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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| 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:xenjacob|xenjacob]] and made on <tt>2007-10-10 06:23:38 UTC</tt>.<br>
| | | de = Reine Stimmungen |
| : The original revision id was <tt>9237531</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 explained=
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| 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.
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| To understand this, you need first to understand the relation of [[FrequencyPitchTutorial|frequency and pitch]]. Kyle Gann is a good thing to read. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].
| | '''Just intonation''' ('''JI''') is an approach to [[musical tuning]] where all [[interval]]s between two notes have [[frequency ratio]]s which are {{W|rational number}}s. For example, a perfect fifth in just intonation can have frequency ratio [[3/2]], a major third [[5/4]], a minor third [[6/5]], and so on. Just intonation is based off of the [[harmonic series]], which is the collection of tones found at integer multiples of a fundamental frequency, and is the set of [[overtone]]s of a note played on a string or pipe instrument. All just intervals can be found as the interval between two notes in the harmonic series; for example, [[5/3]] is the interval between the [[5/1|5th harmonic]] and the [[3/1|3rd harmonic]]. Just intervals with frequency ratios of small numbers, called [[low-complexity just intonation]] (LCJI) intervals, tend to be the most [[consonant]] in the sense that their sounds meld together. |
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| =Just Intonation used=
| | In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit|5-limit tuning]], where the numerator and denominator of any ratio used has 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]]. |
| The use of just intonation could be divided into these two flavors: | |
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| ==Free Style Just==
| | Just intonation contrasts with [[equal temperament]]s in that equal temperaments include intervals with {{W|Irrational number|irrational}} frequency ratios, which are not intervals of just intonation. For example, [[12edo|12-tone equal temperament]] has a frequency ratio of 2<sup>1/12</sup>, which is an irrational number, as a corollary of the {{W|rational root theorem}}. In fact, the only intervals in 12et which are also intervals of just intonation are multiples of the [[2/1|octave]], with a frequency ratio of 2/1. Equal temperaments are often used to approximate just intonation; for example, 12et approximates the perfect fifth [[3/2]], which is 702{{Cent}} in size, with the 7-step interval of 700{{c}}, only 2{{c}} flat. The major third with frequency ratio [[5/4]], which is 386{{c}} in size, is approximated by the 4-step interval of 400{{c}}, at 14{{c}} sharp. The ability of 12et to approximate simple ratios of just intonation is one of the reasons it became popular in the 20th century. Other equal temperaments, such as [[19edo|19et]], [[22edo|22et]], and [[31edo|31et]], also approximate various intervals of just intonation accurately, including higher-limit intervals not approximated well by 12et. |
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| 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]]
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| ==Constrained Just==
| | The structure of just intonation has several implications on music composition. Sequences of intervals that arrive back to the root in equal temperament may not do so in just intonation, and instead reach an interval a [[comma]] above or below the root. For example, going up four perfect fifths, and down a major third and two octaves, arrives back to the root in 12et {{Nowrap| (4 × 700{{c}} – 400{{c}} – 2 × 1200{{c}} {{=}} 0{{c}}) }}, but does not do so in just intonation {{Nowrap| ((3/2)<sup>4</sup> ÷ (5/4) ÷ (2/1)<sup>2</sup> {{=}} [[81/80]] ≠ 1/1) }}. The note reached is instead 81/80 (about 22{{c}}) above the root, rather than being equal to it. The 81/80 comma is known as the ''syntonic comma'', and occurs frequently in 5-limit just intonation. Modifying a simple ratio by a comma often produces a [[wolf interval]]; for example, 3/2 minus a syntonic comma is (3/2) ÷ (81/80) = [[40/27]], which is significantly less consonant than 3/2. Certain chord progressions may also become [[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 include pitch shifts, [[adaptive just intonation]] and [[temperament]]s. |
| (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):
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| //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).
| | == Consonance == |
| | [[File:Major triad 12et saw32.mp3|thumb|A major triad in 12-tone equal temperament.]] |
| | [[File:Major triad ji saw32.mp3|thumb|The same major triad in 5-limit just intonation.]] |
| | 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. |
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| 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.
| | 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. |
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| 3. Other theorists who, in contrast to the above, advocate the use of the [[CombinationProductSet|products]] of a given set of prime numbers, such as Robert Dussaut, Ervin Wilson, and others.
| | Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]]. |
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| 4. Restrictions on the variety of prime numbers used within a system, for example, 3 used with only one other prime ([[3and7JI|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.
