Temperament

Temperament is a method of tuning musical instruments based on approximating certain just intervals with others, in order to maintain the desired harmony between sounds. Process of application of temperament is called tempering, and it is performed in the tempering zone, usually twelve notes within one octave, which is subsequently repeated in remaining octaves.

Temperaments can be defined as sequences of operations or more formally as mathematical functions. Most commonly used can be divided into equal temperaments, well temperaments and regular temperaments.

History

In Medieval Western Europe Pythagorean tuning was the most widely used, in which fifths and fourths are just. At the time, thirds were omitted and considered a dissonance which followed from characteristics of Pythagorean tuning. Instrumental music was based on two-part voice leading - a majestic bourdon and melodic descant. To overcome these limitations, a compromise solution was sought that would allow the use of just thirds without sacrificing the purity of fifths.

That lead to adoption of meantone temperament (meantone tuning) in the end of the 15th century. Fifths began to be narrowed, which led to the complete disappearance of just fifth, instead impression of purity of harmony was achieved by just thirds; differently than before when the pure fifths could not compensate for wide major thirds. This enabled introduction of third-based chords and development of tertian harmony.

The downside of meantone was that not every diatonic scale known in European music in that time could be played in tune on tempered instruments. In practice, meantone was being modified in a various way by tuners to reduce dissonance created by wolf intervals. Some theoreticians since 16th century proposed to resolve the problem by use of equal temperaments like 12, 19 or 31 edo.

Adoption of 12-tone equal temperament became widely accepted practice already in the beginnings of 18th century.