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Just intonation
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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. 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 Elodie Lauten.

Table of contents
1 The diatonic scale in just intonation
2 Why isn't just intonation used much?
3 See also
4 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 G major, the ratio of the frequencies of the notes G:D is 2:3, while that of G:C 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:

from which we get:

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

G A B C D E F# G
 T t s T t T  s

with ratios w.r.t. G of
A 9/8, B 5/4, C 4/3, D 3/2
E 5/3, F# 15/8 and G 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 as the diatonic scale above, produce wolf intervals. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32/27 for C/A, and still worse, a minor tone next to a fourth giving 40/27 for E/A. Moving A down to 10/9 alleviates these difficulties but creates new ones: D/A becomes 27/20, and A/F# becomes 32/27.

You can have more frets on a guitar to handle both A's, 9/8 with G and 10/9 with G so that C/A can be played as 6/5 while D/A can still be played as 3/2. 9/8 and 10/9 are less than 1/53 octave apart, so mechanical and performance considerations have made this approach extremely rare. And the problem of how to tune chords such as C-E-G-A-D is left unresolved.

In Indian music, the basic unaltered diatonic scale is considered to be 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1. This would appear problematic, since (27/16)/(5/4) = 27/20, not 4/3. But Indian music uses melodies over a drone dyad (usually 1/1 and 3/2), so these two pitches would seldom be heard sounding together. Hence Indian practice has had much more use for just intonation than western practice.

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.

See also

musical tuning, microtonal music, mathematics of musical scales, just tuning

External links