Just intonation
Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. Any interval tuned in this way is called a just interval. Another way of considering just intonation is as being based on members of the harmonic series. Thus, although in theory two notes tuned in the frequency ratio 1024:927 might be said to be justly tuned, in practice only ratios using quite small numbers tend to be called just. Intervals used are then capable of being more consonant, but consonance is not always emphasized or a goal in music written with just intonation.
It is possible to tune the familiar diatonic scale or chromatic scale in just intonation but many other justly tuned scales have also been used. Music written in just intonation is most often tonal but need not be, some can loosen these boundaries as in some music of Kraig Grady but which is not as atonal as certain pieces by Ben Johnston which are serial. Composers often impose a limit on how complex the ratios used are: for example, a composer may write in "7-limit JI", meaning that no prime number larger than 7 features in the ratios they use. Under this scheme, the ratio 10:7, for example, would be permitted, but 11:7 would not be, as all non-prime numbers are octaves of, or mathematically and tonally related to, lower primes (example: 12 is an octave of 6, while 9 is a multiple of 3).
Many composers have written in just intonation, including Glenn Branca, Arnold Dreyblatt, Kyle Gann, Lou Harrison, Ben Johnston, Harry Partch, Terry Riley, LaMonte Young, James Tenney, Pauline Oliveros, Stuart Dempster, Kraig Grady and meantone temperament.
If in addition the semitone is altered so that an interval of two semitones is equal to one tone, you get the 12 notes used in modern Western music (see equal temperament), which allows one to travel through twelve equally consonant and dissonant keys.
See also
musical tuning, microtonal music, mathematics of musical scales, just tuning
