Editing Musical tuning (section)

===Systems for the twelve-note chromatic scale<span class="anchor" id="tuning_systems_anchor"></span>===
[[File:Circle progression I IV V I.png|thumb|Comparison of tunings: [[circle progression|I IV<math>{}^6_4</math> V I]]. {{audio|Circle progression just intonation I IV V I.mid|Play just}}, {{audio|Circle progression Pythagorean tuning I IV V I.mid|Play Pythagorean}}, {{audio|Circle progression quarter-comma meantone I IV V I.mid|Play meantone}} (quarter-comma), {{audio|Circle progression Werckmeister temperament I IV V I.mid|Play well temperament}} ([[Werckmeister temperament|Werckmeister]]), and {{audio|Progression en cercle I IV V I.ogg|Play equal temperament}}]]
It is impossible to tune the twelve-note [[chromatic scale]] so that all [[interval (music)|interval]]s are pure. For instance, three pure major thirds stack up to {{small|{{sfrac| 125 | 64 }}}}, which at {{nobr| {{gaps|1|159}} cents}} is nearly a quarter tone away from the octave (1200&nbsp;cents). So there is no way to have both the octave and the major third in just intonation for all the intervals in the same twelve-tone system. Similar issues arise with the fifth {{small|{{sfrac|3|2}}}}, and the minor third {{small|{{sfrac| 6 | 5 }}}}, or any other choice of harmonic-series based pure intervals.

Many different compromise methods are used to deal with this, each with its own characteristics, and advantages and disadvantages.

