Helmholtz-Ellis notation

In music, just intonation (sometimes abbreviated as JI) or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a pure or just interval. Pure intervals are important in music because they correspond to the vibrational patterns found in physical objects which correlate to human perception (see harmonic series). The two notes in any just interval are members of the same harmonic series. Frequency ratios involving large integers such as 1024:729 are not generally said to be justly tuned. "Just intonation is the tuning system of the later ancient Greek modes as codified by Ptolemy; it was the aesthetic ideal of the Renaissance theorists; and it is the tuning practice of a great many musical cultures worldwide, both ancient and modern."

Just intonation can be contrasted and compared with equal temperament, which dominates Western instruments of fixed pitch (e.g., piano or organ) and default MIDI tuning on electronic keyboards. In equal temperament, all intervals are defined as multiples of the same basic interval, or more precisely, the intervals are ratios which are integer powers of the smallest step ratio, so two notes separated by the same number of steps always have exactly the same frequency ratio. However, except for doubling of frequencies (one or more octaves), no other intervals are exact ratios of small integers. Each just interval differs a different amount from its analogous equally-tempered interval.

Justly tuned intervals can be written as either ratios, with a colon (for example, 3:2), or as fractions, with a solidus (3/2). For example, two tones, one at 300 hertz (cycles per second) and the other at 200 hertz, are both multiples of 100 Hz and as such members of the harmonic series built on 100 Hz. Thus 3:2, known as a perfect fifth, may be defined as the musical interval (the ratio) between the second and third harmonics of any fundamental pitch.

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