Just intonation: Difference between revisions

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-27 18:47:47 UTC</tt>.<br>

: The original revision id was <tt>258835780</tt>.<br>

"Just Intonation", as we find it commonly used today, describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of [[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) between the frequencies of pitches.

In languages other than English, the original conceptions of "Just Intonation" are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the "natural scale", that is, intervals derived from the harmonic partials), Intonazione naturale (natural intonation, once again intervals derived from harmonic partials) in Italian, and so on.

Of course, an historical description of something as "natural" does not prove that that thing really is "natural", and defining the concept of "natural", especially in the arts, is always very problematic. Nevertheless, the historical meanings of the terms for what we call "Just Intonation" do claim a "natural" status, and Just Intonation is indeed derived from genuine acoustic phenomena. How important, universal, etc., these phenomena are has been a matter of debate for thousands of years.

The current common usage of describing intervals between pitches by specifying ratios of rational numbers is another way of expressing the "natural scale", for it describes ratios between harmonic partials (in their ideal form). So, contemporary usage of the term is in keeping with historical and international usages. However, just as harmonic vocabulary has expanded over the centuries, so has that which falls under "just intonation" expanded.

Just Intonation 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]].

//7. The use of// //harmonic// //mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.//

  &amp;quot;Just Intonation&amp;quot;, as we find it commonly used today, 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.&lt;br /&gt;

In languages other than English, the original conceptions of &amp;quot;Just Intonation&amp;quot; are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the &amp;quot;natural scale&amp;quot;, that is, intervals derived from the harmonic partials), Intonazione naturale (natural intonation, once again intervals derived from harmonic partials) in Italian, and so on.&lt;br /&gt;

Of course, an historical description of something as &amp;quot;natural&amp;quot; does not prove that that thing really is &amp;quot;natural&amp;quot;, and defining the concept of &amp;quot;natural&amp;quot;, especially in the arts, is always very problematic. Nevertheless, the historical meanings of the terms for what we call &amp;quot;Just Intonation&amp;quot; do claim a &amp;quot;natural&amp;quot; status, and Just Intonation is indeed derived from genuine acoustic phenomena. How important, universal, etc., these phenomena are has been a matter of debate for thousands of years.&lt;br /&gt;

The current common usage of describing intervals between pitches by specifying ratios of rational numbers is another way of expressing the &amp;quot;natural scale&amp;quot;, for it describes ratios between harmonic partials (in their ideal form). So, contemporary usage of the term is in keeping with historical and international usages. However, just as harmonic vocabulary has expanded over the centuries, so has that which falls under &amp;quot;just intonation&amp;quot; expanded.&lt;br /&gt;

Just Intonation 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;em&gt;7. The use of&lt;/em&gt; &lt;em&gt;harmonic&lt;/em&gt; &lt;em&gt;mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.&lt;/em&gt;&lt;br /&gt;