Just intonation: Wikis

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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 solidus (3 ⁄ 2). Colons indicate that division is not done, so it is the preferred usage in music: In practice, two tones, one at 300 Hertz (cycles per second), and the other at 200 hertz is a perfect fifth (3:2).

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 products of small primes are given the name; 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 can be contrasted and compared with equal temperament, which dominates western orchestras and default MIDI tuning. Equal temperament starts by arranging all notes at multiples of the same basic interval, but the intervals themselves are altered slightly, relative to just intonation. Each interval possesses its own degree of alteration. The process results in a tuning system where all intervals will have exactly the same character in any key.

Examples

Just intonation An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.

Equal temperament An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. If you listen to the above file, and then listen to this one, you might be able to hear a slight buzzing in this file.

A pair of major thirds, followed by a pair of full major chords. The first in each pair is in equal temperament; the second is in just intonation. Piano sound.

A pair of major chords. The first is in equal temperament; the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just temperament between the two chords. In the equal temperament chords a roughness or beating can be heard at about 4 hz and about 0.8 Hz. In the just intonation triad this roughness is absent. The square waveform makes the difference between equal and just temperaments more obvious.

History

There were several other systems in use before equal temperament. The Guqin has a musical scale based on harmonic overtone positions. The dots indicate the harmonic positions: 1/8, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 7/8.[1] Pythagorean tuning, perhaps the first tuning system to be theorized in the West,[2] is a system in which all tones can be found using the ratios 3:2 . 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.[3] 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 (Chopin would also write a set of compositions in every key, his 24 Preludes, which in contrast to Bach's Well-Tempered Clavier, made extensive use of the chromatic modulations characteristic of Romantic music. See: Ibid. p. 579.).

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 sufficiently more 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.

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

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
Cents 0 204 386 498 702 884 1088 1200
Step   T t s T t T s
Cent step 204 182 112 204 182 204 112

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
Cents 0 204 316 498 702 814 1018 1200
Step   T s t T s T t
Cent step 204 112 182 204 112 204 182

The minor thirds are correct, and two major 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.

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 produce a twelve tone scale, but it does so by involving ratios of very large numbers, corresponding to natural harmonics very high in the harmonic series that do not occur widely in physical phenomena. This tuning 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
Ratio $\frac{1024}{729}$ $\frac{256}{243}$ $\frac{128}{81}$ $\frac{32}{27}$ $\frac{16}{9}$ $\frac{4}{3}$ $\frac{1}{1}$ $\frac{3}{2}$ $\frac{9}{8}$ $\frac{27}{16}$ $\frac{81}{64}$ $\frac{243}{128}$ $\frac{729}{512}$
Cents 588 90 792 294 996 498 0 702 204 906 408 1110 612

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, sharp (by the ratio of 81:80) [4] 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.

A twelve tone scale can also be created with intervals that are compounded second third and fifth harmonics, called a five-limit tuning. 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 base ratio
cents
5/3
A: 5/3
884
5
E: 5/4
386
15
B: 15/8
1088
45
F: 45/32
590
1 base ratio
cents
1/9
B: 16/9
996
1/3
F: 4/3
498
1
C: 1/1
0
3
G: 3/2
702
9
D: 9/8
204
1/5 base ratio
cents
1/45
G: 64/45
610
1/15
D: 16/15
112
1/5
A: 8/5
814
3/5
E: 6/5
316

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 40/27 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.[5]

Note Sa Ri Ga Ma Pa Dha Ni Sa
Ratio 1/1 9/8 5/4 4/3 3/2 5/3 or 27/16 15/8 2/1
Cents 0 204 386 498 702 884 or 906 1088 1200

Both possible scales appear problematic, if one were to look at it in terms of a polyphonic application. The first would have 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[6] (usually 1/1 and 3/2), these two pitches (27/16 and 5/4) would not 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 irrelevant.

Some accounts of Indian intonation system cite a given 22 Śrutis. According to some musicians, you have a scale of a given 12 pitches and ten in addition (the tonic, Shadja (Sa), and the pure fifth, Pancham (Pa), are inviolate):

Note C D D D D E E E E F F F
Ratio $\frac{1}{1}$ $\frac{256}{243}$ $\frac{16}{15}$ $\frac{10}{9}$ $\frac{9}{8}$ $\frac{32}{27}$ $\frac{6}{5}$ $\frac{5}{4}$ $\frac{81}{64}$ $\frac{27}{20}$ $\frac{4}{3}$ $\frac{45}{32}$
Cents 0 90 112 182 204 294 316 386 408 520 498 590
Note F G A A A A B B B B C
Ratio $\frac{64}{45}$ $\frac{3}{2}$ $\frac{128}{81}$ $\frac{8}{5}$ $\frac{5}{3}$ $\frac{27}{16}$ $\frac{16}{9}$ $\frac{9}{5}$ $\frac{15}{8}$ $\frac{243}{128}$ $\frac{2}{1}$
Cents 610 702 792 814 884 906 996 1018 1088 1110 1200

Where we have two ratios for a given letter name, we have a difference of 81:80, which is known as the syntonic comma.[7] You can see the symmetry, looking at it from the tonic, then the octave.

