Just Intonation, Beating and Why Intervals Sound in Tune

“In tune” sounds like a matter of taste and is mostly a physical event. When two tones sound together and their frequencies stand in a simple whole-number ratio, their overtones land on top of one another and the combined sound holds still. When the ratio is not simple, the overtones nearly coincide, and the small difference between them is heard as a slow pulsation. That pulsation is beating, and it is what listeners mean when they say an interval is out of tune.

#Ratios the ear can check

Just intonation is the system in which intervals are defined by small whole-number frequency ratios. The octave is 2:1, the fifth 3:2, the fourth 4:3, the major third 5:4, the minor third 6:5. Each of these ratios produces overtone coincidences: in a 3:2 fifth, every second overtone of the upper note lines up with every third overtone of the lower one, and in a 5:4 third the fifth overtone of the lower note lines up with the fourth overtone of the upper one. The fewer and lower the coincidences, and the more exactly they line up, the more stable the interval sounds.

Intervals are usually measured in cents, where one equal-tempered semitone is 100 cents and the octave is 1200. A ratio becomes cents through 1200 times the base-2 logarithm of the ratio, which is worth knowing because it lets the two systems be compared directly: the pure fifth of 3:2 is 702.0 cents, and equal temperament gives the same interval 700.

The table is a summary of the whole argument. The perfect fifth and fourth are nearly the same in both systems, the thirds and sixths differ by roughly 14 to 16 cents, and the harmonic seventh of 7:4 sits more than a quarter of a semitone away from the equal-tempered minor seventh that occupies the same staff position.

#Beating is the mechanism

Beating has a rate, and the rate is the arithmetic that makes tempering decisions for composers.

Two tones whose frequencies differ by one cycle per second produce one beat per second. In a fifth, the coincidence that matters in the middle of the keyboard is between the third overtone of the lower note and the second overtone of the upper one. At A3 to E4, the equal-tempered fifth leaves those two partials about 0.7 cycles per second apart, which is a slow, almost imperceptible undulation. In a major third the relevant coincidence is between the fifth overtone of the lower note and the fourth overtone of the upper one, and those partials are higher and the cents error is larger, so the C4 to E4 third beats at roughly 10 cycles per second. The measurement is easy to reproduce: play the third, count the pulses in a second or two, then play the fifth below the same root.

That third is equal-tempered, and it beats. A 5:4 third in the same register has no beat at all in theory. In practice a real piano string is slightly inharmonic, its overtones sit a little sharp of the exact multiples, so even a pure third beats faintly and tuners stretch the octaves to accommodate the effect. The theoretical case and the instrument’s behaviour differ by a small amount, and the direction of the difference is always the same.

The consequence for any twelve-note tuning is a budget. Narrowing the fifth by 2 cents to reach equal temperament costs three quarters of a beat a second at that register. Widening the major third by 13.7 cents costs about ten. A keyboard that wants still, consonant thirds has to protect the 5:4 interval and let the fifths take the damage, which is what meantone temperaments do.

#What just intonation cannot fix

A single fixed set of twelve pitches cannot make every interval pure in every key. The reason is that the same interval can be reached by two different routes that do not agree. Four pure fifths up from C and two octaves down give an E at 407.8 cents, the Pythagorean third. A pure 5:4 third above the same C is 386.3 cents. The difference, 21.5 cents, is the syntonic comma, and no single E can be both. Similarly, twelve pure fifths exceed seven octaves by 23.5 cents, so a chain of fifths cannot be closed at all.

Sequences that stay pure in performance do not solve the problem, they move it. Three pure major thirds stacked upward fall 41 cents short of an octave, the same 128:125 gap that separates G sharp from A flat. A choir or a string quartet that insists on pure thirds on every chord will drift downward as the phrase continues, and the drift is the comma finding its way back in. Unaccompanied ensembles therefore alternate: pure intervals where the chord is held and exposed, slight adjustments where the music has to arrive somewhere.

#Where it survives in practice

Just intonation is not a historical curiosity. It is what unaccompanied ensembles do when they are listening carefully. A barbershop quartet’s dominant seventh chord rings because the seventh is sung near 7:4, and the upper partials of the four voices coincide. Play the same chord the way a keyboard would, and the seventh sits high:

That A flat is about 31 cents sharper than the harmonic seventh a quartet settles on. The gap is not a stylistic preference. It is what the ratio 7:4 costs when a fixed twelve-note tuning cannot supply it, and the ringing sound of the chord is the audience hearing the difference. The same interval is available on any brass instrument as the seventh partial of the harmonic series, which is why brass players tune chords from the series rather than from a keyboard, and why that partial sounds noticeably flat when played against a piano.

String quartets work the same way, with the cello as the reference. The third of a sustained chord is the note most likely to be adjusted, and the leading tone is usually played a little high, because a raised leading tone that resolves upward sounds better when it is close to its destination. In ensemble playing this is a set of habits rather than a theory, and the habit is always the same: find the beat, and slow it down.

Drone-based traditions make the reference pitch explicit. Indian classical music fixes the tonic and fifth on the drone and treats the melodic intonation of other degrees as a variable that can be inflected expressively, which is a different use of the same physics. Turkish and Arabic makam practice includes intervals that sit between the equal-tempered minor and major third, often around 350 cents, close to an 11:9 ratio. Indonesian gamelan ensembles are tuned as a set rather than to a standard: each ensemble has its own tuning, and paired instruments are deliberately tuned a few cycles apart so that they produce a shimmering beat. A piano tuner spends an hour removing exactly that effect.

Composers have also treated just intonation as a system rather than a resource. Harry Partch built instruments for a forty-three-tone scale and wrote for the intonation he had built, and Ben Johnston wrote string quartets that specify just ratios in the score. Both cases depend on performers who can hear the difference and adjust, which makes the intonation part of the performance rather than a setting of the instrument.

#Working with it

For a performer, the useful skill is measuring beats by ear. Play the interval, listen to the top of the sound, and count the pulses. Then change one note by the smallest amount you can and hear the rate fall. When the pulses slow to nothing you are at the pure interval, and the position of that interval in the chord tells you which note to adjust: the third and the seventh carry the intonation of a chord, and the fifth is forgiving enough that it rarely needs attention.

For a composer or arranger, the practical consequences are about register and voicing. The cents error is fixed, but its audibility is not: the same wide third beats slowly in the bass and quickly in the treble, which is why a major third low in a chord sounds thick rather than sharp, and why arrangers keep thirds out of the bottom of a voicing and let the fifth or the octave carry the foundation. Held final chords sound more settled when the third is high enough to beat quickly and low enough not to scream.

A tuner measures the same quantity this article has been describing. The chromatic tuner reports the deviation of a played note from equal temperament in cents, so a fifth reading plus 2 and a third reading minus 14 is the table above, measured live. Hearing the beat and reading the cents are two ways of observing one number.

Questions

Why does a well-tuned piano still sound out of tune?

Equal temperament widens every major third by about 13.7 cents, so a piano's thirds beat audibly by design. A piano tuned to pure thirds would have to abandon pure fifths and would only be playable in a few keys.

Can a fixed-pitch instrument be tuned in just intonation?

Only for one key at a time. Tuning a keyboard so that C major is exactly pure makes the thirds of the remoter keys wider than equal temperament's, so the instrument has to choose which keys it wants to sound best.