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Just intonation

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In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. 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; more complex ratios are often considered to be rational intonation but not necessarily just intonation. Intervals used are then capable of being more consonant.

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 their frequencies form ratios 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

Practical difficulties of use

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 F#:A becomes 27:16.

You can have more frets on a guitar to handle both A's, 9/8 with respect to G and 10/9 with respect to 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 tuning of most complex chords in just intonation is generally ambiguous.

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.

Historical reasons for disuse

There were several other systems in use before equal temperament. Pythagorean tuning was perhaps the first to be theorized, which is a system in which all tones can be found using the ratios 3:2 and 4:3. It is easier to think of this system as a cycle of fifths, but it must be noted that because a series of 12 fifths does not reach the same tone it began with, this system produces "wolf fifths" in the more distant keys (which were consequently unused).

Another system that was popular for keyboards through the Renaissance was meantone temperament. In this system the simpler ratios of 3:2 and 4:3 were compromised in favour of exact 5:4 (major thirds) ratios. Specifically, the fifth (3:2) was slightly narrowed so that a series of four narrowed fifths would produce 5:4 exactly (at some octave transposition). Again, this system is not circular and produced some unplayable keys. (Some keyboards of the 18th century featured split keys differentiating sharp and flat notes to expand the range of useable keys.)

The most common tuning today began as well temperament, which was replaced by the more rigorous equal temperament in the early 20th century. Well temperament largely abandoned just intonation by applying small changes to the intervals so that they became more homogenized and eliminated wolf intervals. In systems of well temperament, and there were many, the goal was to make all keys useable by compromising each of them slightly. Its development was necessary as composers moved toward expression through large harmonic changes (modulation), and required access to a wider realm of tonality. Bach's "Well-Tempered Clavier", a book of compositions in every key, is the most famous example, but the compositions of Chopin, for instance, rely much more on the devices of expression only allowed by well temperament.

Equal Temperament is essentially the most homogenized form of well temperament, in that it tunes an actual circle of fifths by narrowing each by the same amount. In equal temperament, every interval is the same as all other intervals of its type. There are no longer pure and "wolf" fifths, or even good and bad fifths, but simply fifths (or thirds, or seconds, et cetera). Equal temperament is not a form of just intonation.

Today, the dominance of repertoire composed under well tempered systems, the prominence of the piano in musical training, the lack of just-intonation capable instruments, and the way that tuning is not normally a significant part of a musician's education have made equal temperament so prevalent that alternatives are not often discussed.

Despite the obstacles, many today find reasons to pursue just intonation. The purity and stability of its intervals are found quite beautiful by many, but this stability also allows extreme intonational precision as well. The practical study of just intonation can greatly increase ones analytical ability with respect to sound, and yield improvement to musicianship even in well temperament repertoire.

In practice it is very difficult to produce true equal temperament. There are instruments such as the piano where tuning is not dependent on the performer, but these instruments are a minority. The main problem with equal temperament is that its intervals must sound somewhat unstable, and thus the performer has to learn to suppress the more stable just intervals in favour of equal tempered ones. This is counterintuitive, and in small groups, notably string quartets, just intonation is often approached either by accident or design because it is much easier to find (and hear) a point of stability than a point of arbitrary instability.

Singing in just intonation

The human voice is the most pitch-flexible instrument in common use. 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 increasing use of instrumental accompaniment (with its attendant constraints on pitch), most good ensembles naturally tend toward such singing when performing a cappella. Barbershop quartets are a good example for this. 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, but then introduces another problem in that (5:3):(9:8) = 40:27 (a wolf interval), not 3:2. These issues prevented strict just intonation from becoming prevalent in the West, but it thrives in India, where they are largely irrelevant. Synthesis allows ratios to be altered depending on context, so that the third or sixth degree could be alternately sounded in Pythagorean (3-limit) or Ptolemaic (5-limit) tuning to avoid an unjust fifth interval. With non-fixed tunings it is also possible to assign a 7-limit ratio (21:16) to the fourth degree in dominant seventh and leading-tone chords without creating unplayable intervals in other chordal contexts.]

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, Ernesto Rodrigues and La Monte 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).

See also

Instruments that play naturally in just intonation


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