Just intonation - Revision history

Overthink: /* Approximating JI with temperaments */ note microtemperaments

2026-06-22T04:47:06Z

Approximating JI with temperaments: note microtemperaments

← Older revision		Revision as of 04:47, 22 June 2026
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	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 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.		Temperaments also create new structures not found in JI; for example, [[meantone]] temperament (which 12et [[support]]s) tempers out the syntonic comma [[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. There are many temperaments in various limits, including the 5-limit, the [[7-limit]], and beyond, as well as many subgroups.

			[[Microtemperaments]] are very accurate and temper out [[unnoticeable comma]]s, and add minimal damage to the intervals of JI, while also simplifying its structure. For example, [[schismic]] flattens the fifth by around 0.2 cents to equate the [[Pythagorean comma]] with the syntonic comma. Microtemperaments are also very useful in higher prime limits, where JI itself becomes too complex.

	==Instruments==		==Instruments==

Overthink: /* Ways of using JI */ expand this section

2026-06-08T00:51:35Z

Ways of using JI: expand this section

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	; Harmonic limits and subgroups		; 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.		[[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.

	; Restrictions on the denominator or numerator		; 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]])		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]]).

	; Mediants		; Mediants

Overthink: /* Notation */ a few examples; mark to expand

2026-06-06T02:53:23Z

Notation: a few examples; mark to expand

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	== Notation ==		== Notation ==
	There are various [[Musical notation\|notation systems]] for 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==
	{{todo\|cleanup\|inline=1}}		{{todo\|cleanup\|inline=1}}

Overthink: /* See also */ remove somewhat arbitrary inclusions

2026-06-06T02:52:30Z

See also: remove somewhat arbitrary inclusions

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	*[[List of approaches to musical tuning]]		*[[List of approaches to musical tuning]]
	*[[Gallery of just intervals]]		*[[Gallery of just intervals]]
	*[[Gallery of 12-tone just intonation scales]]
	*[[Families of scales]]		*[[Families of scales]]
	*[[boogiewoogiescale\|Boogie woogie scale]]
	*[[:Category:Just intonation]]		*[[:Category:Just intonation]]

	==References==		==References==
	<references />		<references />

Overthink: /* Consonance */ add audio

2026-06-06T02:27:35Z

Consonance: add audio

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	== Consonance ==		== 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.		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.

Overthink: explain in the last paragraph

2026-06-06T02:02:33Z

explain in the last paragraph

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	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 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.

	The structure of just intonation has several implications on music composition. [[~~Wolf interval\|Wolf intervals~~]] and [[~~Comma\|commas~~]], ~~two kinds of dissonant intervals~~, ~~may appear between distantly~~-~~related pitches~~. ~~In addition~~, ~~certain~~ chord progressions ~~are~~ [[Comma pump\|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems ~~without greatly restricting the set of available ratios~~ include pitch shifts, [[adaptive just intonation]] and [[temperament]]. ~~Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently. {{todo\|clarify}}~~		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.

	== Consonance ==		== Consonance ==

Overthink: explain better; add ET paragraph

2026-06-06T01:36:59Z

explain better; add ET paragraph

← Older revision		Revision as of 01:36, 6 June 2026
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	{{Wikipedia}}		{{Wikipedia}}

	'''Just intonation''' ('''JI''') is an approach to [[musical tuning]] which ~~uses tones whose frequencies~~ are ~~whole-~~number ~~ratios of~~ a ~~given fundamental~~ [[~~frequency~~]]. Just intonation ~~includes~~ the [[harmonic series]], which is the collection of tones found at integer multiples of a fundamental frequency~~; all~~ just intervals can be found as the interval between two notes in the harmonic series. Just ratios of small numbers, called ~~'''Low~~-complexity just intonation (LCJI)~~'''~~ intervals, tend to be the most [[~~concordant\|~~consonant]] in the sense that their sounds meld together.		'''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]]~~—intervals~~ where the ~~numerators~~ and ~~denominators~~ of any ratio used ~~have~~ no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].		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<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.

