The Piano Is Slightly Out of Tune on Purpose
Sit down at a well-tuned piano and play a C major chord: C, E and G together. It sounds warm, settled, right. Nobody in the room winces.
But apart from the octaves, none of the notes in that chord are quite where nature would put them. The tuner has nudged them, deliberately, by tiny amounts. The piano you're playing is out of tune, and it's out of tune on purpose.
When two notes agree
To see why, start with what "in tune" really means.
The distance between two notes is called an interval. Some intervals have a special quality: when the two notes are tuned exactly, their sound waves line up so neatly that they blend into one calm, still sound. The cleanest are the octave (C up to the next C) and the fifth (C up to G).
Pull one of the notes slightly off and something changes. The sound starts to shimmer or pulse, a slow wobble that tuners call beating. The further off the note, the faster the beats. A piano tuner listens for exactly this. Beats are how they hear that two strings disagree.
So you'd think the job is simple. Tune every interval until the beating stops.
The gap that won't close
Here's the problem. You can't.
Imagine tuning a piano using nothing but perfect fifths. Start on a low C and tune the G above it so the fifth is pure. Then tune the D above that, then the A, and keep going. After twelve fifths you should arrive back at a C, seven octaves above where you started.
Except you don't, quite. Twelve pure fifths overshoot seven pure octaves by a small amount, so the note you land on is a little sharp of the C it ought to be. Musicians call this leftover the Pythagorean comma, after the ancient Greek tuning tradition linked to Pythagoras.
It's a bit like walking around a block, turning the same perfect corner each time, and finding you've ended up a few steps past your own front door. A violinist can fix this by ear, adjusting each note as they play. On a piano, every note is fixed until the tuner comes back. Something has to give.
Sharing the bill
For centuries, keyboard tuners dealt with the leftover in different ways. Older tunings tended to make the most commonly used keys sound sweet and let the error pile up in keys that were rarely played. Stray into those keys and some chords could sound badly sour.
Equal temperament takes the opposite approach. Instead of keeping some intervals pure and letting others take the hit, it spreads the leftover evenly across all twelve notes of the octave. Every step from one key to the next, white or black, is made exactly the same size.
Think of a group dinner where, instead of working out who ordered what, everyone agrees to split the bill evenly. Nobody pays exactly their share, but nobody gets stung either. In equal temperament every key carries the same small error, so a piece sounds just as good, or just as slightly off, in F sharp as it does in C.
That's the bargain. The octave stays pure. Everything else is a whisker out.
Who pays the most
The error doesn't land equally on every kind of interval, though. Some pay far more than others.
Fifths get off lightly. An equally tempered fifth is a tiny bit narrower than a pure one, so close that most listeners would never notice. Fourths are much the same.
The major third is a different story. That's the C-to-E interval at the heart of every major chord, and on an equally tempered piano it comes out noticeably wide. Hold a major chord and listen carefully, and you can hear it gently pulse.
Which leads to a slightly awkward truth. The chord most of us think of as the very sound of "in tune", the bright, happy major chord, is the one most bent out of shape by the compromise. We've simply heard it that way all our lives.
Even the octaves get stretched
If that were the whole story, a tuner could work from a chart. Real piano strings add another twist.
A piano string is a thick, stiff length of steel wire, and stiff wire doesn't vibrate quite like an ideal, perfectly flexible string. The fainter extra pitches it produces above the main note, called overtones, come out slightly sharp. Physicists call this inharmonicity.
Tuners compensate by stretching. Compared with strict equal temperament, they tune the top of the keyboard a little sharp and the bottom a little flat, so that octaves sound clean against those sharp overtones. Tuners widely treat this stretch as standard practice, and lab studies of piano tones back it up. A piano tuned exactly to the textbook numbers would tend to sound flat at the top, with the bass clashing against the treble.
So even the octave, the one interval equal temperament promises to keep pure, gets nudged on a real piano so that bass and treble agree.
A choice, not a law
None of this was handed down from on high. Equal temperament didn't arrive in a single moment, and it didn't take over overnight. Different tuning systems competed for a long time, and some musicians objected to its beating thirds and preferred older tunings. Equal temperament won out gradually, as music increasingly needed to move freely between all twelve keys.
Instruments without fixed notes don't have to follow it, either. Singers and string players can adjust their notes by ear, and when a string player has to match a piano, they bend their tuning towards its tempered notes.
The sound of agreement
Go back to that C major chord. Every note in it has been shifted a hair from where pure arithmetic would put it, the third most of all, the octaves stretched to suit the steel.
And it still sounds right. Not because it's perfect, but because generations of builders, tuners and listeners agreed to be a little wrong everywhere so that nothing would be badly wrong anywhere. What we hear as "in tune" is that agreement, played back to us every time a key goes down.
Sources
Robin Wilson, "Why are pianos out of tune?", Gresham College lecture (2009). https://www.gresham.ac.uk/watch-now/why-are-pianos-out-tune (accessed 31 Oct 2026)
Jussi Jaatinen and Jukka Pätynen, "Effect of inharmonicity on pitch perception and subjective tuning of piano tones", Journal of the Acoustical Society of America (2022). https://doi.org/10.1121/10.0013572 (accessed 31 Oct 2026)
Haye Hinrichsen, "Revising the Musical Equal Temperament", arXiv (2015). https://arxiv.org/abs/1508.02292 (accessed 31 Oct 2026)
Living Pianos, "Why a piano is never in tune". https://www.livingpianos.com/articles/why-a-piano-is-never-in-tune/ (accessed 31 Oct 2026)
Eric Johnson Pianos, "Equal Temperament, Inharmonicity and Piano Tuning". https://ericjohnsonpianos.com/equal-temperament-and-piano-tuning/ (accessed 31 Oct 2026)
