Real piano strings force a stretched tuning to avoid audible beats

Evidence

If the f0 of a piano note is 400 Hz and inharmonicity results in the second component being stretched to 801 Hz, how many cents sharp will the note an octave above be if it is tuned for no beats between it and the octave below? Tuning for no beats will result in the f0 of the upper note being 801 Hz, slightly greater than an exact octave above 400 Hz, which would be 800 Hz. The frequency ratio (801/800) can be converted to cents using Equation A3.4 in Appendix 3: Number of cents 3986. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1098

Categorization

• DEFINE: Concrete worked example of how inharmonicity propagates up the octave chain, forcing stretched tuning.
• XREF: connects to: Higher piano notes require progressively greater octave stretching — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1097