Why the piano cheats — and you can hear it

A real piano fudges every interval a hair off the pure ratio so all twelve keys work in every key. Play a pure fifth against the piano's fifth and count the beats — the wobble is the compromise, measured in Hz.

two notes, almost the same — count the wobble

how far apart1.96 ¢ · 440.50 Hz vs 440 Hz

0.50

beats / sec

|440.5 − 440|

Set it to zero and the two tones lock into one still sound. Nudge them apart and the loudness starts to throb — exactly 0.50 swells every second, the plain difference of the two frequencies. That throb is a beat.

what the piano actually does to each interval

interval	pure ratio	piano's ratio	off by
Octave	2 : 1	×2.0000	0.0 ¢
Perfect fifth	3 : 2	×1.4983	-2.0 ¢
Perfect fourth	4 : 3	×1.3348	+2.0 ¢
Major third	5 : 4	×1.2599	+13.7 ¢
Minor third	6 : 5	×1.1892	-15.6 ¢
Major sixth	5 : 3	×1.6818	+15.6 ¢

The octave is left perfect. The fifth is shaved by a whisker — barely two cents, you'll strain to hear it. But the major third is stretched almost fourteen cents sharp: press pure then piano and the shimmer is unmistakable. That's the price of twelve keys that work in every key.

For the classroom

Learning goals

When two near-equal frequencies sound together, loudness pulses at the beat rate = |f₁ − f₂| hertz.
Equal temperament detunes each pure ratio by a small, computable number of cents (octave 0, fifth ≈ −2, major third ≈ +14).
Those cents become audible beats — the wider the tempering, the faster the wobble; the size of the beat is the size of the compromise.

Try this

1.Set the machine to 0 (silence-still), then to 'Piano's major third' (~14¢). Have students count the beats per second and compare to the |f₁ − f₂| readout.
2.Play 'pure' then 'piano' for the fifth, then for the major third. Ask which compromise the ear notices more, and connect it back to the ~2¢ vs ~14¢ column.