String stiffness raises the frequency of a piano’s higher modes relative to the fundamental.

Evidence

When a piano string is struck by the hammer, it behaves partly like a bar because it is not completely flexible due to its considerable stiffness. This results in a slight raising in frequency of all the component modes with respect to the fundamental, an effect known as “inharmonic­ ity,” and this effect is greater for the higher modes. — David Howard, Jamie Angus, coustics and Psychoacoustics, 1116

The piano is an instrument in which strings are struck to provide the sound source, and the relationship discussed in the last Section (4.2.1) concerning the modes that will be missing in the sound source is equally relevant here. There is, however, an additional effect that is particularly relevant to the sound source in the piano, and this relates to the fact that the strings of a piano are under very high tension and therefore very hard compared with those on a harpsichord or plucked orchestral stringed instrument. Strings on a piano are struck by a hammer, which is “fired” at the string, from which it immediately bounces back so as not to interfere with the free vibration of the string(s). When a piano string is struck by the hammer, it behaves partly like a bar because it is not completely flexible due to its considerable stiffness. This results in a slight raising in frequency of all the component modes with respect to the fundamental, an effect known as “inharmonicity,” and this effect is greater for the higher modes. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1090

When a piano string is struck by the hammer, it behaves partly like a bar because it is not completely flexible due to its considerable stiffness. This results in a slight raising in frequency of all the component modes with respect to the fundamental, an effect known as “inharmonicity,” and this effect is greater for the higher modes. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1090

Categorization

• DEFINE: Defines inharmonicity — the deviation of partials from perfect integer multiples of the fundamental in real (stiff, thick) strings.
• XREF: Connects to Schoenberg’s and orchestration literature on timbre — stiffness varying by pitch and instrument type accounts for why a piano’s partials never sit exactly on the harmonic series. — connects to: Higher piano notes require progressively greater octave stretching — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1097
• corroborated: 3 independent passages support this claim — David Howard, Jamie Angus, coustics and Psychoacoustics, 1116; David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1090