Musical temperament

A musical temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. These pure intervals are based on a perfect fifth that has an exact 3:2 pitch ratio, but tempering narrows or widens them to accommodate a broader range of tuning needs. Temperaments are especially important for fixed-pitch instruments such as keyboards and guitars, which lack any way to easily make fine pitch adjustments during performance.

Temperaments can be divided into two categories, regular and irregular. Regular temperaments are generated with a finite number of generator intervals, most commonly a tempered perfect fifth, and a period, usually the octave. These include 12-tone equal temperament, the most commonly used temperament today, and more generally meantone temperaments which were historically used. Irregular temperaments cannot be described with such a generation process, and include the historically used well temperaments, which use an unequal sequence of twelve tempered fifths.

The development of well temperament allowed fixed-pitch instruments to play reasonably well in all of the keys. The famous Well-Tempered Clavier by Johann Sebastian Bach takes full advantage of this breakthrough, with pieces written in all 24 major and minor keys. However, while unpleasant intervals (such as the wolf interval) were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own character. This variation led in the 18th century to an increase in the use of 12-tone equal temperament, in which the frequency ratio between each pair of adjacent notes on the keyboard was made equal. In other words, the ratio between two notes that were one octave apart was kept pure, and the twelve notes in between the octave were equally spaced from one another. This allowed music to be transposed between keys without changing the relationship between notes.