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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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-27 18:47:47 UTC</tt>.<br>
| | | de = Reine Stimmungen |
| : The original revision id was <tt>258835780</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">[[toc|flat]]
| | }} |
| ----
| | {{Wikipedia}} |
| =Just Intonation explained=
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| "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.
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| 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.
| | '''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. |
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| In the English language, the term "just" referred to "true, correct", and is still used today in this sense, in the crafts. In printing, to "justify" a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as "natural". | | 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]]. |
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| 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 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. |
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| 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.
| | == Consonance == |
| | 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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| But, first things first. Let us take a look at why the idea of a "natural" or "just" tuning came about, and is still with us.
| | 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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| If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz, we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...
| | ==Ways of using JI== |
| | Here are multiple ways in which musicians and theorists have used just intonation. |
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| The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.*
| | [[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. |
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| Of course we are describing an ideal tone- in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our "fuzzy" and "flawed" reality, spectra adhere to this pattern with impressive consistency.
| | '''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. |
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| In a tuning "according to the natural scale", we have for example a "perfect fifth" as simply the ratio between the third partial and the second partial: " 3:2". In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.
| | '''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]]) |
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| Now, let us play our two example tones together, and we shall see why the German term is "Reine", "pure", and why you'll hear "pure" used in English as well (and in Czech, Slovak, etc.) Let's call our first tone "Do" and our second tone, a perfect fifth higher, "Sol".
| | '''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. |
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| Tone.... Frequencies of Partial Number
| | '''Approximations/alterations of tempered tunings''' <br /> |
| "Do" 100 200 300 400 500 600 700 800 900
| | These are [[Detempering|detemperings]], including [[NEJI]] systems. |
| "Sol" 150 300 450 600 750 900
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| You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of "Tonverschmelzung" in German. It is a very distinctinctive "blending" sound. If our Sol was tuned to, for example, 148 Hertz, its second harmonic component would be at 296 Hertz, and the two tones played together would not "meld together" at 300 Hertz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300-296=4 Hertz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).
| | '''Other approaches'''<br /> |
| | Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[Tritriadic scale|titriadic scales]], and [[combination product sets|product sets]]. |
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| One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.
| | ==Instruments== |
| | {{todo|expand|comment=Expand the instruments section with more examples}} |
| | *The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range. |
| | *Most of [[Harry Partch]]'s instruments were designed to be for just intonation. |
| | ==Music== |
| | {{Main|Music in just intonation}} |
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| There is more to it than this, of course, but the basic principles of Just Intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals were the concern, understanding and defining "Just" was not difficult. These days, though, and going on from these basics, it can get a bit more complicated... | | == Notation == |
| | There are various [[Musical notation|notation systems]] for just intonation. |
| | ==See also== |
| | {{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 /> |
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| If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), 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]].
| | ==Further reading== |
| | | *[http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]] |
| *All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.
| | *[http://nowitzky.hostwebs.com/justint/ Just Intonation] by Mark Nowitzky |
| | | *[http://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann |
| | | *[http://www.kylegann.com/Octave.html Anatomy of an Octave] by Kyle Gann |
| | | *[http://www.dbdoty.com/Words/What-is-Just-Intonation.html What is Just Intonation?] by David B. Doty |
| ="Pure"? "Rational"? Various shades of "Just" Intonation=
| | *[http://lumma.org/tuning/faq/#whatisJI What is "just intonation"?] by Carl Lumma |
| | | *[http://www.dbdoty.com/Words/werntz.html A Response to Julia Werntz] by David B. Doty |
| 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]].
| | *[http://lumma.org/tuning/gws/commaseq.htm Comma Sequences] by Gene Ward Smith |
| | | *[https://casfaculty.case.edu/ross-duffin/just-intonation-in-renaissance-theory-practice/ Just Intonation in Renaissance Theory & Practice] by Ross W. Duffin |
| =Just Intonation in use= | |
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| To start off your exploration of just intonation scales, [[Gallery of 12-tone Just Intonation Scales|this]] is a good place to start.
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| The use of just intonation could be divided into these two flavors:
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| ==Free Style Just==
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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==
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| (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 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). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].//
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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 [[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.//
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| //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.//
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| //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.//
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| //5. Restricting the denominator to one or very few values (the [[OverToneSeries|harmonic series]]).//
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| //6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).//
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| to this can be added
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| //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.//
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| //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//
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| =Variations on 'Just'=
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| [[Regular Temperaments]] are just intonation systems of various [[harmonic limits]] with certain commas 'tempered out'
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| [[AdaptiveJI|Adaptive JI]]
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| =Links=
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| [[hypergenesis58.scl|58 note 11 limit JI]] - hyper-Partchian!
