#Scalesmith's gallery of Just Intonation scales">Scalesmith's gallery of Just Intonation scales</a>

<hr />

<hr />

<h1 id="toc0"><a name="Just Intonation explained"></a>Just Intonation explained</h1>

Just Intonation describes <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">intervals</a> 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 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, 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: (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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow">septimal minor third</a>.<br />

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<em>6. Restricting the numerator to one or a very few values (the <a class="wiki_link" href="/subharmonic%20series">subharmonic series</a> or <a class="wiki_link" href="/aliquot%20scales">aliquot scales</a>).</em><br />

<br />

<h1 id="toc5"><a name="Just Intonation Propaganda?"></a>Just Intonation Propaganda?</h1>

~~<br />~~

Some people feel that there is a certain amount of ideological and propagandistic baggage associated to just intonation, and indeed some people do become infuriated when it is pointed out that the line between JI and non-JI is fuzzy or that <a class="wiki_link" href="/Microtempering">microtempering</a> sounds like JI. There is more than pig-headedness or quasi-religious sentiment involved in such responses, however, as the low prime limit rational numbers have a structure to them which can be grasped and employed by the composer. JI proponents are often reacting against promoters of equal temperament, especially 12-et, who often seem to feel that concern about close approximation to just intervals gets in the way of music.<br />

<br />

<h1 id="~~toc6~~"><a name="Variations on 'Just'"></a>Variations on 'Just'</h1>

<h1 id="toc5"><a name="Variations on 'Just'"></a>Variations on 'Just'</h1>

<a class="wiki_link" href="/Regular%20Temperaments">Regular Temperaments</a> are just intonation systems of various <a class="wiki_link" href="/harmonic%20limits">harmonic limits</a> with certain commas 'tempered out'<br />

<a class="wiki_link" href="/AdaptiveJI">Adaptive JI</a><br />

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<a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Hahn_20distance" rel="nofollow">Hahn distance</a><br />

<br />

<h1 id="~~toc7~~"><a name="Scalesmith's gallery of Just Intonation scales"></a>Scalesmith's gallery of Just Intonation scales</h1>

<h1 id="toc6"><a name="Scalesmith's gallery of Just Intonation scales"></a>Scalesmith's gallery of Just Intonation scales</h1>

<a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales">Gallery of 12-tone Just Intonation Scales</a><br />

<a class="wiki_link" href="/boogiewoogiescale">Boogie woogie scale</a><br />

@@ Line 1: / Line 1: @@
 <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>2010-09-23 13:13:05 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2010-11-01 07:28:18 UTC</tt>.<br>
-: The original revision id was <tt>164916685</tt>.<br>
+: The original revision id was <tt>175265119</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>
@@ Line 35: / Line 35: @@
 //6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).//
-=Just Intonation Propaganda?=
-Some people feel that there is a certain amount of ideological and propagandistic baggage associated to just intonation, and indeed some people do become infuriated when it is pointed out that the line between JI and non-JI is fuzzy or that [[Microtempering|microtempering]] sounds like JI. There is more than pig-headedness or quasi-religious sentiment involved in such responses, however, as the low prime limit rational numbers have a structure to them which can be grasped and employed by the composer. JI proponents are often reacting against promoters of equal temperament, especially 12-et, who often seem to feel that concern about close approximation to just intervals gets in the way of music.
 =Variations on 'Just'=
@@ Line 58: / Line 55: @@
 See also: [[Gallery of Just Intervals]]</pre></div>
 <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:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Just Intonation explained"&gt;Just Intonation explained&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Just Intonation used"&gt;Just Intonation used&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#toc3"&gt; &lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Just Intonation Propaganda?"&gt;Just Intonation Propaganda?&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Variations on 'Just'"&gt;Variations on 'Just'&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Scalesmith's gallery of Just Intonation scales"&gt;Scalesmith's gallery of Just Intonation scales&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&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;Just intonation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#Just Intonation explained"&gt;Just Intonation explained&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Just Intonation used"&gt;Just Intonation used&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#toc3"&gt; &lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Variations on 'Just'"&gt;Variations on 'Just'&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Scalesmith's gallery of Just Intonation scales"&gt;Scalesmith's gallery of Just Intonation scales&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;hr /&gt;
+&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;hr /&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;
   Just Intonation describes &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;intervals&lt;/a&gt; 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 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, 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: (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;br /&gt;
@@ Line 87: / Line 84: @@
 &lt;em&gt;6. Restricting the numerator to one or a very few values (the &lt;a class="wiki_link" href="/subharmonic%20series"&gt;subharmonic series&lt;/a&gt; or &lt;a class="wiki_link" href="/aliquot%20scales"&gt;aliquot scales&lt;/a&gt;).&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Just Intonation Propaganda?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Just Intonation Propaganda?&lt;/h1&gt;
- &lt;br /&gt;
-Some people feel that there is a certain amount of ideological and propagandistic baggage associated to just intonation, and indeed some people do become infuriated when it is pointed out that the line between JI and non-JI is fuzzy or that &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; sounds like JI. There is more than pig-headedness or quasi-religious sentiment involved in such responses, however, as the low prime limit rational numbers have a structure to them which can be grasped and employed by the composer. JI proponents are often reacting against promoters of equal temperament, especially 12-et, who often seem to feel that concern about close approximation to just intervals gets in the way of music.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Variations on 'Just'"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Variations on 'Just'&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Variations on 'Just'"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Variations on 'Just'&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;Regular Temperaments&lt;/a&gt; are just intonation systems of various &lt;a class="wiki_link" href="/harmonic%20limits"&gt;harmonic limits&lt;/a&gt; with certain commas 'tempered out'&lt;br /&gt;
 &lt;a class="wiki_link" href="/AdaptiveJI"&gt;Adaptive JI&lt;/a&gt;&lt;br /&gt;
@@ Line 101: / Line 95: @@
 &lt;a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Hahn_20distance" rel="nofollow"&gt;Hahn distance&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Scalesmith's gallery of Just Intonation scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Scalesmith's gallery of Just Intonation scales&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Scalesmith's gallery of Just Intonation scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Scalesmith's gallery of Just Intonation scales&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales"&gt;Gallery of 12-tone Just Intonation Scales&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/boogiewoogiescale"&gt;Boogie woogie scale&lt;/a&gt;&lt;br /&gt;