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Why Just Intonation?

Been puzzled by references to just intoned or even tempered scales, or wondered exactly what the DX7II's "microtonality" is all about? Robert Rich has the answers.

When treated as more than just a novelty, just intonation can change the way you think about music and the way you play your instrument - but what is it?

ONE OF THE boasts of the Yamaha DX7II was its ability to accommodate alternative tunings or "microtonality". No they're not the result of years of secret research in Yamaha's R&D labs, they're simply alternatives to the familiar western tuning system we're used to hearing music played in. Apart from ethnic tunings - involving such exotic delights as quarter-tones - the most common alternative tuning is just intonation. But what is it, and where did it all begin?

The American composer Harry Partch may have instigated the present interest in tuning in 1949, when he published his book, Genesis of a Music. Since then, composers like Terry Riley, Lou Harrison and LaMonte Young have been writing and performing music in just intonation, and Wendy Carlos has done a great deal recently to publicise the issue. Jon Hassell, Michael Brook and many others have also worked with just intonation. That such a large company as Yamaha should start supporting microtonality shows that something must be happening.

It seems that there's renewed interest in just intonation. Up until this point, however, most of the interest in alternative tunings has come from the avant-garde community - which has perhaps led to the popular impression that alternative tunings sound, well, strange.

In reality, whether a tuning system sounds strange or not depends mostly upon what you do with it. A random tuning does sound strange, but a real tuning system is not random; real tuning is logical, it makes sense. (The scales that we all know and love are based on logical systems too, although musicians do not always understand these systems when tuning their instruments.)

Just intonation is defined as any tuning system whose frequencies all relate to each other in whole numbered ratios, with a preference for ratios expressible in small numbers. For example, if the tonic (or unison) is defined as 1/1, the fifth is 3/2. This means that the frequency of the fifth note of the scale is exactly 1.5 times the frequency of the tonic.

The use of these whole-numbered ratios results in scales whose intervals coincide with the way the ear naturally hears harmony. Unlike equal temperament, just intoned intervals are not equally spaced, but then neither is the natural harmonic series. The commonly used equal tempered system (which is based on an exponential series of incremented multiples of the 12√2) only approximates natural harmony but, because it adopts a standard semitone interval, it has the advantage that all harmonic keys sound equally in tune. In contrast, a just intoned instrument will only be perfectly in tune for one key. It's no coincidence that equal temperament became common when musical structures began to make use of modulations between keys.

So what makes a tuning system sound good or bad? The logic behind it involves overtones, and how they align. (Note that there is some disagreement over this. Some composers feel that overtones play only a minor role. These differences in opinion only really enter the picture when many enharmonic overtones are present, such as in bell tones.)

When two notes sound together in a harmonic relationship (an interval), some of their overtones will match, and some will not. Matching means that the frequency ratio between two notes is expressible in small whole numbers. In general, when lower overtones in the harmonic series align, a greater number of higher overtones will match as well. The lower overtones are also generally louder, so their coincidence will be more evident.

"Just intonation has been around since the beginning of formal music - at least since Pythagoras, possibly since the ancient Babylonians."

Another point to bear in mind is that once a ratio has been reduced to prime numbers, the smaller those numbers are, the "better" the ratio sounds as a chord. So the chords in which lower overtones are aligned will sound better than those in which only the higher overtones coincide.

For example, in the octave (2/1) - which we all know sounds good - all the harmonics of the higher note will match every second harmonic of the lower note. In fact, the second harmonic itself is the octave (see Figure 1). In the case of the fifth note (3/2), the third harmonic of the lower note matches the second harmonic of the upper note. Other intervals are more complex in their alignment, but the process is the same.

Figure 1. Examples of Harmonic Alignment.

The History

JUST INTONATION HAS been around since the beginning of formal music - at least since Pythagoras, possibly since the ancient Babylonians. Just intoned ratios are, in fact, the basis for all harmonic theory. In contrast, equal temperament was developed by contemporaries of Bach (his Well-Tempered Clavier codified the new tuning system, although it was written for well-temperament which is slightly different from the scale we now call equal temperament) and consequently has only been around for about 300 years.

So why are musicians starting to revive just intonation after its 300 year absence? Well, most importantly, a just intoned scale can sound better than an equal tempered one. I say can, because where it provides more natural harmony than equal temperament, it can also create much harsher dissonances. But these dissonances are only part of a wider harmonic vocabulary. For instance, you can choose between at least two good sounding major thirds (using either a ratio of 4/5 or 7/8).

A composer can obtain fantastic special effects, too. How about a melody line that plays hide-and-seek around the harmonics of another note? At last, harmony and timbre can merge. Indeed, one of the most satisfying aspects of just intonation is the fact that harmony suddenly makes sense. This is not the result of intellectualising, but the feeling produced by the music itself. The mind and the senses can work together without conflict.

