Every piano is slightly out of tune, on purpose. Here is the bill.
Twelve-tone equal temperament makes all twelve semitones the same size, so a keyboard can play in every key. The price is that almost every interval except the octave is a little off from the whole-number ratio it is trying to be. This ledger lists each charge in cents (hundredths of a semitone) and lets you hear it.
| Interval | Pure ratio | Pure, cents | Tempered, cents | Charge, cents | Beat, Hz |
|---|---|---|---|---|---|
| Perfect fifth | 3:2 | 701.96 | 700 | −1.96 | 0.37 |
| Perfect fourth | 4:3 | 498.04 | 500 | +1.96 | 0.33 |
| Septimal tritone | 7:5 | 582.51 | 600 | +17.49 | 3.13 |
| Major third | 5:4 | 386.31 | 400 | +13.69 | 2.18 |
| Minor third | 6:5 | 315.64 | 300 | −15.64 | 2.37 |
| Minor sixth | 8:5 | 813.69 | 800 | −13.69 | 2.77 |
| Harmonic seventh | 7:4 | 968.83 | 1000 | +31.17 | 7.00 |
| Octave | 2:1 | 1200.00 | 1200 | +0.00 | 0.00 |
What this site claims
That the fifth is a bargain and the thirds are not. A tempered fifth is off by about 2 cents; a tempered major third by about 14, seven times as much. The claim is about what you can hear: open any interval, press "both", and the pure and tempered notes beat against each other. If a pulse at 2 Hz does not bother you, we are wrong about how much it matters, though not about the numbers.
Open an interval to hear it.