JI, or [[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 [[Benedetti 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''' ('''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.
"Just Intonation", as we find it commonly used today, describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios ([[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) of pitch frequencies.
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]].
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 series), Intonazione naturale (natural intonation, once again intervals derived from harmonic series) in Italian, and so on.
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.
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".
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.
Of course, a historical description of something as "natural" does not prove that something is "natural." Similarly labeling something "natural" without any ground, 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.
== 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.
Specifying ratios of frequencies is another way of expressing the "natural scale", for it describes ratios between partials in the harmonic series (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.
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.
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.
Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]].
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.
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]]
[[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.
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
[[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.
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
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]]).
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
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}}
Tone.... Frequencies of Partial Number
; Approximations/alterations of tempered tunings
"Do" 100 200 300 400 500 600 700 800 900
These are [[Detempering|detemperings]], including [[NEJI]] systems.
"Sol" 150 300 450 600 750 900
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
Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets|product sets]].
One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.
==Approximating JI with temperaments==
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.
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...
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.
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]].
==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}}
*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.
== Notation ==
There are various [[Musical notation|notation systems]] for just intonation, for example [[Helmholtz-Ellis notation]] and the [[Functional Just System]].
{{Todo|expand|inline=1}}
==See also==
{{todo|cleanup|inline=1}}
*[[List of approaches to musical tuning]]
*[[Gallery of just intervals]]
*[[Families of scales]]
*[[:Category:Just intonation]]
==References==
<references />
="Pure"? "Rational"? Various shades of "Just" Intonation=
==Further reading==
*[http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]]
Just Intonation is sometimes distinguished from //rational intonation,// by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]]).
*[http://nowitzky.hostwebs.com/justint/ Just Intonation] by Mark Nowitzky
*[http://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann
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 however, it casually suggests a rank two 7-limit [[Microtempering|microtempering]] system because of 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://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
=Just Intonation in use=
*[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
To start off your exploration of just intonation scales, [[Gallery of 12-tone Just Intonation Scales|this]] is a good place to start.
*[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
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). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].//
//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 and arithmetic// //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'
* [[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]]
* [[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://lumma.org/tuning/faq/#whatisJI|What is "just intonation"?]] by Carl Lumma [[http://www.webcitation.org/65NwFAKLh|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>
JI, or <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> 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="/Benedetti%20height">Benedetti 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 />
&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 (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow">rational numbers</a>) of pitch frequencies.<br />
<br />
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 series), Intonazione naturale (natural intonation, once again intervals derived from harmonic series) in Italian, and so on.<br />
<br />
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 />
<br />
Of course, a historical description of something as &quot;natural&quot; does not prove that something is &quot;natural.&quot; Similarly labeling something &quot;natural&quot; without any ground, 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 />
<br />
Specifying ratios of frequencies is another way of expressing the &quot;natural scale&quot;, for it describes ratios between partials in the harmonic series (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 />
<br />
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 />
<br />
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 />
<br />
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 />
<br />
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 />
<br />
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 />
<br />
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 />
<br />
Tone.... Frequencies of Partial Number<br />
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 />
<br />
One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.<br />
<br />
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 />
<br />
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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Just Intonation is sometimes distinguished from <em>rational intonation,</em> by requiring that the ratios be lower than some arbitrary complexity (as for example measured by <a class="wiki_link" href="/Tenney%20height">Tenney height</a>).<br />
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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 however, it casually suggests a rank two 7-limit <a class="wiki_link" href="/Microtempering">microtempering</a> system because of 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 />
<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 />
(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 />
<em>7. The use of</em> <em>harmonic and arithmetic</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>
<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 />
<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://lumma.org/tuning/faq/#whatisJI" rel="nofollow">What is &quot;just intonation&quot;?</a> by Carl Lumma <a class="wiki_link_ext" href="http://www.webcitation.org/65NwFAKLh" 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>
Just intonation (JI) is an approach to musical tuning where all intervals between two notes have frequency ratios which are rational numbers. 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 overtones 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 5th harmonic and the 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.
In the context of Western music theory prior to the 20th century, the term just intonation used alone usually refers to 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.[1] 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.
Just intonation contrasts with equal temperaments in that equal temperaments include intervals with irrational frequency ratios, which are not intervals of just intonation. For example, 12-tone equal temperament has a frequency ratio of 21/12, which is an irrational number, as a corollary of the rational root theorem. In fact, the only intervals in 12et which are also intervals of just intonation are multiples of the 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 ¢ in size, with the 7-step interval of 700 ¢, only 2 ¢ flat. The major third with frequency ratio 5/4, which is 386 ¢ in size, is approximated by the 4-step interval of 400 ¢, at 14 ¢ 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 19et, 22et, and 31et, also approximate various intervals of just intonation accurately, including higher-limit intervals not approximated well by 12et.
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 (4 × 700 ¢ – 400 ¢ – 2 × 1200 ¢ = 0 ¢), but does not do so in just intonation ((3/2)4 ÷ (5/4) ÷ (2/1)2 = 81/80 ≠ 1/1). The note reached is instead 81/80 (about 22 ¢) 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 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 temperaments.
A major triad in 12-tone equal temperament.The same major triad in 5-limit just intonation.
LCJI intervals achieve consonance through alignment of partials if the interval has 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.
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.
Similar logic may be used for instruments with timbres not aligning with the harmonic series; see timbral tuning.
Ways of using JI
Here are multiple ways in which musicians and theorists have used just intonation.
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.
Harmonic limits and subgroups
Harmonic limits, 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. Subgroups name a list of allowable prime numbers used. For example, the 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 forms, etc.
The use of harmonic and arithmetic 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.[citation needed]
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 temperaments. A popular choice is equal temperaments; for example the predominant 12et, which is widely used to approximate 5-limit JI. Other equal temperaments exist, for example 19et, 22et, and 31et. Besides equal temperaments, other temperaments exist, such as regular temperaments and well temperaments.
Temperaments also create new structures not found in JI; for example, meantone temperament (which 12et supports) 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.
Instruments
The 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.
