Just intonation: Difference between revisions

Line 9:

'''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}}

@@ Line 9: / Line 9: @@
 {{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}}