(This is just one example of "explaining" a 22-Śruti scale of tones. There are many takes on this, just as there are many ears.)

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 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 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.

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 John Adams, David Beardsley, Glenn Branca, Martin Bresnick, Wendy Carlos, Lawrence Chandler, Tony Conrad, Stuart Dempster, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Ben Johnston, Elodie Lauten, György Ligeti, Douglas Leedy, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, Marc Sabat, Wolfgang von Schweinitz, 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). Yuri Landman derived a just intoned musical scale from a initially considered atonal prepared guitar playing technique based on adding a third bridge under the strings. When this bridge is positioned in the noded positions of the harmonic series the volume of the instrument increases and the overtone becomes clear and has a consonant relation to the complementary opposed string part creating a harmonic multiphonic tone.[8]

Staff Notation

Ex. 1: Legend of the HE Accidentals

Recently some composers have been developing notation methods for Just Intonation using the conventional five-line staff. James Tenney, amongst others, preferred to combine JI ratios with cents deviations from the equal tempered pitches, indicated in a legend or directly in the score, allowing performers to readily use electronic tuning devices if desired. Beginning in the 1960s, Ben Johnston had proposed an alternative approach, redefining the understanding of conventional symbols (the seven "white" notes, the sharps and flats) and adding further accidentals, each designed to extend the notation into higher prime limits. Johnston‘s method is based on a diatonic C Major scale tuned in JI, in which the interval between D (9/8 above C) and A (5/3 above C) is one Syntonic Comma less than a Pythagorean perfect fifth 3:2. To write a perfect fifth, Johnston introduces a pair of symbols representing this comma, + and -. Thus, a series of perfect fifths beginning with F would proceed C G D A+ E+ B+. The three conventional white notes A E B are tuned as Ptolemaic major thirds (5:4) above F C G respectively. Johnston introduces new symbols for the septimal, undecimal, tridecimal, and further prime extensions to create an accidental based exact JI notation for what he has named "Extended Just Intonation".

In the years 2000-2004, Marc Sabat and Wolfgang von Schweinitz worked together in Berlin to develop a different accidental based method, the Extended Helmholtz-Ellis JI Pitch Notation.[9] Following the method of notation suggested by Helmholtz in his classic "On the Sensations of Tone as a Physiological Basis for the Theory of Music", incorporating Ellis' invention of cents, and following Johnston's step into "Extended JI", Sabat and Schweinitz consider each prime dimension of harmonic space to be represented by a unique symbol. In particular they take the conventional flats, naturals and sharps as a Pythagorean series of perfect fifths. Thus, a series of perfect fifths beginning with F proceeds C G D A E B F# and so on. The advantage for musicians is that conventional reading of the basic fourths and fifths remains familiar. (Such an approach has also been advocated by Daniel James Wolf and Joe Monzo, who refers to it by the acronym HEWM (Helmholtz-Ellis-Wolf-Monzo).) In the Sabat-Schweinitz design, Syntonic Commas are marked by arrows attached to the flat, natural or sharp sign, Septimal Commas using Giuseppe Tartini's symbol, and Undecimal Quartertones using the common practice quartertone signs (a single cross and backwards flat). For higher primes, additional signs have been designed. To facilitate quick estimation of pitches, cents indications may be added (downward deviations below and upward deviations above the respective accidental). The convention used is that the cents written refer to the tempered pitch implied by the flat, natural, or sharp sign and the note name. A complete legend and fonts for the notation (see samples) are open source and available from PLAINSOUND MUSIC EDITION.

References

1. ^ The harmonic scale of Guqin clarified in the essay 3rd Bridge Helix by Yuri Landman on furious.com
2. ^ The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts. See: West, M.L.. The Babylonian Musical Notation and the Hurrian Melodic Texts, Music & Letters vol. 75 no. 2 (May 1994). pp. 161-179.
3. ^ Grout, Donald Jay and Claude Palisca. A History of Western Music, sixth edition. W.W. Norton & Company Inc., New York, 2000. p. 349. ISBN 0-393-97527-4
4. ^ Danielou, Alain (1968). The Ragas of Northern Indian Music. Barrie & Rockliff, London. ISBN 0214156893.
5. ^ Bagchee, Sandeep. Nad: Understanding Raga Music. BPI (India) PVT Ltd. pp. 23. ISBN 81-86982-07-8.
6. ^ Bagchee, Sandeep. Nad: Understanding Raga Music. BPI (India) PVT Ltd. pp. 16. ISBN 81-86982-07-8.
7. ^ Danielou, Alain (1968). The Ragas of Northern Indian Music. Barrie & Rockliff, London. ISBN 0214156893.
8. ^ 3rd Bridge Helix by Yuri Landman on furious.com
9. ^ see article "The Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmethode für die natürlichen Intervalle" in "Mikrotöne und Mehr - Auf György Ligetis Hamburger Pfaden", ed. Manfred Stahnke, von Bockel Verlag, Hamburg 2005 ISBN 3-932696-62-X