Just intonation: Difference between revisions

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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 21/12, 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. [[~~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)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.

== Consonance ==

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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.
-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 ==