Finite wire thickness raises higher piano partials, producing stretched harmonics

Evidence

Piano wires have finite thickness, making them not quite ideal; the effect is to slightly raise the frequency of the higher modes compared to multiples of the lowest frequency. Higher frequencies bend the wires more sharply, which is resisted in the same way that a metal rod resists bending, a force that is in addition to the tension force that we took into account when deriving the harmonic modes of an ideal stretched string. Real string modes have stretched partials. — Eric J. Heller, Why You Hear What You Hear- An…, 1820

Categorization

• corroborated: 2 independent passages support this claim — Eric J. Heller, Why You Hear What You Hear- An…, 1820
• XREF: connects to: Physically correct octaves actually cause audible beats in a real piano; Stretched partials give the piano its characteristic musical color; Octave stretching on pianos emerges directly from string inharmonicity — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1096 (+3 more)

Broader Connections

• Octave stretching on pianos emerges directly from string inharmonicity #expands (0.95) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1096
• A string’s stiffness raises its higher-frequency modes above the ideal values #corroborates (0.90) — Eric J. Heller, Why You Hear What You Hear- An…, 860
• Piano octave stretching grows with pitch as strings shorten #qualifies (0.80)