Piano inharmonicity and stretched tuning
PAPER III — PHYSICS · QUESTION 32 OF 35
In plain terms: The free university acoustics page explaining why a real string’s overtones drift sharp — the physics the question wants.
The question as set
Why does a struck piano string not sound the harmonic series it is drawn to sound? Quantify the inharmonicity contributed by bending stiffness, and say what stretched tuning is for.
The question in Paper III · Its section on the reference page
The question in context
This is question 32 of the 35 set in Paper III — Physics of The Truly Hardest Exam in the World, the examination of The College of All Minds. In the College’s subject index it is filed under Acoustics and Hearing and the ear. Like every question of the College, it carries three references of record, verified live at publication and described twice — technically, and in plain terms — so the ground can be judged before it is walked. The response required is an argument, not a survey, of at most three thousand words, as technical as the question demands and no more.
The references of record
The mechanism stated: stiffness adding a bending term to the wave equation, partials sharpened above harmonic ratios.
In plain terms: The free university acoustics page explaining why a real string’s overtones drift sharp — the physics the question wants.
The inharmonicity coefficient derived and quantified: partial frequencies as n·f₀√(1+Bn²), with B from radius, tension, and Young’s modulus.
In plain terms: The free Stanford text giving the formula that puts a number on the effect.
Acoustics and vibration animations
The modes made visible: ideal against stiff strings, and the consequence for tuning a stretched octave.
In plain terms: The free animated demonstrations that show what the equations are describing.
Background: Inharmonicity · Stretched tuning