Editing Meantone temperament (section)

==Notable meantone temperaments==
[[Image:Meantone.jpg|500px|thumb|right|Figure 1. Comparison between [[Pythagorean tuning]] (blue), [[equal-tempered]] (black), [[quarter-comma meantone]] (red) and third-comma meantone (green). For each, the common origin is arbitrarily chosen as C. The values indicated by the scale at the left are deviations in cents with respect to equal temperament.]]

[[Quarter-comma meantone]], which tempers each of the twelve [[perfect fifths]] by {{sfrac| 1 | 4 }} of a [[syntonic comma]], is the best known type of meantone temperament, and the term ''meantone temperament'' is often used to refer to it specifically. Four ascending fifths (as {{sc|'''C G D A E'''}}) tempered by {{nobr|{{sfrac| 1 | 4 }} comma}} (and lowered by two octaves) produce a just [[major third]] ({{sc|'''C E'''}}) (with ratio {{nobr|{{math|5 : 4}}}}), which is one syntonic comma (or about 22 [[Cent (music)|cents]]) narrower than the Pythagorean third that would result from four [[perfect fifth]]s.

It was commonly used from the early 16th century till the early 18th, after which twelve-tone equitemperament eventually came into general use. For church organs and some other keyboard purposes, it continued to be used well into the 19th century, and is sometimes revived in early music performances today. [[Quarter-comma meantone]] can be well approximated by a division of the octave into [[31 equal temperament|31&nbsp;equal steps]].

It proceeds in the same way as [[Pythagorean tuning]]; i.e., it takes the fundamental (say, {{sc|'''C'''}}) and goes up by six successive fifths (always adjusting by dividing by powers of {{math| 2 }} to remain within the octave above the fundamental), and similarly down, by six successive fifths (adjusting back to the octave by multiplying by {{nobr|powers of {{math|2}} ).}} However, instead of using the {{sfrac| 3 | 2 }} ratio, which gives [[perfect fifths]], this must be divided by the fourth root of {{sfrac| 81 | 80 }}, which is the [[syntonic comma]]: the ratio of the Pythagorean third {{sfrac| 81 | 64 }} to the just major third 
{{sfrac| 5 | 4 }}. Equivalently, one can use {{math|{{radic|5 |4}}}} instead of {{sfrac| 3 | 2 }}, which produces the same slightly reduced fifths. This results in the interval {{sc|'''C E'''}} being a [[Just intonation|just major third]] {{sfrac| 5 | 4 }}, and the intermediate seconds ({{sc|'''C D'''}}, {{sc|'''D E'''}}) dividing {{sc|'''C E'''}} uniformly, so {{sc|'''D C'''}} and {{sc|'''E D'''}} are equal ratios, whose square is {{sfrac| 5 | 4 }}. The same is true of the major second sequences {{sc|'''F G A'''}} and {{sc|'''G A B'''}}.

However, there is a residual gap in [[Quarter-comma meantone | quarter-comma meantone tuning]] between the last of the upper sequence of six fifths and the last of the lower sequence; e.g. between {{sc|'''F'''}}{{music|#}} and {{sc|'''G'''}}{{music|b}} if the starting point is chosen as  {{sc|'''C'''}}, which, adjusted for the octave, are in the ratio of {{sfrac| 125 | 128 }} or {{math | -41.06}} cents. This is in the sense opposite to the Pythagorean comma (i.e. the upper end is flatter than the lower one) and nearly twice as large.

In [[third-comma meantone]], the fifths are tempered by {{nobr|{{sfrac| 1 | 3 }}}} of a syntonic comma. It follows that three descending fifths (such as {{sc|'''A D G C'''}}) produce a [[Just intonation|just minor third]] ({{sc|'''A C'''}}) of ratio {{sfrac| 6 | 5 }}, which is nearly one syntonic comma wider than the minor third resulting from Pythagorean tuning of three [[perfect fifth]]s. Third-comma meantone can be very well approximated by a division of the octave into [[19 equal temperament|19&nbsp;equal steps]].