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'''Just intonation''' ('''JI''') is an approach to [[musical tuning]] where [[pitch]]es are chosen in a way such that every [[interval]] can be expressed as a whole-number [[ratio]] of the [[frequencies]] of pitches. '''Just intervals''' naturally occur in the [[harmonic series]] as intervals between any two [[harmonic]]s of a fundamental tone produced with a harmonic [[timbre]]. For instance, an interval with a frequency ratio of [[3/2]] appears between the 2nd and 3rd harmonics. Just intonation is particularly efficient when used with harmonic instruments, because it allows the tuning and the timbre to reinforce each other.


In theory, there are infinitely many just intervals, because each possible {{w|fraction}} corresponds to a just interval. In practice, however, additional constraints are used to reduce the number of intervals to a reasonable amount, but also in many cases to prioritize [[consonant]] intervals. Usual constraints include [[subgroup]]s of [[generator]]s (including [[prime limit]]s), common denominators or numerators (as used in [[primodality]]), and [[complexity]] limits (usually [[height]] limits). Multiple constraints can be applied at the same time as well, such as the intersection of a prime limit and an [[odd limit]].
'''Just intonation''' ('''JI''') or '''Rational intonation''' ('''RI''') is an approach to [[musical tuning]] which uses the most [[concordant]] (or melding) intervals, which are found at rational ratios of [[Frequency|frequencies]].
 
Just intervals are precisely those intervals which achieve concordance 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 where every comprising interval is a concordance. Just intervals are also precisely those intervals which achieve concordance by melding with each other by forming different [[Harmonic|harmonics]] of the same unplayed root. (JI multi-note chords formed from harmonics of the same root can be the most concordant chords.)


In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit]] tuning. ''Extended just intonation'', a term coined by [[Ben Johnston]], usually refers to higher prime limits,<ref>[https://marsbat.space/pdfs/EJItext.pdf Sabat, Marc. ''On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation'']</ref> such as the [[7-limit]], the [[11-limit]] and the [[13-limit]]. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].
In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit]] tuning. ''Extended just intonation'', a term coined by [[Ben Johnston]], usually refers to higher prime limits,<ref>[https://marsbat.space/pdfs/EJItext.pdf Sabat, Marc. ''On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation'']</ref> such as the [[7-limit]], the [[11-limit]] and the [[13-limit]]. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].


The structure of just intonation has several implications on music composition. [[Wolf interval]]s and [[comma]]s, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[comma pump]]s, 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 features or as problems to be solved. Examples of approaches that try to solve these problems include pitch shifts, [[adaptive just intonation]] and [[temperament]].
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 features or as problems to be solved. Examples of approaches that try to solve these problems include pitch shifts, [[adaptive just intonation]] and [[temperament]].
 
== Just intonation explained ==
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 {{w|Pitch (music)|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 [[harmonic series]].
 
In languages other than English, the original conceptions of "just intonation" are more obviously retained in the terms used in those languages: German ''Reine Stimmung'' (pure, that is, beatless, tuning), Ukrainian ''Натуральний стрій'' and French ''gamme naturelle'' (both referring to the "natural scale", that is, intervals derived from the harmonic series), Italian ''intonazione naturale'' (natural intonation, once again intervals derived from harmonic series), and so on.
 
In the English language, the term "just" once 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".
 
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.
 
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.


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.
==Just intonation in use==
Here are multiple ways in which musicians and theorists have used just intonation.


If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), 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...
[[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.


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<ref>All manner of [[bell]]s, [[gong]]s, [[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.</ref>.  
'''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.


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.
'''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]])


In just intonation, the perfect fifth is simply the ratio between the second and third harmonics: 2:3. In our example tone, that would be the ratio of 200 Hz to 300 Hz. A just intonation perfect fifth above our original tone would have a fundamental frequency 3/2 times the fundamental frequency of our original tone. 3/2 times 100 Hz gives us 150 Hz.
'''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.


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 and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "So".
'''Approximations/alterations of tempered tunings''' <br />
These are [[Detempering|detemperings]], including [[NEJI]] systems.  


<pre>Tone  Frequencies of partials (Hz)
'''Other approaches'''<br />
Do    100  200  300  400  500  600  700  800  900  ...
Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[Tritriadic scale|titriadic scales]], and [[combination product sets|product sets]].
So    150  300  450  600  750  900  1050  1200  1350  ...</pre>


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 distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, 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 Hz, 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).
==Instruments==
 
{{todo|expand|comment=Expand the instruments section with more examples}}
One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.
*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.
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 was 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...
==Music==
 
== Just intonation in use ==
To start off your exploration of just intonation scales, the [[Gallery of 12-tone Just Intonation Scales]] is a good place to start.
 
Look at [[Musical notation|notation systems]] for just intonation.
 