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| [[Hahn distance]]
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| [[Gallery of Just Intervals]]
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| [[Gallery of 12-tone Just Intonation Scales]]
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| [[boogiewoogiescale|Boogie woogie scale]] | |
| [[Arnold Dreyblatt]]
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| [[Gallery of pentatonics]]
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| [[FiniteSubsetJI]]
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| [[List of root-3rd-P5 triads in JI]]
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| =Articles=
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| * [[http://en.wikipedia.org/wiki/Just_intonation|Wikipedia article on just intonation]] | |
| * [[http://nowitzky.hostwebs.com/justint/|Just Intonation]] by Mark Nowitzky [[http://www.webcitation.org/5xeAm2lPL|Permalink]]
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| * [[http://www.kylegann.com/tuning.html|Just Intonation Explained]] by Kyle Gann [[http://www.webcitation.org/5xe2iC7Nq|Permalink]] | |
| * [[http://www.kylegann.com/Octave.html|Anatomy of an Octave]] by Kyle Gann [[http://www.webcitation.org/5xe30LCev|Permalink]] | |
| * [[http://www.dbdoty.com/Words/What-is-Just-Intonation.html|What is Just Intonation?]] by David B. Doty [[http://www.webcitation.org/5xe3MeWVq|Permalink]] | |
| * [[http://www.dbdoty.com/Words/werntz.html|A Response to Julia Werntz]] by David B. Doty [[http://www.webcitation.org/5xe38KWx4|Permalink]] | |
| * [[http://lumma.org/tuning/gws/commaseq.htm|Comma Sequences]] by Gene Ward Smith [[http://www.webcitation.org/5xe4rPLZ0|Permalink]]</pre></div> | |
| <h4>Original HTML content:</h4>
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| <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:WikiTextTocRule:16:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- 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="#x&quot;Pure&quot;? &quot;Rational&quot;? Various shades of &quot;Just&quot; Intonation">&quot;Pure&quot;? &quot;Rational&quot;? Various shades of &quot;Just&quot; Intonation</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Just Intonation in use">Just Intonation in use</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- 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: -->
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Just Intonation explained"></a><!-- ws:end:WikiTextHeadingRule:0 -->Just Intonation explained</h1>
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| &quot;Just Intonation&quot;, as we find it commonly used today, 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.<br />
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| <br />
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| In languages other than English, the original conceptions of &quot;Just Intonation&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 &quot;natural scale&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.<br />
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| <br />
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| In the English language, the term &quot;just&quot; referred to &quot;true, correct&quot;, and is still used today in this sense, in the crafts. In printing, to &quot;justify&quot; a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as &quot;natural&quot;.<br />
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| <br />
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| Of course, an historical description of something as &quot;natural&quot; does not prove that that thing really is &quot;natural&quot;, and defining the concept of &quot;natural&quot;, especially in the arts, is always very problematic. Nevertheless, the historical meanings of the terms for what we call &quot;Just Intonation&quot; do claim a &quot;natural&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.<br />
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| <br />
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| The current common usage of describing intervals between pitches by specifying ratios of rational numbers is another way of expressing the &quot;natural scale&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 &quot;just intonation&quot; expanded.<br />
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| <br />
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| But, first things first. Let us take a look at why the idea of a &quot;natural&quot; or &quot;just&quot; tuning came about, and is still with us.<br />
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| <br />
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| If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz, we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...<br />
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| <br />
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| The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of &quot;music of the spheres&quot; and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.*<br />
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| <br />
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| Of course we are describing an ideal tone- in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our &quot;fuzzy&quot; and &quot;flawed&quot; reality, spectra adhere to this pattern with impressive consistency.<br />
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| <br />
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| In a tuning &quot;according to the natural scale&quot;, we have for example a &quot;perfect fifth&quot; as simply the ratio between the third partial and the second partial: &quot; 3:2&quot;. In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.<br />
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| <br />
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| Now, let us play our two example tones together, and we shall see why the German term is &quot;Reine&quot;, &quot;pure&quot;, and why you'll hear &quot;pure&quot; used in English as well (and in Czech, Slovak, etc.) Let's call our first tone &quot;Do&quot; and our second tone, a perfect fifth higher, &quot;Sol&quot;.<br />
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| <br />
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| Tone.... Frequencies of Partial Number<br />
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| &quot;Do&quot; 100 200 300 400 500 600 700 800 900<br />
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| &quot;Sol&quot; 150 300 450 600 750 900<br />
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| <br />
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| You see that the tones share the frequencies of some of the partials. These partials will &quot;meld&quot; when our Do and Sol are played together. This goes by the wonderful name of &quot;Tonverschmelzung&quot; in German. It is a very distinctinctive &quot;blending&quot; sound. If our Sol was tuned to, for example, 148 Hertz, its second harmonic component would be at 296 Hertz, and the two tones played together would not &quot;meld together&quot; at 300 Hertz, but would &quot;beat&quot;. That is, we would hear a throbbing sound, the &quot;beat rate&quot; of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300-296=4 Hertz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).<br />