| | == Ways of using JI == |
| | Here are multiple ways in which musicians and theorists have used just intonation. |
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| 5. Restricting the denominator to one or very few values (the [[OverToneSeries|harmonic series]]).
| | ; [[Free style JI]] |
| | [[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. |
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| 6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).//
| | ; Harmonic limits and subgroups |
| | [[Harmonic limit]]s, also known as ''prime limits'', set a limit for the highest prime number in the factorization of any ratio used; for example, western music is based off the [[5-limit]]. Lower limits tend to be more familiar and consonant, while higher limits contain more exotic harmony. [[Subgroup]]s name a list of allowable prime numbers used. For example, the [[2.3.7 subgroup|2.3.7-subgroup]] consists of all intervals with only primes 2, 3, and 7 in the numerator and denominator. (A harmonic limit is also a type of subgroup, though they are less commonly stated as such.) Different subgroups each contain their own unique structures, including commas, temperaments, [[scale form]]s, etc. |
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| =Just Intonation Propaganda= | | ; Restrictions on the denominator or numerator |
| | 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 very few values" (the [[subharmonic series]], [[IFDO]]s/undertone scales), or both ([[Tonality diamond|tonality diamonds]]). |
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| //Insert fair discussion of JI proselytizing and psychoacoustics psychobabble right here!//
| | ; Mediants |
| | The use of harmonic and arithmetic [[mediant (operation)|mediants]] as was common with the [[ancient Greek music|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.{{citation needed}} |
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| =Variations on 'Just'=
| | ; Approximations/alterations of tempered tunings |
| [[Regular Temperaments]] are just intonation systems of various [[harmonic limits]] with certain commas tempered out
| | These are [[Detempering|detemperings]], including [[NEJI]] systems. |
| [[AdaptiveJI|Adaptive JI]] schemes usually temper out these commas over time, resulting in | |
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| **Broken** links to JI theory pages on [[http://moinmoin.riters.com/microtonal|another microtonal wiki]], which await transfer to this wiki:
| | ; Other approaches |
| [[http://moinmoin.riters.com/microtonal/index.cgi/58Note11LimitJI|58 note 11 limit JI]] - hyper-Partchian!
| | Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets|product sets]]. |
| [[http://moinmoin.riters.com/microtonal/index.cgi/Reduction|Reduction]] | |
| [[http://moinmoin.riters.com/microtonal/index.cgi/Comma_20sequences|Comma sequences]] | |
| [[http://moinmoin.riters.com/microtonal/index.cgi/Hahn_20distance|Hahn distance]]
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| ==Scalesmith's gallery of Just Intonation scales== | | ==Approximating JI with temperaments== |
| [[boogiewoogiescale|Boogie woogie scale]] | | There are a lot of JI intervals, and it's difficult to keep track of all of them. As such, people often use simpler systems to approximate JI intervals, known as [[temperament]]s. A popular choice is [[equal temperament]]s; for example the predominant [[12edo|12et]], which is widely used to approximate [[5-limit]] JI. Other equal temperaments exist, for example [[19edo|19et]], [[22edo|22et]], and [[31edo|31et]]. Besides equal temperaments, other temperaments exist, such as [[regular temperament]]s and [[well temperament]]s. |
| [[Arnold Dreyblatt]] | | |
| [[Gallery of pentatonics]] | | Temperaments also create new structures not found in JI; for example, [[meantone]] temperament (which 12et [[support]]s) tempers out 81/80, making [[5/4]] the same as the major third obtained by stacking four fifths, [[81/64]]; this structural feature is often assumed without thinking in western music. |
| [[FiniteSubsetJI]]</pre></div> | | |
| <h4>Original HTML content:</h4>
| | ==Instruments== |
| <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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Just Intonation explained"></a><!-- ws:end:WikiTextHeadingRule:0 -->Just Intonation explained</h1>
| | {{todo|expand|comment=Expand the instruments section with more examples}} |
| Describe 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, and you will be speaking Just Intonation.<br />
| | *The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range. |
| <br />
| | *Most of [[Harry Partch]]'s instruments were designed to be for just intonation. |
| To understand this, you need first to understand the relation of <a class="wiki_link" href="/FrequencyPitchTutorial">frequency and pitch</a>. Kyle Gann is a good thing to read. A transparent illustration and one of just intonation's acoustic bases is the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>.<br />
| | ==Music== |
| <br />
| | {{Main|Music in just intonation}} |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><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 />
| | == Notation == |
| <br />
| | There are various [[Musical notation|notation systems]] for just intonation, for example [[Helmholtz-Ellis notation]] and the [[Functional Just System]]. |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Just Intonation used-Free Style Just"></a><!-- ws:end:WikiTextHeadingRule:4 -->Free Style Just</h2>
