Meantone temperaments are musical temperaments, that is a variety of tuning systems, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them narrower than a perfect fifth), in order to push the thirds closer to pure. Meantone temperaments are constructed similarly to Pythagorean tuning, as a stack of equal fifths, but they are temperaments in that the fifths are not pure.
Notable meantone temperaments
Equal temperament, obtained by making all semitones the same size, each equal to one-twelfth of an octave (with ratio the 12th root of 2 to one (:1), narrows the fifths by about 2 cents or 1/12 of a Pythagorean comma, and produces thirds that are only slightly better than in Pythagorean tuning. Equal temperament is roughly the same as 1/11 comma meantone tuning.
Quarter-comma meantone, which tempers the fifths by 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 C–G–D–A–E) tempered by 1/4 comma produce a perfect major third (C–E), one syntonic comma narrower than the Pythagorean third that would result from four perfect fifths. Quarter-comma meantone has been practiced from the early 16th century to the end of the 19th. It can be approximated by a division of the octave in 31 equal steps.
This proceeds in the same way as Pythagorean tuning; i.e., it takes the fundamental (say, C) and goes up by six successive fifths (always adjusting by dividing by powers of 2 to remain within the octave above the fundamental), and similarly down, by six successive fifths, while adjusting the octave by multiplying by powers of 2. However, instead of using the 3/2 ratio, which gives "perfect" fifths, this must be multiplied by the fourth root of (80/81), (which is the "syntonic comma": the ratio of a just major third (5/4) to a Pythagorean one (81/64)) or, equivalently, one can use 5^(1/4), to slightly reduce the fifths. This results in the interval C-E being a "perfect third" (5/4), and the intermediate seconds (C-D, D-E) dividing C-E uniformly, so D-C and E-D are equal ratios whose square is 5/4. The same is true of the major second sequences F-G-A and G-A-B. However, there is still a "comma" in meantone tuning (i.e. the F and the G are not the same), which is actually larger than the Pythagorean one, and in the opposite sense.
In third-comma meantone, the fifths are tempered by 1/3 comma, and three descending fifths (such as A–D–G–C) produce a perfect minor third (A–C) one syntonic comma wider than the Pythagorean one that would result from three perfect fifths. Third-comma meantone can be approximated extremely well by a division of the octave in 19 equal steps.
The tone as a mean
The name "meantone temperament" derives from the fact that all such temperaments have only one size of the tone, between the major tone (8:9) and minor tone (9:10) of just intonation, which differ by a syntonic comma. In any regular system (i.e. with all fifths but one of the same size) the tone (as C–D) is reached after two fifths (as C–G–D), while the major third is reached after four fifths: the tone therefore is exactly half the major third.
This is one sense in which the tone is a mean.
In the case of quarter-comma meantone, in addition, where the major third is made narrower by a syntonic comma, the tone is also half a comma narrower than the major tone of just intonation, or half a comma wider than the minor tone: this is another sense in which the tone in quarter-tone temperament may be considered a mean tone, and it explains why quarter-comma meantone is often considered the meantone temperament properly speaking.
Meantone temperaments
"Meantone" can receive the following equivalent definitions: The meantone is the geometric mean between the major whole tone (9:8 in just intonation) and the minor whole tone (10:9 in just intonation). The meantone is the mean of its major third (for instance the square root of 5:4 in quarter-comma meantone).