Temperament

Temperament is a method of tuning musical instruments based on approximating harmonic targets (almost always just intervals) with other intervals, in order to maintain the desired harmony between sounds while simultaneously simplifying melodic structure. For example, in standard tuning, we approximate the harmonic target of 3/2 with the 7-semitone interval, and the target of 5/4 with the 4-semitone interval. The process of application of temperament is called tempering.

Temperaments can be defined as sequences of operations or more formally as mathematical functions. Most commonly-used temperaments can be divided into well temperaments and regular temperaments. Equal temperaments, a very popular tuning approach, are a subset of regular temperaments which chop a given interval (such as the octave) into any number of equally large morsels.

History

In Medieval Western Europe, Pythagorean tuning was the most widely used, in which fifths and fourths are tuned to just intonation. At the time, thirds were omitted and considered a dissonance, which followed from the characteristics of Pythagorean tuning (the diatonic major third is rather complex). Instrumental music was based on two-part voice leading. 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 the adoption of meantone temperament in the end of the 15th century. Fifths began to be narrowed, which led to the complete disappearance of the just fifth, and instead the impression of purity of harmony was achieved by just thirds. This enabled the 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 various ways by tuners to reduce the dissonance created by wolf intervals. Some theoreticians since the 16th century proposed to resolve the problem by the use of equal temperaments like 12edo, 19edo, and 31edo.

12-tone equal temperament became widely accepted practice by the beginning of the 18th century.