Just intonation - Revision history

FloraC at 13:11, 23 August 2026

2026-08-23T13:11:55Z

← Older revision		Revision as of 13:11, 23 August 2026
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	In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit\|5-limit tuning]], where the numerator and denominator of any ratio used has no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].		In 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 [[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)<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.		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.
Line 44:		Line 46:
	; Other approaches		; Other approaches
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.

			== Approximating JI with temperaments ==
			{{Main\|Temperament}}
			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 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 ==

Domin: The structure of 12edo is irrelevant to topic of just intonation.

2026-08-23T12:22:12Z

The structure of 12edo is irrelevant to topic of just intonation.

← Older revision		Revision as of 12:22, 23 August 2026
Line 12:		Line 12:

	In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit\|5-limit tuning]], where the numerator and denominator of any ratio used has no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].		In 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 [[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)<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.		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.

Domin: This is irrelevant to the topic of just intonation and contains factual error. If this section is considered to contain valuable correct the error and include it in appropriate page.

2026-08-23T12:05:35Z

This is irrelevant to the topic of just intonation and contains factual error. If this section is considered to contain valuable correct the error and include it in appropriate page.

← Older revision		Revision as of 12:05, 23 August 2026
Line 46:		Line 46:
	; Other approaches		; Other approaches
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.

	~~== 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 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 ==

Domin: Undo revision 236116 by Overthink (talk) Temperaments are not relevant to the topic of just intonation.

2026-08-23T12:04:34Z

Undo revision 236116 by Overthink (talk) Temperaments are not relevant to the topic of just intonation.

← Older revision		Revision as of 12:04, 23 August 2026
Line 48:		Line 48:

	== Approximating JI with temperaments ==		== Approximating JI with temperaments ==
	~~{{Main\|Temperament}}~~
	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.

Overthink: /* Approximating JI with temperaments */ add this

2026-08-20T22:39:24Z

Approximating JI with temperaments: add this

← Older revision		Revision as of 22:39, 20 August 2026
Line 48:		Line 48:

	== Approximating JI with temperaments ==		== Approximating JI with temperaments ==
			{{Main\|Temperament}}
	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.

Overthink: Undo revision 236043 by Domin (talk) It's only a brief section, and someone reading this page might not be fully committed to using JI by itself (which has many caveats). Explain on the talk page what you mean by "factual error"

2026-08-20T22:38:23Z

Undo revision 236043 by Domin (talk) It's only a brief section, and someone reading this page might not be fully committed to using JI by itself (which has many caveats). Explain on the talk page what you mean by "factual error"

← Older revision		Revision as of 22:38, 20 August 2026
Line 46:		Line 46:
	; Other approaches		; Other approaches
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.

			== 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 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 ==

Domin: This is irrelevant to the topic of just intonation and contains factual error. If this section is considered to contain valuable correct the error and include it in appropriate page.

2026-08-18T06:17:17Z

This is irrelevant to the topic of just intonation and contains factual error. If this section is considered to contain valuable correct the error and include it in appropriate page.

← Older revision		Revision as of 06:17, 18 August 2026
Line 46:		Line 46:
	; Other approaches		; Other approaches
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.

	~~== 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 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 ==

FloraC: + category

2026-08-09T07:06:37Z

+ category

← Older revision		Revision as of 07:06, 9 August 2026
Line 1:		Line 1:
	{{~~interwiki~~		{{Interwiki
	\| en = Just intonation		\| en = Just intonation
	\| de = Reine Stimmung		\| de = Reine Stimmung
Line 47:		Line 47:
	Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.		Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.

	==Approximating JI with temperaments==		== Approximating JI with temperaments ==
	There are a lot of JI intervals, and it's difficult to keep track of all of them. As such, people often use simpler systems to approximate JI intervals, known as [[temperament]]s. A popular choice is [[equal temperament]]s; for example the predominant [[12edo\|12et]], which is widely used to approximate [[5-limit]] JI. Other equal temperaments exist, for example [[19edo\|19et]], [[22edo\|22et]], and [[31edo\|31et]]. Besides equal temperaments, other temperaments exist, such as [[regular temperament]]s and [[well temperament]]s.		There 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.

Line 54:		Line 54:
	[[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.		[[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 ==
	{{~~todo~~\|expand\|comment=Expand the instruments section with more examples}}		{{Todo\|expand\|comment=Expand the instruments section with more examples}}
	*The [[Kalimba#Array mbira\|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range.
	*Most of [[Harry Partch]]'s instruments were designed to be for just intonation.		* The [[Kalimba #Array mbira\|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range.
	==Music==		* Most of [[Harry Partch]]'s instruments were designed to be for just intonation.

			== Music ==
	{{Main\|Music in just intonation}}		{{Main\|Music in just intonation}}

	== Notation ==		== Notation ==
	There are various [[~~Musical~~ notation\|notation systems]] for just intonation, for example [[~~Helmholtz-Ellis~~ notation]] and the [[Functional Just System]].		There are various [[musical notation\|notation systems]] for just intonation, for example [[Helmholtz–Ellis notation]] and the [[Functional Just System]].
	{{Todo\|expand\|inline=1}}		{{Todo\|expand\|inline=1}}

	==See also==		== See also ==
	{{~~todo~~\|cleanup\|inline=1}}		{{Todo\|cleanup\|inline=1}}
	*[[List of approaches to musical tuning]]
	*[[Gallery of just intervals]]		* [[List of approaches to musical tuning]]
	*[[Families of scales]]		* [[Gallery of just intervals]]
	*[[:Category:Just intonation]]		* [[Families of scales]]

	==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 -->

Fredg999: Linking

2026-07-27T20:36:44Z

Linking

← Older revision		Revision as of 20:36, 27 July 2026
Line 45:		Line 45:

	; Other approaches		; 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 set]]s.

	==Approximating JI with temperaments==		==Approximating JI with temperaments==

Fredg999: Interwiki

2026-07-19T06:40:31Z

Interwiki

← Older revision		Revision as of 06:40, 19 July 2026
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	\| en = Just intonation		\| en = Just intonation
	\| de = Reine Stimmung		\| de = Reine Stimmung
	\| es = Entonación ~~Justa~~		\| es = Entonación justa
	\| ja = 純正律		\| ja = 純正律
	\| ko = 순정률		\| ko = 순정률

← Older revision		Revision as of 12:22, 23 August 2026
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	In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit\|5-limit tuning]], where the numerator and denominator of any ratio used has no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].		In 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 [[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)<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.		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.