Every Major Third on a Piano Is Fourteen Cents Sharp

The reference is a convention, not a constant

Everything above hangs on a chosen value for one note, and 440 Hz is a modern agreement rather than a fact about sound. Some orchestras tune slightly higher for brightness, historical performance often uses lower pitches, and the alternative reference of 432 Hz that circulates online sits 32 cents below 440 — about a third of a semitone, which is clearly audible next to an instrument tuned the usual way. Changing the reference moves every note by the same ratio and leaves every interval untouched, so a piece transposed this way sounds identical in structure and slightly different in colour.

A piano is not tuned to these numbers either

Real strings are stiff, and a stiff string's overtones sit slightly above exact multiples of its fundamental. A tuner who set every note to the frequencies computed here would produce an instrument whose octaves beat audibly, because the ear judges an octave against the overtone rather than the fundamental. The fix is stretch tuning: the top of the keyboard is set progressively sharp and the bottom progressively flat, by an amount that depends on the instrument. The arithmetic here gives the target for a synthesiser or a tuning app and the starting point for a piano, not the finished result.

Frequently asked questions

What is a cent?

A hundredth of an equal-tempered semitone, so twelve hundred of them make an octave. It exists because pitch is perceived multiplicatively — the same musical distance is the same ratio at any height — and adding cents is easier than multiplying ratios. Trained listeners notice about five cents on a sustained tone and considerably less on two notes sounding together, where the beating gives the difference away. That is why the two-cent error in a tempered fifth passes unnoticed and the fourteen-cent error in a third does not.

Why twelve notes rather than some other number?

Because twelve happens to be the smallest division of the octave that gets close to the simple ratios that matter most. Stacking twelve equal steps lands within two cents of a pure fifth, which is remarkably good and much better than eleven or thirteen would manage; the fifth is the interval the ear is most sensitive to, so a system that gets it nearly right can be wrong about the rest. Other divisions exist and are used — nineteen and thirty-one both give a better third — but each needs an instrument built for it, and twelve is the one the keyboard was standardised on.