| | {{Todo|expand|inline=1}} |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><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 -&gt; <a class="wiki_link" href="/FreeStyleJI">FreeStyleJI</a><br />
| | ==See also== |
| <br />
| | {{todo|cleanup|inline=1}} |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Just Intonation used-Constrained Just"></a><!-- ws:end:WikiTextHeadingRule:8 -->Constrained Just</h2>
| | *[[List of approaches to musical tuning]] |
| (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, &quot;Differential Coherence&quot;, <em>1/1</em> vol. 11, no. 2: p.1):<br />
| | *[[Gallery of just intervals]] |
| <br />
| | *[[Families of scales]] |
| <em>1. The principle of &quot;<a class="wiki_link" href="/harmonic%20limits">harmonic limits</a>,&quot; which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's &quot;psycharithmes&quot; and his ordering by complexity; Gioseffe Zarlino's five-limit &quot;senario,&quot; and the like; Helmholtz's theory of consonance with its &quot;blending of partials,&quot; which, like the others, results in giving priority to the lowest prime numbers).<br />
| | *[[:Category:Just intonation]] |
| <br />
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| 2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the &quot;monophonic&quot; system of Harry Partch's <a class="wiki_link" href="/tonality%20diamond">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.<br />
| | ==References== |
| <br />
| | <references /> |
| 3. Other theorists who, in contrast to the above, advocate the use of the <a class="wiki_link" href="/CombinationProductSet">products</a> of a given set of prime numbers, such as Robert Dussaut, Ervin Wilson, and others.<br />
| | |
| <br />
| | ==Further reading== |
| 4. Restrictions on the variety of prime numbers used within a system, for example, 3 used with only one other prime (<a class="wiki_link" href="/3and7JI">7</a>, 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.<br />
| | *[http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]] |
| <br />
| | *[http://nowitzky.hostwebs.com/justint/ Just Intonation] by Mark Nowitzky |
| 5. Restricting the denominator to one or very few values (the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>).<br />
| | *[http://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann |
| <br />
| | *[http://www.kylegann.com/Octave.html Anatomy of an Octave] by Kyle Gann |
| 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 />
| | *[http://www.dbdoty.com/Words/What-is-Just-Intonation.html What is Just Intonation?] by David B. Doty |
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| | *[http://lumma.org/tuning/faq/#whatisJI What is "just intonation"?] by Carl Lumma |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Just Intonation Propaganda"></a><!-- ws:end:WikiTextHeadingRule:10 -->Just Intonation Propaganda</h1>
| | *[http://www.dbdoty.com/Words/werntz.html A Response to Julia Werntz] by David B. Doty |
| <br />
| | *[http://lumma.org/tuning/gws/commaseq.htm Comma Sequences] by Gene Ward Smith |
| <em>Insert fair discussion of JI proselytizing and psychoacoustics psychobabble right here!</em><br />
| | *[https://casfaculty.case.edu/ross-duffin/just-intonation-in-renaissance-theory-practice/ Just Intonation in Renaissance Theory & Practice] by Ross W. Duffin |
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| <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Variations on 'Just'"></a><!-- ws:end:WikiTextHeadingRule:12 -->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> schemes usually temper out these commas over time, resulting in<br />
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| <br />
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| <strong>Broken</strong> links to JI theory pages on <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal" rel="nofollow">another microtonal wiki</a>, which await transfer to this wiki:<br />
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| <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/58Note11LimitJI" rel="nofollow">58 note 11 limit JI</a> - hyper-Partchian!<br />
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| <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Reduction" rel="nofollow">Reduction</a><br />
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| <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Comma_20sequences" rel="nofollow">Comma sequences</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 />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Variations on 'Just'-Scalesmith's gallery of Just Intonation scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->Scalesmith's gallery of Just Intonation scales</h2>
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| <a class="wiki_link" href="/boogiewoogiescale">Boogie woogie scale</a><br />
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| <a class="wiki_link" href="/Arnold%20Dreyblatt">Arnold Dreyblatt</a><br />
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| <a class="wiki_link" href="/Gallery%20of%20pentatonics">Gallery of pentatonics</a><br />
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| <a class="wiki_link" href="/FiniteSubsetJI">FiniteSubsetJI</a></body></html></pre></div>
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