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:~~hstraub~~|~~hstraub~~]] and made on <tt>2010-06-~~07 08~~:19:48 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-07-24 17:10:02 UTC</tt>.<br>

: The original revision id was <tt>~~147370937~~</tt>.<br>

: The original revision id was <tt>153886137</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>

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----

=Just Intonation explained=

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 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 [[Microtempering|microtempering]] 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 [[http://en.wikipedia.org/wiki/Septimal_minor_third|septimal minor third]].

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 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 [[Microtempering|microtempering]] 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 [[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 frequency and [[http://en.wikipedia.org/wiki/Pitch_%28music%29|pitch]]. Kyle Gann's //[[http://www.kylegann.com/tuning.html|Just Intonation Explained]]// is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].

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=Scalesmith's gallery of Just Intonation scales=

[[~~otones12~~-24]]

[[Gallery of 12-tone Just Intonation Scales]]

[[boogiewoogiescale|Boogie woogie scale]]

[[Arnold Dreyblatt]]

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<hr />

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

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 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 />

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 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 />

<br />

If you are used to speaking only in note names, you may need to study the relation between frequency and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow">pitch</a>. Kyle Gann's <em><a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow">Just Intonation Explained</a></em> is one good reference. 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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<br />

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

<a class="wiki_link" href="/~~otones12~~-24">~~otones12~~-24</a><br />

<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 />

<a class="wiki_link" href="/Arnold%20Dreyblatt">Arnold Dreyblatt</a><br />

<a class="wiki_link" href="/Gallery%20of%20pentatonics">Gallery of pentatonics</a><br />

<a class="wiki_link" href="/FiniteSubsetJI">FiniteSubsetJI</a></body></html></pre></div>

@@ 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:hstraub|hstraub]] and made on <tt>2010-06-07 08:19:48 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-07-24 17:10:02 UTC</tt>.<br>
-: The original revision id was <tt>147370937</tt>.<br>
+: The original revision id was <tt>153886137</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 10: / Line 10: @@
 ----
 =Just Intonation explained=
-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 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 [[Microtempering|microtempering]] 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 [[http://en.wikipedia.org/wiki/Septimal_minor_third|septimal minor third]].
+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 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 [[Microtempering|microtempering]] 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 [[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 frequency and [[http://en.wikipedia.org/wiki/Pitch_%28music%29|pitch]]. Kyle Gann's //[[http://www.kylegann.com/tuning.html|Just Intonation Explained]]// is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].
@@ Line 51: / Line 51: @@
 =Scalesmith's gallery of Just Intonation scales=
-[[otones12-24]]
+[[Gallery of 12-tone Just Intonation Scales]]
 [[boogiewoogiescale|Boogie woogie scale]]
 [[Arnold Dreyblatt]]
@@ Line 61: / Line 61: @@
 &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 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 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;
+  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 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;
 &lt;br /&gt;
 If you are used to speaking only in note names, you may need to study the relation between frequency and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;pitch&lt;/a&gt;. 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; is one good reference. 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;
@@ Line 102: / Line 102: @@
 &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;a class="wiki_link" href="/otones12-24"&gt;otones12-24&lt;/a&gt;&lt;br /&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;
 &lt;a class="wiki_link" href="/Arnold%20Dreyblatt"&gt;Arnold Dreyblatt&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Gallery%20of%20pentatonics"&gt;Gallery of pentatonics&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/FiniteSubsetJI"&gt;FiniteSubsetJI&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>