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clarified relation between JI and harmonic series
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = Just intonation
: This revision was by author [[User:guest|guest]] and made on <tt>2007-10-10 05:39:20 UTC</tt>.<br>
| de = Reine Stimmungen
: The original revision id was <tt>9236813</tt>.<br>
| es = Entonación Justa
: The revision comment was: <tt></tt><br>
| ja = 純正律
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| ko = 순정률
<h4>Original Wikitext content:</h4>
| ro = Intervale raționale
<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 defined=
}}
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.
{{Wikipedia}}


=Overtone Series=
'''Just intonation''' ('''JI''') is an approach to [[musical tuning]] which uses tones whose frequencies are whole-number ratios of a given fundamental [[frequency]]. Just intonation includes the [[harmonic series]], which is the collection of tones found at integer multiples of a fundamental frequency; all just intervals can be found as the interval between two notes in the harmonic series. Just ratios of small numbers, called '''Low-complexity just intonation (LCJI)''' intervals, tend to be the most [[concordant|consonant]] in the sense that their sounds meld together.  
(Insert theory/description here...)


==[[OverToneSeries|Compositions]]==
In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit]] tuning--intervals where the numerators and denominators of any ratio used have no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].


=Freestyle Just=
The structure of just intonation has several implications on music composition. [[Wolf interval|Wolf intervals]] and [[Comma|commas]], two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[Comma pump|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, [[adaptive just intonation]] and [[temperament]]. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently.
= =
=Adaptive Just Intonation=
Compose music such that each pitch you choose is chosen from the infinitely large pool of rational numbers. Daunting!


==Software==  
== Consonance ==
* Chuckk Hubbard's [[http://www.badmuthahubbard.com/jisequencer.html|No-scale JI Sequencer]]
LCJI intervals achieve consonance through alignment of [[Partial|partials]] if the interval has [[Harmonic timbre|harmonic timbre]]. In fact, alignment of partials is a stronger effect with harmonic timbre: if partials align at frequency n, they will also align at every multiple of n; and in addition, two notes whose partials align with the same root note will also have partials aligning with each other. This allows for the construction of just-intonation chords of more than two notes where every comprising interval is a consonance.  
* [[http://www.math.tu-dresden.de/%7Emutabor/|Mutabor]]


== ==
Low-complexity JI intervals and chords also achieve consonance by being the ratios between harmonics of a (possibly unplayed) fundamental even if they do not have harmonic timbre.
==[[AdaptiveJI|Compositions]]==


=Constraints=  
==Ways of using JI==
Create subsets of the infinitely large pool of ratios by setting constraints. Here are six ways (from Jacques Dudon, "Differential Coherence", //1/1// vol. 11, no. 2: p.1):
Here are multiple ways in which musicians and theorists have used just intonation.


//1. The principle of "[[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).
[[Free style JI|'''Free style JI''']] <br />
[[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.


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 Harry Partch's [[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.
'''Harmonic limits and subgroups'''<br />
[[Harmonic limit|Harmonic limits]] set a limit for the highest prime number in the factorization of any ratio used. [[Subgroup|Subgroups]] name a list of allowable prime numbers used.


3. Other theorists who, in contrast to the above, advocate the use of the products of a given set of prime numbers, such as Robert Dussaut, Ervin Wilson, and others.
'''Restrictions on the denominator or numerator'''<br />
Some approaches restrict "the denominator to one or very few values"<ref name=":0">From Jacques Dudon, "Differential Coherence", ''1/1'' vol. 11, no. 2: p.1).</ref> (the [[harmonic series]], [[isoharmonic chord]]s, [[AFDO]]s/[[overtone scale]]s, [[primodality]], [[Ringer scale|ringer scales]]), the "numerator to one or a very few values" (the [[subharmonic series]], [[IFDO]]s/undertone scales), or both ([[Tonality diamond|tonality diamonds]])


4. Restrictions on the variety of prime numbers used within a system, for example, 3 used with only one 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.
'''Mediants'''<br />
The use of harmonic and arithmetic [[Mediant (operation)|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.


5. Restricting the denominator to one or very few values (the [[OverToneSeries|harmonic series]]).
'''Approximations/alterations of tempered tunings''' <br />
These are [[Detempering|detemperings]], including [[NEJI]] systems.  


6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).//
'''Other approaches'''<br />
Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[Tritriadic scale|titriadic scales]], and [[combination product sets|product sets]].


**Broken** links to JI theory pages on [[http://moinmoin.riters.com/microtonal|another microtonal wiki]], which await transfer to this wiki:
==Instruments==
[[http://moinmoin.riters.com/microtonal/index.cgi/58Note11LimitJI|58 note 11 limit JI]] - hyper-Partchian!
{{todo|expand|comment=Expand the instruments section with more examples}}
[[http://moinmoin.riters.com/microtonal/index.cgi/Reduction|Reduction]]
*The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range.
[[http://moinmoin.riters.com/microtonal/index.cgi/Comma_20sequences|Comma sequences]]
*Most of [[Harry Partch]]'s instruments were designed to be for just intonation.
[[http://moinmoin.riters.com/microtonal/index.cgi/Hahn_20distance|Hahn distance]]
==Music==
{{Main|Music in just intonation}}