The main ones are:
; [[Just intonation]] : [[File:Prelude 1, Just intonation.ogg|thumb|Prelude No. 1, C major, BWV 846, from the [[Well-Tempered Clavier]] by [[Johann Sebastian Bach]]. Played in just intonation.]]
: In just intonation, the frequencies of the scale notes are related to one another by simple numeric ratios, a common example of this being {{math| {{small|{{sfrac| 1 | 1 }}}}, {{small|{{sfrac| 9 | 8 }}}}, {{small|{{sfrac| 5 | 4 }}}}, {{small|{{sfrac| 4 | 3 }}}}, {{small|{{sfrac| 3 | 2 }}}}, {{small|{{sfrac| 5 | 3 }}}}, {{small|{{sfrac| 15 | 8 }}}}, {{small|{{sfrac| 2 | 1 }}}} }} to define the ratios for the seven notes in a C&nbsp;major scale, plus the return to the tonic an octave up on the 8th ([[octave|"perfect 8th" or octave]]). In this example, though many intervals are pure, the interval from D to A ({{small|{{sfrac| 5 | 3 }}}} to {{small|{{sfrac| 9 | 8 }}}}) is {{small|{{sfrac| 40 | 27 }}}} instead of the expected {{small|{{sfrac| 3 | 2 }}}}. The same issue occurs with most just intonation tunings. This can be dealt with to some extent using alternative pitches for the notes. Even that, however, is only a partial solution, as an example makes clear: If one plays the sequence C G D A E C in just intonation, using the intervals {{small|{{math|{{sfrac| 3 | 2 }}}}}}, {{small|{{sfrac| 3 | 4 }}}}, and {{small|{{sfrac| 4 | 5 }}}}, then the second C in the sequence is higher than the first by a [[syntonic comma]] of {{small|{{math|{{sfrac| 81 | 80 }}}}}}. This is the infamous "[[comma pump]]". Each time around the comma pump, the pitch continues to spiral upwards. This shows that it is impossible to keep to any small fixed system of pitches if one wants to stack musical intervals this way. So, even with ''adaptive tuning'', the musical context may sometimes require playing musical intervals that are not pure. Instrumentalists with the ability to vary the pitch of their instrument may micro-adjust some of the intervals naturally; there are also systems for adaptive tuning in software ([[microtuner]]s). Harmonic fragment scales form a rare exception to this issue. In tunings such as {{math|1:1, 9:8, 5:4, 3:2, 7:4, 2:1}}, all the pitches are chosen from the harmonic series (divided by powers of 2 to reduce them to the same octave), so all the intervals are related to each other by simple numeric ratios. 
; [[Pythagorean tuning]] : [[File:Prelude 1, Pythagorean tuning.ogg|thumb|Prelude No. 1, C major, BWV 846, from the Well-Tempered Clavier by Johann Sebastian Bach. Played in Pythagorean tuning.]]
: A Pythagorean tuning is technically both a type of just intonation and a zero-[[syntonic comma|comma]] meantone tuning, in which the frequency ratios of the notes are all derived from the number ratio&nbsp;3:2. Using this approach for example, the 12&nbsp;notes of the Western chromatic scale would be tuned to the following ratios: {{math| 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 729:512, 3:2, 128:81, 27:16, 16:9, 243:128, 2:1&nbsp;.}} Also called "3-limit" because it uses no prime factors other than 2 and 3, this Pythagorean system was of primary importance in Western musical development in the Medieval and Renaissance periods. As with nearly all just intonation systems, it has a [[wolf interval]]. In the example given, it is the interval between the {{math| 729:512 }} and the{{math|  256:243 }} (F{{sup|{{music|#}}}} to D{{sup|{{music|b}}}}, if one tunes the {{small|{{math|{{sfrac| 1 | 1 }}}}}} to C). The major and minor thirds are also impure, but at the time when this system was at its zenith, the third was considered a dissonance, so this was of no concern. See also: [[Shí-èr-lǜ]].
; [[Meantone temperament]] : [[File:Prelude 1, Meantone temperament.ogg|thumb|Prelude No. 1, C major, BWV 846, from the [[Well-Tempered Clavier]] by [[Johann Sebastian Bach]]. Played in meantone temperament.]]
: A system of tuning that averages out pairs of ratios used for the same interval (such as 9:8 and 10:9). The best known form of this temperament is [[quarter-comma meantone]], which tunes major thirds justly in the ratio of 5:4 and divides them into two whole tones of equal size – this is achieved by flattening the fifths of the Pythagorean system slightly (by a quarter of a [[syntonic comma]]). However, the fifth may be flattened to a greater or lesser degree than this and the tuning system retains the essential qualities of [[meantone temperament]]. Historical examples include [[19 equal temperament|{{nobr|{{small|{{sfrac| 1 | 3 }}}} comma}}]] and {{nobr|{{small|{{sfrac| 2 | 7 }}}} comma}} meantone.
; [[Well temperament]] : [[File:Prelude 1, Werckmeister temperament.ogg|thumb|Prelude No. 1, C major, BWV 846, from the Well-Tempered Clavier by Johann Sebastian Bach. Played in well temperament.]]
: Any one of a number of systems where the ratios between intervals are unequal, but approximate to ratios used in just intonation. Unlike meantone temperament, the amount of divergence from just ratios varies according to the exact notes being tuned, so that C–E is probably tuned closer to a 5:4 ratio than, say, D{{sup|{{Music|b}}}}–F. Because of this, well temperaments have no [[wolf interval]]s.
; [[Equal temperament]] : [[File:Prelude 1, Equal temperament.ogg|thumb|Prelude No. 1, C major, BWV 846, from the Well-Tempered Clavier by Johann Sebastian Bach. Played in equal temperament.]]
: The standard twelve-tone equal temperament is a special case of meantone temperament (extended eleventh-comma), in which the twelve notes are separated by [[logarithm]]ically equal distances (100&nbsp;cents): [//upload.wikimedia.org/wikipedia/commons/b/b3/Et_scale.ogg A harmonized C major scale in equal temperament] (.ogg format, 96.9&nbsp;KB). This is the most common tuning system used in Western music, and is the standard system used as a basis for [[Piano tuning|tuning a piano]]. Since this scale divides an octave into twelve equal-ratio steps and an octave has a frequency ratio of two, the frequency ratio between adjacent notes is then the [[twelfth root of two]], {{nobr|{{math| 2{{sup| {{small|1/12}} }} ≋ 1.05946309}} ... .}} However, the octave can be divided into other than 12&nbsp;equal divisions, some of which may be more harmonically pleasing, as far as thirds, sixths, and [[harmonic seventh]]s (via augmented sixths) are concerned, such as [[19 equal temperament]] (extended {{small|{{nobr|{{sfrac| 1 | 3 }}}} comma}} meantone), [[31 equal temperament]] (extended quarter-comma meantone) and [[53 equal temperament]] (extended Pythagorean tuning).

Tuning systems that are not produced with exclusively just intervals are usually referred to as ''[[Musical temperament|temperaments]]''.