Tempered Tuning

A system of tuning in which the intervals deviate from the "pure" (i.e., acoustically correct)  intervals of the Pythagorean system and just intonation.  The deviations are necessary because these two systems, although perfect within a small range of tones (mainly those of the C-major scale), become increasingly inadequate with the successive introduction of the chromatic tones.  For instance, the acoustically perfect fifth might well be used to obtain a succession of five or six fifths, c,g,d,a,e,b.  However, if tones such as f#, c#, g#, d# are added in the same manner, the resulting tones cannot be satisfactorily used for melodies such as  d e f# g.  Moreover, the twelfth tone of the succession of fifths, b#, is noticeably higher than the tone c it would represent in our system of notation.  Thus it is necessary to devise methods that, instead of being perfect in the simple keys and intolerably wrong in the others, spread the inevitable inaccuracy over all the tones and keys.  The most consistent realization of this principle is the "equal temperament" universally used today.

The principle of equal temperament is to divide the octave into twelve equal semitones.  In equal temperament no interval other than the octave is acoustically correct or pure.  The deviation of the fifth by 2 cents (100 cents = a "well-tempered" semitone) is too small to be perceived.  With the thirds, the difference is considerably greater.  The well-tempered major third (400 cents) is 14 cents larger (sharper) than the "pure" major third (386 cents).

The introduction of equal temperament into musical practice was very slow.  The system was not adopted in Germany until c. 1800, and in France and England c. 1850.