# 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; in other words, the two notes are members of the same harmonic series. 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.

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## The diatonic scale in just intonation

It is possible to tune the familiar diatonic scale or chromatic scale in just intonation, in many ways, all of which make certain chords purely tuned and as consonant as possible, and others considerably more dissonant and indeed seeming out-of-tune to modern ears (see below for more on this).

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 (perfect fifth) is 2:3, while that of G:C (perfect fourth) 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:

• 6:5 = Ts (minor third)
• 5:4 = Tt (major third)
• 4:3 = Tts (perfect fourth)
• 3:2 = TTts (perfect fifth)
• 2:1 = TTTttss (octave)

It gives rise to the following scale in the key of G (this is only one possibility):

```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?

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 (for instance, A could be 4:3 below D (making it 9/8, if G is 1) or 4:3 above E (making it 10/9, if G is 1) but not both at the same time, so one of the fourths in the chord will have to be an out-of-tune wolf interval). However the frets may be removed entirely -- this, unfortunately, makes in-tune fingering of many chords exceedingly difficult, due to the construction and mechanics of the human hand -- and the problem of how to tune chords such as C-E-G-A-D in just intonation remains unresolved.

For many instruments tuned in just intonation, you can't change keys without retuning your instrument. For instance, if you tune a piano to just intonation intervals and a minimum of wolf intervals for the key of G, then only one other key (typically E-flat) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes modulation within a piece, or playing a repertoire of pieces in different keys impractical to impossible.

Synthesizers have proven a valuable tool for composers wanting to experiment with just intonation. Many commercial synthesizers provide the ability to use built-in just intonation scales or to program your own. Wendy Carlos used a system on her 1986 album Beauty in the Beast, where one electronic keyboard was used to play the notes, and another used to instantly set the root note to which all intervals were tuned, which allowed for modulation. On her 1987 lecture album Secrets of Synthesis there are audible examples of the difference in sound between traditional equal temperament and just intonation.

If the value of the major and minor tones are adjusted so that they are both equal, one gets a meantone temperament. This eliminates the wolf intervals from the diatonic scale without too much damage to the pure consonances; hence meantone tuning was the primary keyboard tuning used in Western music from about 1480 to 1780. A 12-note keyboard in meantone is limited to six major keys (typically B-flat, F, C, G, D, and A) with no wolf intervals; it was not uncommon during this era to find keyboard with split keys so that more than 12 notes, and thus more than six wolfless major keys, could be accessed. 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 keys with only 12 notes on an instrument. The adventurous harmonic paths blazed by composers such as Beethoven and Schubert require such a closed system, though in their day keyboard tunings were probably somewhat unequal as in a well temperament. Regardless, the influence of such composers was such as to lead to the total abandonment of meantone temperament, which held out longest (until the 1850s) only in England and Spain. 12-note equal temperament does quite a bit more damage to the consonances than meantone temperament; today, many musicians, after exposure to tunings with purer harmonies, refuse to return to 12-note equal temperament due to its noticeably less-pure consonances.

## Singing in just intonation

The human voice is the most pitch-flexible instrument of all. Pitch can be varied with no restraints and adjusted in the midst of performance, without needing to retune (as even with the otherwise very flexible string instruments). Although the explicit use of just intonation fell out of favour concurrently with the incrasing use of instrumental accompaniment (with its attendant constraints on pitch), most good ensembles naturally tend toward such singing when performing a cappella. Two exemplary contemporary ensembles that meticulously tune their singing in accordance with just intonation (whenever indicated) are The Hilliard Ensemble and Orlando Consort.

## Bagpipe tuning

In bagpipe tuning, just intonation is possible, and more and more readily used. In the past old instruments seem to have used a folk scale with many irregular intervals, e.g. the fourth sharp. Today though, because much of the tuning is done by ear, and presumably because many pipers are exposed to other music, the process of tuning the chanter (melody pipe) to the drones lends itself to just intonation.

## Non-western tuning

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 (a wolf interval), not 4/3. But Indian music uses melodies over a drone dyad (usually 1/1 and 3/2), so these two pitches (27/16 and 5/4) would seldom be heard sounding together. See sargam and swara. The just scale with the ratios 1/1, 9/8, 5/4, 4/3, 3/2, *5/3*, 15/8, 2/1 gives (5/3):(5/4) = 4/3 (a perfect fourth), and allows these notes to sound together in a consonant fashion.

## Western composers who specified just intonation

Most composers don't specify how instruments are to be tuned, although historically most have assumed one tuning system which was common in their time; in the 20th century most composers assumed equal temperament would be used. However, a few have specified just intonation systems for some or all of their compositions, including Glenn Branca, Wendy Carlos, Stuart Dempster, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Ben Johnston, Elodie Lauten, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, James Tenney, and LaMonte Young. Eivind Groven is often considered a just intonation composer but just intonation purists will disagree. His tuning system was in fact schismatic temperament, which is indeed capable of far closer approximations to just intonation consonances than 12-note equal temperament or even meantone temperament, but still alters the pure ratios of just intonation slightly in order to achieve a simpler and more flexible system than true just intonation.

Music written in just intonation is most often tonal but need not be; some music of Kraig Grady uses just intonation scales designed by Erv Wilson explicitly for a consonant form of atonality, and Ben Johnston's Sonata for Microtonal Piano (1964) uses serialism to achieve an atonal result. 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 a double octave of 3, while 9 is a square of 3).

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