Equal, meantone, just — why your piano is slightly out of tune

An arithmetic problem nobody can fix

Stack twelve pure fifths upward from C — C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯ — and you land on a note that ought to be a C again, seven octaves higher. It is not. It misses by just under a quarter of a semitone, 23.46 cents to be exact.

The reason is arithmetic and cannot be circumvented. A pure fifth is the ratio 3:2, an octave 2:1. Twelve fifths give (3/2)¹² ≈ 129.746; seven octaves give 2⁷ = 128. Those two numbers are not equal and cannot be, because a power of three is never a power of two. The difference is called the Pythagorean comma.

From this it follows that a keyboard with twelve notes per octave cannot have all its intervals pure. The error has to be put somewhere. The entire history of temperament is the history of the question where exactly.

Three answers from three centuries

Just intonation fixes the intervals of a single key exactly according to the simple ratios: third 5:4, fifth 3:2. In that key everything is flawless. Modulate and it becomes unbearable — individual intervals end up more than 20 cents off. For unaccompanied choirs just intonation is nevertheless the normal case, because singers adjust pitch continuously.

Meantone temperament, dominant from the sixteenth into the eighteenth century, distributes the error differently: it makes the major thirds pure and narrows every fifth slightly in exchange. The result sounds wonderfully warm in the common keys. The price is one single, appallingly out-of-tune fifth, the so-called wolf fifth, usually between G♯ and E♭. Pieces that touch it were simply unplayable on such instruments.

Equal temperament, standard since the nineteenth century, distributes the comma perfectly evenly: every semitone is exactly the twelfth root of two. No fifth is pure, no third is pure, but all keys are equally good — and equally imperfect. The fifth sits 2 cents flat, which nobody hears. The major third sits 14 cents sharp, which you certainly do hear once you have noticed it.

What this means day to day

Tune a piano so that every string reads exactly zero cents and you have an equal-tempered instrument — and a C major triad whose third sits 14 cents sharp. That chord beats slightly, and that is normal. It is not a fault of the instrument but the compromise built into the system.

With instruments of variable pitch it looks different. String quartets, brass ensembles and vocal groups play and sing thirds lower than a piano would have them — automatically, without discussion, because the ear corrects towards the absence of beating. That is exactly why an a cappella chord often sounds purer than the same chord on a grand piano.

And it is why combining piano and choir is acoustically demanding. The choir pulls towards just intonation, the piano stays equal-tempered, and the two rub against each other on the thirds. Experienced choral directors therefore prefer to give chords without the third in rehearsal, or drop the instrument entirely once the intonation is secure.

The guitar adds a difficulty of its own: the frets sit equal-tempered, but finger pressure raises string tension by differing amounts depending on gauge and position. That is why an open E major chord sounds slightly off on many guitars although every open string reads zero cents — the only remedies are intonation at the saddle and, for the major third G♯, a G string deliberately tuned a touch flat.

From practice

The demonstration that reliably convinces takes thirty seconds and needs a guitar. Play the harmonic over the seventh fret of the A string (a pure fifth above the fundamental, therefore a pure E) together with the open high E string. Both should be the same E. They are not quite: the pure fifth lies about 2 cents above the equal-tempered one, which at 330 Hz gives a slow beat of about 0.4 Hz, once every two and a half seconds, audible in a quiet room if you listen closely. That is not the comma but the deviation of a single fifth, about one twelfth of it. The whole comma of about 23 cents only arises when twelve fifths are stacked on top of each other.

With advanced students the historical route through a score is worth taking. Bach’s “Well-Tempered Clavier” of 1722 goes through all twenty-four keys — which would have been impossible on a meantone instrument. “Well-tempered” expressly does not mean “equal”, but an unevenly distributed temperament in which all keys are playable while each retains its own character. That B♭ major sounded different from B major was an audible fact for Bach’s audience, not an imagining.

A typical misconception among learners: they take the tuner as the standard of purity. A tuner measures against the equal-tempered grid, that is, against a compromise. When a wind player in an ensemble plays their third 12 cents below the display, they are not tuning wrongly — they are tuning purely. Saying that sentence out loud once spares a good deal of rehearsal argument.

How a piano tuner solves it in practice

A tuner does not start at an arbitrary key but first lays the so-called temperament: roughly one octave in the middle of the instrument, usually F to F. Only those twelve notes are tuned by calculation; everything else is then derived from them in pure octaves, and octaves can be tuned beat-free and therefore unambiguously.

Within the temperament nothing is tuned pure; everything is tuned by beat rates. In that register an equal-tempered fifth beats about once every second and a half, while a major third beats several times a second. The tuner counts those beats and then cross-checks: a rising chain of thirds has to get steadily faster. If it does not, there is an error in the chain, and it can be narrowed down to the exact note.

One practical peculiarity is added on top: piano strings are stiff and therefore produce partials that lie fractionally above the whole-number multiple. So the bass is tuned slightly flat and the treble slightly sharp relative to the calculation — this is known as stretch. A piano reading exactly zero cents across all seven octaves actually sounds wrong. Here the deviation is the correction.