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.