Octave stretching on pianos emerges directly from string inharmonicity

Evidence

The notes of a piano are usually tuned to equal temperament (see Chapter 3), and octaves are then tuned by minimizing the beats between pairs of notes an octave apart. When tuning two notes an octave apart, the components that give rise to the strongest sensation of beats are the first harmonic of the upper note and the second harmonic of the lower note. These are tuned in unison to minimize the beats between the notes. This results in the f0 of the lower note being slightly lower than half the f0 of the higher note due to the inharmonicity between the first and second components of the lower note. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1096

Categorization

• XREF: This explains why a piano’s treble octaves run sharp relative to pure math — connects to any listening experience of piano tuning offsets. — connects to: The inharmonicity of real strings is a desirable feature, not a defect; Higher modes of a lumped bead string are not equally spaced, ruining periodicity; A string’s stiffness raises its higher-frequency modes above the ideal values — Eric J. Heller, Why You Hear What You Hear- An…, 860 (+10 more)
• corroborated: 3 independent passages support this claim — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1096

Broader Connections

• Piano octave stretching grows with pitch as strings shorten #expands (0.95)
• Musical intervals trace directly to harmonic-series frequency ratios #qualifies (0.75) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 811