Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of whole 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.

Justly tuned intervals are usually written either as ratios, with a colon (for example, 3:2), or as fractions, with a slash (3/2). Sometimes a technical distinction is made between the two styles, but in general they are equivalent and interchangeable.

Although in theory two notes tuned in an arbitrary frequency ratio such as 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.

Just intonation is usually compared to equal temperament, the tuning system that is by far the most common in the West, which arranges all notes at multiples of the same basic interval. This results in a tuning system where all intervals will sound equally out of tune in any key—all major thirds will have exactly the same character, for example—but the intervals themselves are detuned slightly.

The diatonic scale

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 and stable as possible, and the other chords not accommodated sound considerably less stable.

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 to D (perfect fifth) is 2/3, while that of G to C (perfect fourth) is 3/4. Three basic intervals can be used to construct any interval involving the prime numbers 2, 3 and 5 (known as 5-limit just intonation).

• s = 16:15 Semitone
• t = 10:9 Minor tone
• T = 9:8 Major tone

which combine to form

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

A just diatonic scale may be derived as follows. Suppose we insist that the chords F-A-C, C-E-G, and G-B-D be just major triads (then A-C-E and E-G-B are just minor triads, but D-F-A is not).

Then we obtain this scale:

Note	C		D		E		F		G		A		B		C
Ratio	1/1		9/8		5/4		4/3		3/2		5/3		15/8		2/1
Step		T		t		s		T		t		T		s

The major thirds are correct, and two minor thirds are right, but D-F is not.

Another way to do it is as follows. We can insist that the chords D-F-A, A-C-E, and E-G-B be just minor triads (then F-A-C and C-E-G are just major triads, but G-B-D is not).

Then we get the following scale:

Note	A		B		C		D-		E		F		G		A
Ratio	1/1		9/8		6/5		4/3		3/2		8/5		9/5		2/1
Step		T		s		t		T		s		T		t

The major thirds are correct, and two minor thirds are right, but B-D is not.

If we compare with the scale above, we see that six notes can be lined up, but one note, D, has changed its value. It is evidently not possible to get all six chords mentioned correct.

There are other possibilities; instead of lowering D, we can raise A. But this breaks something else.

The twelve tone scale

There are several ways to create a just tuning of the twelve tone scale.

The oldest known form of tuning, Pythagorean tuning, can easily produce a twelve tone scale. It uses ratios involving only powers of 3 and 2, creating a cycle of just perfect fifths, as follows:

note	G♭	D♭	A♭	E♭	B♭	F	C	G	D	A	E	B	F♯
fraction	${\frac {1024}{729}}$	${\frac {256}{243}}$	${\frac {128}{81}}$	${\frac {32}{27}}$	${\frac {16}{9}}$	${\frac {4}{3}}$	${\frac {1}{1}}$	${\frac {3}{2}}$	${\frac {9}{8}}$	${\frac {27}{16}}$	${\frac {81}{64}}$	${\frac {243}{128}}$	${\frac {729}{512}}$

Between enharmonic notes at either end of the cycle is a difference of about 24 cents, known as the Pythagorean comma. This twelve tone scale is fairly close to Equal temperament, but it does not offer much advantage for tonal harmony because only the perfect intervals (fourth, fifth, and octave) are simple enough to sound pure. Major thirds, for instance, receive the rather unstable interval of 81/64, well sharp of the preferred 5/4. The primary reason for its use is that it is extremely easy to tune, as its building block, the perfect fifth, is the simplest interval after the octave and unison.

Twelve tone scales can also be created with intervals that are compounded second third and fifth harmonics. Only factors 2, 3 and 5 are used in the construction summarized in the table below:

factor	1/(3*3)	1/3	1	3	3*3
5		5/3 A: 5/3	5 E: 5/4	15 B: 15/8	45 F♯: 45/32
1	1/9 B♭: 16/9	1/3 F: 4/3	1 C: 1	3 G: 3/2	9 D: 9/8
1/5	1/45 G♭: 64/45	1/15 D♭: 16/15	1/5 A♭: 8/5	3/5 E♭: 6/5

Starting with C in the centre of this diagram, horizontally a relationship by 3 is applied and vertically by 5. To all fractions outside the range of 1 to 2, powers of 2 are used to bring the tones within the same octave. Between the ends of the chain, F♯ and G♭, the enharmonic comma is less than 20 cents. This system has the advantage of making available pure thirds (5/4 and 6/5) as well as fifths, but also contains many intervals that are not (e.g. D to A is 45/24 rather than 3/2, or B♭ to D is the Pythagorean 81/64 rather than 5/4) which practically limits modulation to a narrow range of keys.

Indian scales

In Indian music, the just diatonic scale described above is used, though there are different possibilities for the 6th pitch (Dha), and further modifications may be made to all pitches excepting Sa and Pa.

	Sa		Re		Ga		Ma		Pa		Dha		Ni		Sa
Scale 1	1/1		9/8		5/4		4/3		3/2		27/16		15/8		2/1
Scale 2	1/1		9/8		5/4		4/3		3/2		5/3		15/8		2/1

Both possible scales appear problematic. The first has a problem because (27/16)/(5/4) = 27/20, which is a wolf interval, being uncomfortably close to the purer 4:3. However, because Indian music uses melodies over a drone dyad (usually 1/1 and 3/2), these two pitches (27/16 and 5/4) would seldom be heard sounding together. See Swara and Śruti (music).

The alternative, using 5/3 for Dha gives (5/3)/(5/4) = 4/3, and allows these notes to sound together in a consonant fashion, but then introduces another problem as (5/3)/(9/8) = 40/27, which is another wolf interval, this time close to 3/2. These wolf intervals are incompatible with much western music, but in Indian music they are largely irrelevant.

Practical difficulties

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 F:D, and still worse, a minor tone next to a fourth giving 40:27 for A:D. Moving D down to 10/9 alleviates these difficulties but creates new ones: G:D becomes 27:20, and B:G 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.

Singing

The human voice is among the most pitch-flexible instruments 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 a cappella ensembles naturally tend toward just intonation because of the comfort of its stability. Barbershop quartets are a good example of this.

Two exemplary contemporary ensembles that meticulously tune their singing in accordance with just intonation (whenever indicated) are The Hilliard Ensemble and Orlando Consort.

History

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 usable 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 usable 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 fact 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 one's 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.

Western composers

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 Aleksander Baath, David Beardsley, Glenn Branca, Wendy Carlos, Tony Conrad, Stuart Dempster, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Ben Johnston, Elodie Lauten, Douglas Leedy, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, Adam Silverman, James Tenney, Ernesto Rodrigues, Daniel James Wolf, 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 and Daniel James Wolf 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

• Equal temperament
• Mathematics of musical scales
• Meantone temperament
• Microtonal music
• Microtuner
• Musical tuning
• Pythagorean tuning
• Pythagorean intervals
• Semitone
• Well temperament
• List of meantone intervals
• Whole-tone scale
• Regular number

Instruments designed for just intonation

• 3rd Bridge
• Moodswinger
• Bagpipe
• Natural horn
• Trumpet marine
• Pedal Steel Guitar
• Long string instrument

External links

• The Wilson Archives
• Just Intonation Explained by Kyle Gann
• Just Intonation Network
• The Chrysalis Foundation -- Just Intonation: Two Definitions
• Medieval Music and Arts Foundation
• Why does our musical scale have twelve notes?
• Why does Just Intonation sound so good?
• Art of the States: microtonal/just intonation works using just intonation by American composers
• Music Novatory - Just Intonation
• Dante Rosati's 21 Tone Just Intonation guitar
• Just Intonation by Mark Nowitzky