A string’s stiffness raises its higher-frequency modes above the ideal values

Evidence

A real string has thickness and resists bending of its own accord, independent of the tension. For low-frequency, long-wavelength vibrations, this does not make much difference, since the curvatures of the string are low. However, at higher frequencies and shorter wavelengths, the bending is more severe, since so many wavelengths are packed into the length of the string. This extra resistance to bending sends the frequencies of a real string higher than they would be for the ideal string. — Eric J. Heller, Why You Hear What You Hear- An…, 860

Categorization

• DEFINE: Explains the physical mechanism behind inharmonicity of real strings — bending stiffness adds a restoring force beyond tension that grows with frequency.
• XREF: This is the physical basis of the inharmonicity of piano strings — the reason piano tuners stretch their tuning, a classic connection between physics and music practice. — connects to: Higher modes of a lumped bead string are not equally spaced, ruining periodicity; A tiny extra mass ruins a string’s harmonic spacing and makes it sound sour; String stiffness raises component mode frequencies above the harmonic series, creating inharmonicity (+9 more)
• corroborated: 3 independent passages support this claim — Eric J. Heller, Why You Hear What You Hear- An…, 860

Broader Connections

• String stiffness raises component mode frequencies above the harmonic series, creating inharmonicity #corroborates (1.00)
• Real string stiffness slightly raises wave velocity with frequency, affecting timbre. #corroborates (0.85)
• String stiffness makes higher frequencies travel faster, changing timbre #corroborates (0.85) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 103
• Octave stretching on pianos emerges directly from string inharmonicity #expands (0.80) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1096
• Instrument stiffness versus tension determines how widely its modes are spaced #qualifies (0.75)