==Scalesmith's gallery of Just Intonation scales==
== Notation ==
[[boogiewoogiescale|Boogie woogie scale]]
There are various [[Musical notation|notation systems]] for just intonation.
[[Arnold Dreyblatt]]
==See also==
Gallery of pentatonics
{{todo|cleanup|inline=1}}
*[[List of approaches to musical tuning]]
*[[Gallery of just intervals]]
*[[Gallery of 12-tone just intonation scales]]
*[[Families of scales]]
*[[boogiewoogiescale|Boogie woogie scale]]
*[[:Category:Just intonation]]
==References==
<references />


==[[FiniteSubsetJI|Compositions]]== </pre></div>
==Further reading==
<h4>Original HTML content:</h4>
*[http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]]
<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 defined"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Just Intonation defined&lt;/h1&gt;
*[http://nowitzky.hostwebs.com/justint/ Just Intonation] by Mark Nowitzky
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;
*[http://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann
&lt;br /&gt;
*[http://www.kylegann.com/Octave.html Anatomy of an Octave] by Kyle Gann
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Overtone Series"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Overtone Series&lt;/h1&gt;
*[http://www.dbdoty.com/Words/What-is-Just-Intonation.html What is Just Intonation?] by David B. Doty
(Insert theory/description here...)&lt;br /&gt;
*[http://lumma.org/tuning/faq/#whatisJI What is "just intonation"?] by Carl Lumma
&lt;br /&gt;
*[http://www.dbdoty.com/Words/werntz.html A Response to Julia Werntz] by David B. Doty
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Overtone Series-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;a class="wiki_link" href="/OverToneSeries"&gt;Compositions&lt;/a&gt;&lt;/h2&gt;
*[http://lumma.org/tuning/gws/commaseq.htm Comma Sequences] by Gene Ward Smith
&lt;br /&gt;
*[https://casfaculty.case.edu/ross-duffin/just-intonation-in-renaissance-theory-practice/ Just Intonation in Renaissance Theory & Practice] by Ross W. Duffin
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Freestyle Just"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Freestyle Just&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt; &lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Adaptive Just Intonation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Adaptive Just Intonation&lt;/h1&gt;
Compose music such that each pitch you choose is chosen from the infinitely large pool of rational numbers. Daunting!&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Adaptive Just Intonation-Software"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Software&lt;/h2&gt;
&lt;ul&gt;&lt;li&gt;Chuckk Hubbard's &lt;a class="wiki_link_ext" href="http://www.badmuthahubbard.com/jisequencer.html" rel="nofollow"&gt;No-scale JI Sequencer&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.math.tu-dresden.de/%7Emutabor/" rel="nofollow"&gt;Mutabor&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt; &lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Adaptive Just Intonation-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;a class="wiki_link" href="/AdaptiveJI"&gt;Compositions&lt;/a&gt;&lt;/h2&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Constraints"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Constraints&lt;/h1&gt;
Create subsets of the infinitely large pool of ratios by setting constraints. Here are six ways (from Jacques Dudon, &amp;quot;Differential Coherence&amp;quot;, &lt;em&gt;1/1&lt;/em&gt; vol. 11, no. 2: p.1):&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;1. The principle of &amp;quot;&lt;a class="wiki_link" href="/harmonic%20limits"&gt;harmonic limits&lt;/a&gt;,&amp;quot; which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's &amp;quot;psycharithmes&amp;quot; and his ordering by complexity; Gioseffe Zarlino's five-limit &amp;quot;senario,&amp;quot; and the like; Helmholtz's theory of consonance with its &amp;quot;blending of partials,&amp;quot; which, like the others, results in giving priority to the lowest prime numbers).&lt;br /&gt;
&lt;br /&gt;
2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the &amp;quot;monophonic&amp;quot; system of Harry Partch's &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt;. 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.&lt;br /&gt;
&lt;br /&gt;
3. Other theorists who, in contrast to the above, advocate the use of the products of a given set of prime numbers, such as Robert Dussaut, Ervin Wilson, and others.&lt;br /&gt;
&lt;br /&gt;
4. Restrictions on the variety of prime numbers used within a system, for example, 3 used with only one 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.&lt;br /&gt;
&lt;br /&gt;
5. Restricting the denominator to one or very few values (the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;).&lt;br /&gt;
&lt;br /&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;strong&gt;Broken&lt;/strong&gt; links to JI theory pages on &lt;a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal" rel="nofollow"&gt;another microtonal wiki&lt;/a&gt;, which await transfer to this wiki:&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/58Note11LimitJI" rel="nofollow"&gt;58 note 11 limit JI&lt;/a&gt; - hyper-Partchian!&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Reduction" rel="nofollow"&gt;Reduction&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Comma_20sequences" rel="nofollow"&gt;Comma sequences&lt;/a&gt;&lt;br /&gt;
&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:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Constraints-Scalesmith's gallery of Just Intonation scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Scalesmith's gallery of Just Intonation scales&lt;/h2&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;
Gallery of pentatonics&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Constraints-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;a class="wiki_link" href="/FiniteSubsetJI"&gt;Compositions&lt;/a&gt;&lt;/h2&gt;
&lt;/body&gt;&lt;/html&gt;</pre></div>
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