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: [[Free style JI]]
 
=== Constrained just ===
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):
 
# ''The principle of "[[harmonic limit]]s", 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]].''
# ''Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of {{w|Harry Partch}}'s {{w|tonality diamond}}. This, incidentally, 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.''
# ''Other theorists who, in contrast to the above, advocate the use of [[combination product sets|products sets]] of given arrays of prime numbers, such as [[Ervin Wilson]], Robert Dussaut, and others.''
# ''[[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.''
# ''Restricting the denominator to one or very few values (the [[harmonic series]], [[isoharmonic chord]]s, [[AFDO]]s/[[overtone scale]]s).''
# ''Restricting the numerator to one or a very few values (the [[subharmonic series]],  [[IFDO]]s/undertone scales).''
 
To this may be added:
# ''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.''
# ''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.''
# ''Choosing some set of relatively high overtones (disregarding prime limit or subgroups), and using each overtone as a root for extended harmony within the set ([[primodality]], [[neji]]s).''
# ''The use of a [[tritriadic scale]] as described by John Chalmers in 1986.''
 
== Instruments ==
Any tunable instrument can be tuned to just intonation, with the exception of fretted instruments like guitars, in which each string's notes have a "baked in" tuning. Even those instruments can be refretted to just intonation. A few noteworthy examples:
* All fretless string instruments.
* The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a 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}}
{{Main|Music in just intonation}}


== See also ==
== Notation ==
* [[Harmonic series]]
There are various [[Musical notation|notation systems]] for just intonation.
* [[Overtone scale]]
==See also==
* [[Gallery of just intervals]]
{{todo|cleanup|inline=1}}
* [[Gallery of 12-tone just intonation scales]]
*[[List of approaches to musical tuning]]
* [[Helmholtz–Ellis notation]]
*[[Gallery of just intervals]]
* [[:Category:Just intonation]]
*[[Gallery of 12-tone just intonation scales]]
 
*[[Families of scales]]
== References ==
*[[boogiewoogiescale|Boogie woogie scale]]
*[[:Category:Just intonation]]
==References==
<references />
<references />


== Further reading ==
==Further reading==
* [http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]]
*[http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]]
* [http://nowitzky.hostwebs.com/justint/ Just Intonation] by Mark Nowitzky
*[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/tuning.html Just Intonation Explained] by Kyle Gann
* [http://www.kylegann.com/Octave.html Anatomy of an Octave] 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
*[http://www.dbdoty.com/Words/What-is-Just-Intonation.html What is Just Intonation?] by David B. Doty
* [http://lumma.org/tuning/faq/#whatisJI What is "just intonation"?] by Carl Lumma
*[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
*[http://www.dbdoty.com/Words/werntz.html A Response to Julia Werntz] by David B. Doty
* [http://lumma.org/tuning/gws/commaseq.htm Comma Sequences] by Gene Ward Smith
*[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
*[https://casfaculty.case.edu/ross-duffin/just-intonation-in-renaissance-theory-practice/ Just Intonation in Renaissance Theory & Practice] by Ross W. Duffin
 
[[Category:Just intonation| ]] <!-- main article -->
[[Category:Terms]]

Revision as of 19:23, 14 May 2025

English Wikipedia has an article on:

Just intonation (JI) or Rational intonation (RI) is an approach to musical tuning which uses the most concordant (or melding) intervals, which are found at rational ratios of frequencies.

Just intervals are precisely those intervals which achieve concordance 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 where every comprising interval is a concordance. Just intervals are also precisely those intervals which achieve concordance by melding with each other by forming different harmonics of the same unplayed root. (JI multi-note chords formed from harmonics of the same root can be the most concordant chords.)

In the context of Western music theory prior to the 20th century, the term just intonation used alone usually refers to 5-limit tuning. Extended just intonation, a term coined by Ben Johnston, usually refers to higher prime limits,[1] such as the 7-limit, the 11-limit and the 13-limit. The practice of just intonation without any particular constraint is sometimes referred to as rational intonation (RI) or as free style JI.

The structure of just intonation has several implications on music composition. Wolf intervals and commas, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are 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 features or as problems to be solved. Examples of approaches that try to solve these problems include pitch shifts, adaptive just intonation and temperament.

Just intonation in use

Here are multiple ways in which musicians and theorists have used just intonation.

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.

Harmonic limits and subgroups
Harmonic limits set a limit for the highest prime number in the factorization of any ratio used. Subgroups name a list of allowable prime numbers used.

Restrictions on the denominator or numerator
Some approaches restrict "the denominator to one or very few values"[2] (the harmonic series, isoharmonic chords, AFDOs/overtone scales, primodality, ringer scales), the "numerator to one or a very few values" (the subharmonic series, IFDOs/undertone scales), or both (tonality diamonds)

Mediants
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.

Approximations/alterations of tempered tunings
These are detemperings, including NEJI systems.

Other approaches
Other approaches include Meru scales, titriadic scales, and product sets.

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.

Music

Main article: Music in just intonation

Notation

There are various notation systems for just intonation.

See also

Todo: cleanup

References

  1. ↑ Sabat, Marc. On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation
  2. ↑ From Jacques Dudon, "Differential Coherence", 1/1 vol. 11, no. 2: p.1).

Further reading

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