Meantone temperament

The most common form of meantone temperament tunes all the major thirds to the just ratio of 5:4 (so, for instance, if G is tuned to 400Hz, B is tuned to 500Hz). This is achieved by tuning the perfect fifth a quarter of a syntonic comma flatter than the just ratio of 3:2. It is this that gives the system its name, quarter comma meantone.

This system gives whole tones in the ratio 2:sqrt(5), semitones in the ratio ratio 5^(5/4):8, and perfect fifths in the ratio of 1:5^(1/4), which is 1.495349.., compared with perfect fifth of 3:2, which is 1.5. One of the fifths will be a wolf interval, that is an interval which is so badly out of tune, it is not usable. This is because twelve perfect fifths, each flattened by a quarter of a syntonic comma, does not add up to an exact number of octaves.

The term meantone temperament is sometimes used when referring specifically to quarter tone meantone. However, because there are similar systems which tune the third to a slightly different size which are also frequently called meantone.

These systems, like quarter comma meantone, are also named after the amount which the size of the perfect fifths is flatter than a 3:2. So there is also 1/5 comma meantone and 2/7 comma meantone, for example.

Quarter comma meantone is particularly suitable for music which treats the third as a consonant interval. In practice, this means most western music since the 16th century. However, because of the wolf interval, it is not possible to play in tune in all keys. For this reason, well temperaments and eventually equal temperament, became more popular.