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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''') 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.


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|5-limit tuning]], where the numerator and denominator of any ratio used has no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].


The 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]].
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.


== Just intonation explained ==
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.
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.
== 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.  


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".
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.


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.
Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]].


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.
== Ways of using JI ==
Here are multiple ways in which musicians and theorists have used just intonation.


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.
; [[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.


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...
; Harmonic limits and subgroups
[[Harmonic limit]]s 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]]. [[Subgroup]]s name a list of allowable prime numbers used.


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>.
; 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]])


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.
; 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}}


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.
; Approximations/alterations of tempered tunings
These are [[Detempering|detemperings]], including [[NEJI]] systems.  


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".
; Other approaches
Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets|product sets]].


<pre>Tone  Frequencies of partials (Hz)
==Approximating JI with temperaments==
Do    100  200  300  400  500  600  700  800  900  ...
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.
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).
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.


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}}
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...
*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.
== Just intonation in use ==
==Music==
To start off your exploration of just intonation scales, the [[Gallery of 12-tone Just Intonation Scales]] is a good place to start.
{{Main|Music in just intonation}}
 
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 ==
== Notation ==
{{Main|Music in just intonation}}
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 ==
==See also==
* [[Harmonic series]]
{{todo|cleanup|inline=1}}
* [[Overtone scale]]
*[[List of approaches to musical tuning]]
* [[Gallery of just intervals]]
*[[Gallery of just intervals]]
* [[Gallery of 12-tone just intonation scales]]
*[[Families of scales]]
* [[Helmholtz–Ellis notation]]
*[[:Category:Just intonation]]
* [[:Category:Just intonation]]


== References ==
==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]]

Latest revision as of 02:53, 6 June 2026

English Wikipedia has an article on:

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.

Consonance

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.

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; for example, western music is based off the 5-limit. 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 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.[citation needed]

Approximations/alterations of tempered tunings

These are detemperings, including NEJI systems.

Other approaches

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

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 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.

Music

Main article: Music in just intonation

Notation

There are various notation systems for just intonation, for example Helmholtz-Ellis notation and the Functional Just System.

Todo: expand

See also

Todo: cleanup

References

  1. ↑ From Ben Johnston "A Notation System for Extended Just Intonation." Maximum Clarity, 2006, p. 77
  2. ↑ From Jacques Dudon, "Differential Coherence", 1/1 vol. 11, no. 2: p.1).

Further reading

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