Just intonation
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-04-02 10:02:41 UTC.
- The original revision id was 216432328.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[toc|flat]] ---- =Just Intonation explained= Just Intonation 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. 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]]. =Just Intonation used= The use of just intonation could be divided into these two flavors: ==Free Style Just== = = Lou Harrison used this term; it means that you choose just-intonation pitches, from the set of all possible just intervals (not from a mode or scale), as you use them in music. Dedicated page -> [[FreeStyleJI]] ==Constrained Just== (In need of a better name maybe) Here are six ways that musicians and theorists have constrained the field of potential just ratios (from Jacques Dudon, "Differential Coherence", //1/1// vol. 11, no. 2: p.1): //1. The principle of "[[Harmonic Limit|harmonic limits]]," which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers).// //2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [[http://en.wikipedia.org/wiki/Harry_Partch|Harry Partch]]'s [[http://en.wikipedia.org/wiki/Pitch_%28music%29|tonality diamond]]. This, incidentially, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1,3 5 7 9, 11, or their octaves.// //3. Other theorists who, in contrast to the above, advocate the use of [[http://en.wikipedia.org/wiki/Hexany|products sets]] of given arrays of prime numbers, such as [[http://en.wikipedia.org/wiki/Erv_Wilson|Ervin Wilson]],////Robert Dussaut,// //and others.// //4. [[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.// //5. Restricting the denominator to one or very few values (the [[OverToneSeries|harmonic series]]).// //6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).// to this can be added //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.// //8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [[http://anaphoria.com/wilsonintroMERU.html|Meru scales]] are a good example as well as Jacques Dudon// =Variations on 'Just'= [[Regular Temperaments]] are just intonation systems of various [[harmonic limits]] with certain commas 'tempered out' [[AdaptiveJI|Adaptive JI]] =Links= [[hypergenesis58.scl|58 note 11 limit JI]] - hyper-Partchian! [[Hahn distance]] [[Gallery of Just Intervals]] [[Gallery of 12-tone Just Intonation Scales]] [[boogiewoogiescale|Boogie woogie scale]] [[Arnold Dreyblatt]] [[Gallery of pentatonics]] [[FiniteSubsetJI]] =Articles= * [[http://www.webcitation.org/5xe2iC7Nq|Just Intonation Explained]] by Kyle Gann * [[http://www.webcitation.org/5xe30LCev|Anatomy of an Octave]] by Kyle Gann * [[http://www.webcitation.org/5xe3MeWVq|What is Just Intonation?]] by David B. Doty * [[http://www.webcitation.org/5xe38KWx4|A Response to Julia Werntz]] by David B. Doty * [[http://www.webcitation.org/5xe4rPLZ0|Comma Sequences]] by Gene Ward Smith
Original HTML content:
<html><head><title>Just intonation</title></head><body><!-- ws:start:WikiTextTocRule:16:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Just Intonation explained">Just Intonation explained</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Just Intonation used">Just Intonation used</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#toc3"> </a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Variations on 'Just'">Variations on 'Just'</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Articles">Articles</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> <!-- ws:end:WikiTextTocRule:25 --><hr /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Just Intonation explained"></a><!-- ws:end:WikiTextHeadingRule:0 -->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 /> <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 /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Just Intonation used"></a><!-- ws:end:WikiTextHeadingRule:2 -->Just Intonation used</h1> The use of just intonation could be divided into these two flavors:<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Just Intonation used-Free Style Just"></a><!-- ws:end:WikiTextHeadingRule:4 -->Free Style Just</h2> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --> </h1> Lou Harrison used this term; it means that you choose just-intonation pitches, from the set of all possible just intervals (not from a mode or scale), as you use them in music. Dedicated page -> <a class="wiki_link" href="/FreeStyleJI">FreeStyleJI</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Just Intonation used-Constrained Just"></a><!-- ws:end:WikiTextHeadingRule:8 -->Constrained Just</h2> (In need of a better name maybe) Here are six ways that musicians and theorists have constrained the field of potential just ratios (from Jacques Dudon, "Differential Coherence", <em>1/1</em> vol. 11, no. 2: p.1):<br /> <br /> <em>1. The principle of "<a class="wiki_link" href="/Harmonic%20Limit">harmonic limits</a>," which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers).</em><br /> <br /> <em>2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow">Harry Partch</a>'s <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow">tonality diamond</a>. This, incidentially, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1,3 5 7 9, 11, or their octaves.</em><br /> <br /> <em>3. Other theorists who, in contrast to the above, advocate the use of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexany" rel="nofollow">products sets</a> of given arrays of prime numbers, such as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Ervin Wilson</a>,</em><em>Robert Dussaut,</em> <em>and others.</em><br /> <br /> <em>4. <a class="wiki_link" href="/Just%20intonation%20subgroups">Restrictions on the variety of prime numbers</a> used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.</em><br /> <br /> <em>5. Restricting the denominator to one or very few values (the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>).</em><br /> <br /> <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 /> to this can be added<br /> <em>7. The use of</em> <em>harmonic</em> <em>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.</em><br /> <br /> <em>8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's <a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMERU.html" rel="nofollow">Meru scales</a> are a good example as well as Jacques Dudon</em><br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Variations on 'Just'"></a><!-- ws:end:WikiTextHeadingRule:10 -->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 /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:12 -->Links</h1> <a class="wiki_link" href="/hypergenesis58.scl">58 note 11 limit JI</a> - hyper-Partchian!<br /> <a class="wiki_link" href="/Hahn%20distance">Hahn distance</a><br /> <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intervals</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><br /> <br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Articles"></a><!-- ws:end:WikiTextHeadingRule:14 -->Articles</h1> <ul><li><a class="wiki_link_ext" href="http://www.webcitation.org/5xe2iC7Nq" rel="nofollow">Just Intonation Explained</a> by Kyle Gann</li><li><a class="wiki_link_ext" href="http://www.webcitation.org/5xe30LCev" rel="nofollow">Anatomy of an Octave</a> by Kyle Gann</li><li><a class="wiki_link_ext" href="http://www.webcitation.org/5xe3MeWVq" rel="nofollow">What is Just Intonation?</a> by David B. Doty</li><li><a class="wiki_link_ext" href="http://www.webcitation.org/5xe38KWx4" rel="nofollow">A Response to Julia Werntz</a> by David B. Doty</li><li><a class="wiki_link_ext" href="http://www.webcitation.org/5xe4rPLZ0" rel="nofollow">Comma Sequences</a> by Gene Ward Smith</li></ul></body></html>