Higher modes of a lumped bead string are not equally spaced, ruining periodicity

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 (#expands, via linked claim)

Categorization

• XREF: connects to: The inharmonicity of real strings is a desirable feature, not a defect; A string’s stiffness raises its higher-frequency modes above the ideal values — Eric J. Heller, Why You Hear What You Hear- An…, 860; String stiffness raises component mode frequencies above the harmonic series, creating inharmonicity (+2 more)

Related

• A tiny extra mass ruins a string’s harmonic spacing and makes it sound sour (similarity 0.93)
• A string’s stiffness raises its higher-frequency modes above the ideal values (similarity 0.92)
• The inharmonicity of real strings is a desirable feature, not a defect (similarity 0.87)
• Instrument stiffness versus tension determines how widely its modes are spaced (similarity 0.85)
• String stiffness raises the frequency of a piano’s higher modes relative to the fundamental. (similarity 0.85)
• Slightly mistuned piano strings produce richer, longer-lasting sound (similarity 0.84)
• String stiffness makes higher frequencies travel faster, changing timbre (similarity 0.84)
• Real string stiffness slightly raises wave velocity with frequency, affecting timbre. (similarity 0.84)
• Waveforms become non-periodic because higher normal-mode frequencies are not equally spaced (similarity 0.83)
• Finite wire thickness raises higher piano partials, producing stretched harmonics (similarity 0.82)
• Real strings show slightly sharp partials because tight curvature stiffens them like bars (similarity 0.82)
• Increasing string tension raises the frequency of a single-bead oscillator (similarity 0.82)
• Any initial shape repeats periodically only if all modes are harmonic (similarity 0.81)
• Each piano string has two independent polarization modes doubling the coupled-mode count (similarity 0.81)
• When a string’s linear density is fixed, pitch is varied by tension and length (similarity 0.81)

Broader Connections

• String stiffness raises component mode frequencies above the harmonic series, creating inharmonicity #expands (0.95)
• A string’s stiffness raises its higher-frequency modes above the ideal values #expands (0.90) — Eric J. Heller, Why You Hear What You Hear- An…, 860
• A tiny extra mass ruins a string’s harmonic spacing and makes it sound sour #exemplifies (0.85)
• Octave stretching on pianos emerges directly from string inharmonicity #expands (0.80)