Just intonation: Difference between revisions

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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 of a harmonic sound. 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 [[Wikipedia:Fraction|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]].

In theory, there are infinitely many just intervals, because each possible [[Wikipedia:Fraction|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]].

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

The structure of just intonation has several implications on music composition. [[Wolf ~~intervals~~]] and [[~~commas~~]], 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. The first approach leads mainly to [[adaptive just intonation]], while the second leads to [[temperament]].

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. The first approach leads mainly to [[adaptive just intonation]], while the second leads to [[temperament]].

== Just intonation explained ==

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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 of a harmonic sound. 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 [[Wikipedia:Fraction|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]].
+In theory, there are infinitely many just intervals, because each possible [[Wikipedia:Fraction|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]].
 In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit]] tuning. ''Extended just intonation'', a term coined by [[Ben Johnston]], usually refers to higher prime limits,<ref>[https://marsbat.space/pdfs/EJItext.pdf Sabat, Marc. ''On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation'']</ref> such as the [[7-limit]], the [[11-limit]] and the [[13-limit]]. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].
-The structure of just intonation has several implications on music composition. [[Wolf intervals]] and [[commas]], 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. The first approach leads mainly to [[adaptive just intonation]], while the second leads to [[temperament]].
+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. The first approach leads mainly to [[adaptive just intonation]], while the second leads to [[temperament]].
 == Just intonation explained ==