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A vibrating string creates harmonics. Ideally the second harmonic (an octave above the fundamental) is exactly twice the fundamental in pitch and half the fundamental in length. However, materials like steel can exhibit some inharmonic characteristics. For example, the node created by the vibrating string is a point with zero amplitude, essentially a spot on the string that doesn’t vibrate, and theoretically has no length. However, since steel has some rigidity the node point also has some dimension. The result is that these nodes actually shorten the length of the partials and cause them to be sharper than the ideal harmonic pitch. With each higher harmonic, the number of nodes increases, and the string length of the partial is shortened even more from the ideal length, resulting in an even sharper pitch. Guitar makers take this into account when they set frets on steel-string guitars and minutely lengthen the distance between frets compared to nylon-string guitars, which have smaller nodes. Piano tuners also account for the steel strings and often tune each successive octave to the second, third, or fourth harmonics of the lower octave. This essentially stretches the tuning sharp from middle C to the highest C on the piano, ¬between 20 to 30 cents even as much as 40 cents (40% of a half step), and 15 to 20 cents flat from the A below middle C down to the lowest A on the piano. These pitch changes (from the ideal) are fairly subtle from octave to octave, but tuning a piano this way ensures that notes in one octave when played simultaneously with notes in another octave will sound perfectly in tune.
