in music, the mathematical expression of a given system of pitch relationships. A tuning system is characterized by a series of tones expressed as a set of whole or fractional numbers, each of which shows the ratio of the frequency of the upper tone to that of the lower tone of an interval. Tuning systems are important in the construction and tuning of musical instruments with fixed or semifixed pitch.

The monodic music of a number of European peoples typically uses Pythagorean tuning, which is br.sed on a perfect fifth, having a pitch ratio of 3 : 2. Beginning approximately in the 16th century, just intonation was widely used in polyphonic music. It was based on the perfect fifth and the major third, having a pitch ratio of 5 : 4. In the early 18th century, the intensive development of tonal relationships led to the creation of equal-temperament tuning, in which a perfect octave (2 : 1) is divided into 12 equal semitones. In actual performance on instruments that have undetermined tuning, such as the violin and cello, intonation is in zones.