Why Can’t Every Fifth on a Piano Be Tuned Pure?

A pure fifth has the frequency ratio 3:2 and sounds stable and well fused. Starting on C and tuning twelve consecutive pure fifths eventually returns to the note name C; on a keyboard, that endpoint should also equal seven octaves above the start. The arithmetic does not close: (3/2) to the twelfth power is slightly larger than 2 to the seventh power. The gap is called the Pythagorean comma. If the earlier fifths remain pure, the accumulated error must be placed somewhere, producing a conspicuously rough 'wolf' interval.

Twelve-tone equal temperament takes a different approach. It preserves the octave and divides it into twelve semitones with equal frequency ratios. Every fifth is therefore slightly narrower than 3:2, spreading the discrepancy rather than concentrating it. The cost is that, apart from octaves, familiar intervals are not perfectly pure; major thirds differ particularly noticeably from just intonation. The benefit is that D-sharp and E-flat can share one key, music can modulate, and every key has the same interval structure.

This does not mean a piano tuner merely applies a formula. Real piano strings are stiff, so their partials sit slightly above ideal integer multiples, and octaves at the extremes are usually stretched. Equal temperament supplies the organising principle; beat rates, strings and the sound of the whole instrument still guide the practical tuning.

The useful listening judgement is not that a piano is 'out of tune', but that tuning chooses among incompatible demands. The piano accepts a tiny impurity in every fifth to make the entire keyboard navigable. Slight beating within chords is therefore not a system failure; it is part of the audible texture left by this coordinated compromise.

https://www.phys.unsw.edu.au/jw/tartini-temperament.html
https://www.animations.physics.unsw.edu.au/jw/Tartini-tones-temperament.html