The air sings: sympathetic vibrations
Chords sound rotten on an out-of-tune piano. I’ve written about why we think that might be: harmonics of the detuned lower notes end up a small distance from those of the upper notes, rather than directly overlapping them.
My piano was tuned this weekend. Playing it after the technician had left, the sound’s transformation startled me: how rich and powerful my piano had suddenly become! I remembered, then, a less obvious but absolutely critical effect of correct tuning: sympathetic vibrations of a note’s upper harmonics.
When you play a middle C, the resulting wave carries pitches of higher frequencies according to the harmonic series: C5, G5, C6, E6, and so on. The wave, emanating from middle C’s string, will then excite the strings corresponding to the other pitches—providing the dampers on those strings are lifted (by pressing its key or the damper pedal). The sympathetic vibrations in these higher strings increase the volume of their respective harmonics, coloring the resulting tone.
In other words, not every C major chord composed of a C4, E4, and G4 will sound the same. The resulting sound will depend on which dampers are lifted in the octaves above! If the piano is out in tune, though, the frequencies of the strings in the harmonic series won’t align, so there will be little sympathetic vibration.
Here’s a little video demonstrating this principle in action: