Equal Temperament vs Just Intonation: Why the Piano Is Deliberately Out of Tune

Pure intervals do not fit inside an octave. Every tuning system is a decision about where to hide the leftovers.

Short answer: just intonation tunes intervals to small whole-number ratios, so their partials line up exactly and chords lock without beating, but the resulting scale only works in the key it was built for. Equal temperament divides the octave into twelve identical steps of exactly 100 cents, so every key is equally usable and every interval except the octave is slightly impure. On a modern piano the fifth is about 2 cents narrow and the major third about 13.7 cents wide. That is not a manufacturing defect. It is the price of being able to play in all twelve keys on one instrument.

The problem: the arithmetic does not close

Start from two facts that both sound reasonable. An octave is a frequency ratio of 2:1. A perfect fifth is 3:2. Now stack twelve fifths: multiply 3/2 by itself twelve times and you get about 129.746. Stack seven octaves instead: two to the seventh is 128. Those should be the same note. They differ by a ratio of about 1.0136, which is 23.46 cents, the Pythagorean comma. Nearly a quarter of a semitone has appeared out of nowhere.

It happens again with thirds. Four pure fifths, reduced into one octave, give a major third of 81:64. A pure major third is 5:4, that is 80:64. The gap between them, 81:80, is 21.51 cents, the syntonic comma. Pure fifths and pure thirds are not compatible.

This is not a defect of Western music. It is arithmetic: powers of two, powers of three and powers of five never coincide. Any instrument with fixed pitches has to absorb the leftovers somewhere, and the history of tuning is the history of arguing about where.

Just intonation: pure ratios, one key at a time

Just intonation takes the ratio side of the trade. Intervals are set to small whole-number ratios drawn from the harmonic series, so that partials of the two notes coincide instead of clashing. The result is unmistakable when you hear it: the chord stops shimmering and settles. There is nothing mystical about the effect. If the fifth partial of the lower note lands exactly on the fourth partial of the upper note, there is nothing left to beat against, and the roughness disappears.

The cost arrives the moment you change key. A keyboard tuned justly in C will have a usable D major triad only by accident, and somewhere in the circle there will be a fifth left so badly wide or narrow that it is unusable, traditionally called a wolf. The Huygens-Fokker Foundation, which exists to document exactly this territory, describes just intonations as systems built on purity that quickly become complicated, and often demand new instruments to realise properly.

Equal temperament: everything slightly wrong, nothing unusable

Twelve-tone equal temperament takes the flexibility side. Every semitone is the twelfth root of two, which is exactly 100 cents by definition, and the comma is spread evenly across the whole circle so that no key is worse than any other. The fifths are all narrowed by about 2 cents. Nobody notices. The thirds pay for it.

Two cents is below the threshold most listeners can detect on a single sustained comparison, so the tempered fifth passes unnoticed. Fourteen to sixteen cents is a different matter, and it is why an equal-tempered major third sounds restless in a way that a barbershop third does not.

What that actually sounds like, in beats

The most concrete way to hear temperament is to count beats. Take A4 at 440 Hz. The third harmonic of that A is 1320 Hz. In equal temperament, E5 sits at 659.26 Hz, whose second harmonic is 1318.51 Hz. The two are 1.49 Hz apart, so a tempered fifth on A pulses about one and a half times per second. Slow, gentle, and easy to miss unless you are told to listen for it.

Now the major third. The fifth harmonic of A4 is 2200 Hz. In equal temperament, C sharp 5 sits at 554.37 Hz, whose fourth harmonic is 2217.46 Hz. That is 17.5 Hz apart, which is far too fast to count; you perceive it as roughness rather than pulsing. Tune the C sharp down to 550 Hz, a pure 5:4, and its fourth harmonic lands exactly on 2200. The beating stops. That is the whole difference between the two systems, audible in one adjustment.

Everything in between

Equal temperament was not an obvious winner, and for centuries it was not the default. Meantone temperaments narrow the fifths more aggressively in order to buy nearly pure major thirds, which works beautifully in a limited set of keys and catastrophically outside them. Well temperaments distribute the error unevenly so that common keys are sweeter and remote ones are merely tolerable, which is where the old idea of key character comes from. Extended systems such as 31-tone equal temperament close the circle at a different number of steps and get very close to pure thirds while remaining fully modulatory.

Grove's survey of temperaments treats all of these as members of one family: each is a rule for distributing an unavoidable error. When early-music ensembles choose a historical temperament, they are not being pedantic; they are restoring a set of key colours that the composer could hear and that equal temperament flattens out.

The piano is off twice over

Temperament is only half the story on a piano. Real strings are stiff, so their partials are not exact multiples of the fundamental: the second partial is slightly sharper than twice the fundamental, and the error grows for higher partials. That is called inharmonicity, and it means that a piano tuned to mathematically exact octaves would sound flat in the treble and sharp in the bass to a listener attending to partials rather than fundamentals.

Tuners therefore stretch the octaves, widening them progressively towards both ends of the keyboard. The average shape of that stretch across skilled technicians is known as the Railsback curve, and acoustics work has shown it follows quantitatively from string inharmonicity plus the ear's sensitivity to roughness. So the piano is not tuned to the arithmetic, and it should not be: it is tuned to what the instrument's own partials demand.

Who still tunes purely

Anyone who can bend a pitch. Unaccompanied choirs drift towards pure thirds because the chord audibly locks when they do, which is also why a choir singing with a piano and a choir singing alone sound subtly different. Barbershop quartets do it deliberately and treat the resulting ring as the point of the style. String players adjust every note by ear, tuning leading notes high and thirds low depending on harmonic context. Many traditions built on a sustained drone tune to the drone, which is just intonation by another name. And a whole strand of contemporary composition works with ratios that equal temperament cannot reach at all.

What this means for your ear

When you train intervals with a synthesised reference, you are almost always hearing equal temperament. That is the right default, because it is what keyboards, guitars, tuners and most recordings use. But it is worth knowing that the major third you have memorised is about 14 cents wider than the one a good choir will hand you, and that a singer who lands on the pure version is not flat. Interval names are categories; tuning systems decide where inside the category the note actually sits.

Practically, this is an argument for training in cents rather than in adjectives. If you know that 2 cents is inaudible, 14 cents is a colour, and 25 cents is an error, you can talk about intonation precisely instead of arguing about whether something sounded nice. Our guide to cents covers the unit, and how small a difference you can hear covers where your own threshold currently sits.

Keep reading

• What cents are: the unit that makes all of this measurable.
• Beats and tuning by ear: how tuners turn a pitch error into a countable pulse.
• Consonance and dissonance: why lined-up partials sound smooth in the first place.
• A=440 and concert pitch: the other tuning decision, and why orchestras ignore it.

Sources: Temperaments, Grove Music Online, Oxford University Press; Huygens-Fokker Foundation, Tuning systems and Microtonality; Huygens-Fokker Foundation, List of intervals and Logarithmic interval measures; Nave, Equal temperament and Just temperament and others, HyperPhysics, Georgia State University; Truax (ed.), Equal temperament and Intonation, Handbook for Acoustic Ecology, Simon Fraser University; Monzo, 12-edo, Tonalsoft Encyclopedia; Giordano (2015), Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance, Journal of the Acoustical Society of America 138(4).