Real strings produce inharmonic partials whose deviation grows steeply with thickness

Evidence

Any practical string must have a finite radius, and the effect is given in Equation 4.2c. This is the effect of inharmonicity, or the amount by which the actual frequencies of the modes vary from integer multiples of the fundamental. … It can be seen that inharmonicity increases as the square of the component mode (n2) and as the fourth power of the string radius (r4), and that it decreases with increased tension and as the square of increased length. Inharmonicity can be kept low if the strings are thin (small r), long (large L) and under high tension (high T). The effect would therefore be particularly marked for bass strings if they were simply made thicker (larger r) to give them greater mass, since the variation is to the fourth power of r. Therefore, in many stringed instruments, including pianos, guitars and violins, the bass strings are wrapped with wire to increase their mass without increasing the underlying core string’s radius (r). — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1095

Categorization

• DEFINE: Defines inharmonicity precisely and explains why wire-wrapping bass strings avoids it.
• XREF: connects to: String instruments naturally produce inharmonic partial spectra. — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 2044; Most vibrating objects produce inharmonic partials, unlike the ideal string — Eric J. Heller, Why You Hear What You Hear- An…, 805; String stiffness raises component mode frequencies above the harmonic series, creating inharmonicity (+3 more)

Broader Connections

• Real strings show slightly sharp partials because tight curvature stiffens them like bars #expands (0.92) — Eric J. Heller, Why You Hear What You Hear- An…, 876
• String stiffness raises component mode frequencies above the harmonic series, creating inharmonicity #expands (0.90)
• 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
• String instruments naturally produce inharmonic partial spectra. #expands (0.85) — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 2044
• Most vibrating objects produce inharmonic partials, unlike the ideal string #expands (0.78) — Eric J. Heller, Why You Hear What You Hear- An…, 805
• Higher modes of a lumped bead string are not equally spaced, ruining periodicity #qualifies (0.72)