"Unlike equal temperament, just intoned intervals are not equally spaced but then, neither is the natural harmonic series."

Before I get too carried away, I should mention that tunings don't always conform to their abstract ideal when converted into music. In acoustic instruments especially, physical tuning does not always fit the theory but, because of its natural basis, it's much easier to tune to just intonation than equal temperament. String players naturally approximate just intervals in their playing, as do horn players, singers and others who have fine control over the pitch of their instrument.

Why, then, did theorists bother to create equal temperament in the first place? It all has to do with that wonderful invention, the keyboard. At the time that black and white ivories were starting to replace the levers on the front of organs, composers were devising ways to make more complicated music. One of the major complications involved key changes.

Just intonation does not accommodate key changes very easily; you need either 20-40 notes per octave, or a way to redefine the frequency of each key on the keyboard whilst playing; effectively, the ability to bend notes. Note-bending poses no problem with fretless string or wind instruments - players do it naturally - nor should individual note-bending or flexible octaves present a problem to electronic keyboards, although designers have rarely bothered to include these capabilities. On a mechanical keyboard, though, the problem was nearly insurmountable. The solution was to adapt the scale. Since all the notes in equal temperament are the same distance apart, it doesn't matter what key the music is played in - it all observes the same harmonic relationships.

The inventors of tempered scales didn't claim that they sounded better - they admitted that tempered scales were compromises - but equal temperament works pretty well. In fact, equal temperament is so convenient that not everyone who uses it would benefit from changing to just intonation. A musician has to learn new playing techniques for just intonation, and composers must take care when creating harmonies, as a misplaced just intoned chord can sound pretty painful. Key changes pose a special challenge, but not an insurmountable one.

Theoretically, electronic keyboards promise freedom of intonation at last, but in practise only a few companies have seen fit to stray beyond the confines of equal temperament. The rest of you, take heed.

Play in a Day

THE BEST WAY to learn about something is to try it, and the same is true of just intonation. Microtonality is a feature of the TX81Z as well as the DX7II, though the DX's pitch resolution of 1.2 cents leaves a lot to be desired and the TX's is even worse at around 1.6 cents. Other retunable instruments include the Prophet 5 (rev. 3.0 to 3.3, starting at serial number 1300 around 1980), the Synergy, Kurzweil and Synclavier.

"The biggest difficulty in playing with just intonation is that not many fretted or keyboard instruments are retunable."

Figure 2. TX81Z/DX7II Tuning Preset #2 "Pure Major".
Note Cents* Ratio
C 0.0 1/1
C# 70.7 25/24
D 203.9 9/8
D# 315.6 6/5
E 386.3 5/4
F 498.0 4/3
F# 568.7 25/18
G 702.0 3/2
G# 772.6 25/16
A 884.4 5/3
A# 1017.6 9/5
B 1088.3 15/8

*Cent values are theoretical Actual cents.

On a DX7II or TX81Z, you need to go into Performance Memory edit and call up tuning preset 2. The manual doesn't give the ratios for this or any of the tunings, so they appear here in Figure 2. Now pick a thin, bright, sustained sound with no vibrato (thin sounds are more transparent to tuning differences). Holding down the C (1/1), play one note at a time slowly up the scale. To get a direct comparison with equal temperament, you'll have to lift up on the keys, then switch the tuning preset to 1 - equal temperament - before holding down the keys again. Because of the poor tuning resolution, you will hear slow beating in the harmonics of some of the just intoned chords. But if you listen to the approximations of these chords in equal temperament, you will notice that this beating is usually so pronounced that it gives indistinct harmonics. A good just intoned chord will "lock" into the harmonics so well that subharmonics are clearly audible - actual phantom harmonies whose overtones are the very notes you are playing.

It takes a while to learn the capabilities of just intonation, but its beauty can be immediately appreciated. Thinking in terms of true harmonic relationships can open the mind as well as the ears to the logic hidden within musical structures. And with luck, the capabilities of our instruments will soon match the abilities of our ears and minds to hear and enjoy harmony.

Further Reading:
On the Sensation of Tone, Helmholtz, Hermann; Dover, New York, 1954.

Intervals, Scales and Temperaments, Lloyd, Lloyd S.; Boyle, Hugh; MacDonald & Co., London, 1963.

Genesis of a Music, Partch, Harry; DaCapo Press, New York, 1949.

1/1 - Journal of the Just Intonation Network (published quarterly). (Contact Details).

Previous Article in this issue

Living in the Past

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Eno Sense

Music Technology - Copyright: Music Maker Publications (UK), Future Publishing.


Music Technology - Feb 1988

Donated & scanned by: Mike Gorman

Feature by Robert Rich

Previous article in this issue:

> Living in the Past

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> Eno Sense

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