justintonation

Justintonation, more commonly written as just intonation, is a tuning system in which the frequencies of notes are related by simple whole-number ratios. In this approach, the size of intervals is derived from the harmonic series, and pure intervals such as the octave, perfect fifth, and major third are formed by small-integer ratios relative to a chosen reference pitch. A given tonic or key determines a consonant set of intervals that are as close as possible to these ratios.

Historically, just intonation has roots in ancient and medieval theory and was developed further in Renaissance and early Baroque music. It was widely used before the spread of equal temperament, and it remains influential in early music performance, vocal ensembles, and some traditional and non-Western musical practices. In modern Western music, equal temperament and other temperaments have become dominant for keyboarded instruments, but just intonation is still studied and applied in specialized contexts and in computer-based tuning.

Principles of the system center on intervals built from simple ratios. Common reference intervals include the octave (2:1), the perfect fifth (3:2), and the major third (5:4); a minor third (6:5) is also used in many 5-limit tunings. Because the ratios cannot be maintained identically across all keys on fixed-pitch instruments, practitioners use various approaches such as 5-limit or higher-limit tunings, meantone temperaments, or context-specific tunings that preserve consonances within a given key while acknowledging compromises when modulating.

In contemporary practice, just intonation can be realized by vocal ensembles, fretted or stringed instruments, and electronic instruments or software that support key-dependent tunings, enabling precise intervals within a chosen tonal context.