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| One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.<br />
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| There is more to it than this, of course, but the basic principles of Just Intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals were the concern, understanding and defining &quot;Just&quot; was not difficult. These days, though, and going on from these basics, it can get a bit more complicated...<br />
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| If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), 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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| *All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are &quot;stretched&quot; according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic &quot;noise&quot;. A breathily played flute has a large addition of inharmonic material, a &quot;jinashi&quot; shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.<br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x&quot;Pure&quot;? &quot;Rational&quot;? Various shades of &quot;Just&quot; Intonation"></a><!-- ws:end:WikiTextHeadingRule:2 -->&quot;Pure&quot;? &quot;Rational&quot;? Various shades of &quot;Just&quot; Intonation</h1>
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| Just Intonation 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 />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Just Intonation in use"></a><!-- ws:end:WikiTextHeadingRule:4 -->Just Intonation in use</h1>
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| To start off your exploration of just intonation scales, <a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales">this</a> is a good place to start.<br />
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| The use of just intonation could be divided into these two flavors:<br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Just Intonation in use-Free Style Just"></a><!-- ws:end:WikiTextHeadingRule:6 -->Free Style Just</h2>
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| <a class="wiki_link" href="/Lou%20Harrison">Lou Harrison</a> 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 />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Just Intonation in use-Constrained Just"></a><!-- ws:end:WikiTextHeadingRule:8 -->Constrained Just</h2>
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| (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 />
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| <em>1. The principle of &quot;<a class="wiki_link" href="/Harmonic%20Limit">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). See <a class="wiki_link" href="/3-limit">3-limit</a>, <a class="wiki_link" href="/5-limit">5-limit</a>, <a class="wiki_link" href="/7-limit">7-limit</a>, <a class="wiki_link" href="/11-limit">11-limit</a>, <a class="wiki_link" href="/13-limit">13-limit</a>.</em><br />
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| <em>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 <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 />
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| <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 />
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| <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 />
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| <em>5. Restricting the denominator to one or very few values (the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>).</em><br />
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| <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 />
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| to this can be added<br />
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| <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 />
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| <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 />
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| <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Variations on 'Just'"></a><!-- ws:end:WikiTextHeadingRule:10 -->Variations on 'Just'</h1>
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| <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 />
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| <a class="wiki_link" href="/AdaptiveJI">Adaptive JI</a><br />
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| <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:12 -->Links</h1>
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| <a class="wiki_link" href="/hypergenesis58.scl">58 note 11 limit JI</a> - hyper-Partchian!<br />
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| <a class="wiki_link" href="/Hahn%20distance">Hahn distance</a><br />
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| <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intervals</a><br />
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| <a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales">Gallery of 12-tone Just Intonation Scales</a><br />
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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><br />
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| <a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI">List of root-3rd-P5 triads in JI</a><br />
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| <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Articles"></a><!-- ws:end:WikiTextHeadingRule:14 -->Articles</h1>
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| <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Just_intonation" rel="nofollow">Wikipedia article on just intonation</a></li><li><a class="wiki_link_ext" href="http://nowitzky.hostwebs.com/justint/" rel="nofollow">Just Intonation</a> by Mark Nowitzky <a class="wiki_link_ext" href="http://www.webcitation.org/5xeAm2lPL" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow">Just Intonation Explained</a> by Kyle Gann <a class="wiki_link_ext" href="http://www.webcitation.org/5xe2iC7Nq" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.kylegann.com/Octave.html" rel="nofollow">Anatomy of an Octave</a> by Kyle Gann <a class="wiki_link_ext" href="http://www.webcitation.org/5xe30LCev" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.dbdoty.com/Words/What-is-Just-Intonation.html" rel="nofollow">What is Just Intonation?</a> by David B. Doty <a class="wiki_link_ext" href="http://www.webcitation.org/5xe3MeWVq" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.dbdoty.com/Words/werntz.html" rel="nofollow">A Response to Julia Werntz</a> by David B. Doty <a class="wiki_link_ext" href="http://www.webcitation.org/5xe38KWx4" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://lumma.org/tuning/gws/commaseq.htm" rel="nofollow">Comma Sequences</a> by Gene Ward Smith <a class="wiki_link_ext" href="http://www.webcitation.org/5xe4rPLZ0" rel="nofollow">Permalink</a></li></ul></body></html></pre></div>
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