	The structure of just intonation has several implications on music composition. [[Wolf interval\|Wolf intervals]] and [[Comma\|commas]], two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[Comma pump\|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, [[adaptive just intonation]] and [[temperament]]. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently. {{todo\|clarify}}		The structure of just intonation has several implications on music composition. [[Wolf interval\|Wolf intervals]] and [[Comma\|commas]], two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[Comma pump\|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, [[adaptive just intonation]] and [[temperament]]. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently. {{todo\|clarify}}

FloraC: /* Ways of using JI */ formatting and request citation

2025-12-27T17:59:50Z

Ways of using JI: formatting and request citation

← Older revision		Revision as of 17:59, 27 December 2025
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	Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]].		Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]].

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

	[[Free style JI~~\|'''Free style JI'''~~]] ~~<br />~~		; [[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.		[[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~~'''<br />~~		; Harmonic limits and subgroups
	[[Harmonic limit~~\|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]]. [[Subgroup]]s name a list of allowable prime numbers used.		[[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.

	~~'''~~Restrictions on the denominator or numerator~~'''<br />~~		; 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]])		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]])

	~~'''~~Mediants~~'''<br />~~		; Mediants
	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.		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}}

	~~'''~~Approximations/alterations of tempered tunings~~''' <br />~~		; Approximations/alterations of tempered tunings
	These are [[Detempering\|detemperings]], including [[NEJI]] systems.		These are [[Detempering\|detemperings]], including [[NEJI]] systems.

	~~'''~~Other approaches~~'''<br />~~		; Other approaches
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets\|product sets]].		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets\|product sets]].

Overthink: /* Ways of using JI */ mention 5-limit

2025-12-27T10:29:39Z

Ways of using JI: mention 5-limit

← Older revision		Revision as of 10:29, 27 December 2025
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	'''Harmonic limits and subgroups'''<br />		'''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.		[[Harmonic limit\|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]]. [[Subgroup]]s name a list of allowable prime numbers used.

	'''Restrictions on the denominator or numerator'''<br />		'''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]])		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]])

	'''Mediants'''<br />		'''Mediants'''<br />

Overthink: brief temperament section

2025-12-27T10:21:55Z

brief temperament section

← Older revision		Revision as of 10:21, 27 December 2025
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	'''Other approaches'''<br />		'''Other approaches'''<br />
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets\|product sets]].		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product sets\|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 [[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==		==Instruments==

← Older revision		Revision as of 01:36, 6 June 2026
Line 9:		Line 9:
	{{Wikipedia}}		{{Wikipedia}}

	'''Just intonation''' ('''JI''') is an approach to [[musical tuning]] which ~~uses tones whose frequencies~~ are ~~whole-~~number ~~ratios of~~ a ~~given fundamental~~ [[~~frequency~~]]. Just intonation ~~includes~~ the [[harmonic series]], which is the collection of tones found at integer multiples of a fundamental frequency~~; all~~ just intervals can be found as the interval between two notes in the harmonic series. Just ratios of small numbers, called ~~'''Low~~-complexity just intonation (LCJI)~~'''~~ intervals, tend to be the most [[~~concordant\|~~consonant]] in the sense that their sounds meld together.		'''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]]~~—intervals~~ where the ~~numerators~~ and ~~denominators~~ of any ratio used ~~have~~ no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].		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<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.

	The structure of just intonation has several implications on music composition. [[Wolf interval\|Wolf intervals]] and [[Comma\|commas]], two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[Comma pump\|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, [[adaptive just intonation]] and [[temperament]]. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently. {{todo\|clarify}}		The structure of just intonation has several implications on music composition. [[Wolf interval\|Wolf intervals]] and [[Comma\|commas]], two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[Comma pump\|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, [[adaptive just intonation]] and [[temperament]]. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently. {{todo\|clarify}}