Just intonation: Difference between revisions

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'''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|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]].

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 2^1/12, 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.

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)⁴ ÷ (5/4) ÷ (2/1)² {{=}} [[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.

[[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.  

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 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.

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]]).

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}}

==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.

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.

==Instruments==

*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 are various [[Musical notation|notation systems]] for just intonation, for example [[Helmholtz-Ellis notation]] and the [[Functional Just System]].

 

==See also==

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*[[Gallery of just intervals]]

*[[Families of scales]]

*[[:Category:Just intonation]]

 

==References==

<references />

 

==Further reading==

*[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://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann

*[http://www.kylegann.com/Octave.html Anatomy of an Octave] by Kyle Gann

*[http://www.dbdoty.com/Words/What-is-Just-Intonation.html What is Just Intonation?] by David B. Doty

*[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://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