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 greater 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. 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, and Elodie Lauten.

Table of contents

1 The diatonic scale in just intonation
2 Why isn't just intonation used much?
3 External links

The diatonic scale in just intonation

The prominent notes of a given scale are tuned so that the ratios of their frequencies are comprised of relatively small integers. For example, in the key of F major, the ratio of the frequencies of the notes F:C is 2:3, while that of F:Bb is 3:4.

All ratios that involve the prime numbers of 2, 3 and 5 can be built out of the following 3 basic intervals:

s=16/15 Semitone
t=10/9 Minor Tone
T=9/8 Major Tone

from which we get:

6/5 = Ts
5/4 = Tt
4/3 = Tts
3/2 = TTts
2/1 = TTTttss

It gives rise to the following scale in the key of F:

F G A BbC D E F T t s T t T s

with ratios w.r.t. F of

G 9/8, A 5/4, Bb 4/3, C 3/2 D 5/3, E 15/8 and F 2/1

Why isn't just intonation used much?

For many instruments tuned in just intonation, you can't change keys without retuning your instrument. Also, some fixed just intonation scales and systems, such , produce wolf intervals. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32/27 for Bb/G. (You can have more frets on a guitar to handle both G's, 9/8 with F and 10/9 with F so that Bb/G will become 6/5 while C-G is still 3/2)

If the value of the major and minor tones are adjusted so that they are both equal, one gets a 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.