Stretched partials give the piano its characteristic musical color

Evidence

Piano wires have finite thickness, making them not quite ideal; the effect is to slightly raise the frequency of the higher modes compared to multiples of the lowest frequency. Higher frequencies bend the wires more sharply, which is resisted in the same way that a metal rod resists bending, a force that is in addition to the tension force that we took into account when deriving the harmonic modes of an ideal stretched string. Real string modes have stretched partials. — Eric J. Heller, Why You Hear What You Hear- An…, 1820 (#corroborates, via linked claim)

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 (#exemplifies, via linked claim)

Stretch tune. Modulates the Railsback effect. Stretch tuning makes the low range slightly flatter and the high range slightly sharper so you hear an in-tune piano across the full range. — Eric J. Heller, Why You Hear What You Hear- An…, 1849 (#expands, via linked claim)

Categorization

• XREF: connects to: Finite wire thickness raises higher piano partials, producing stretched harmonics — Eric J. Heller, Why You Hear What You Hear- An…, 1820; Octave stretching on pianos emerges directly from string inharmonicity — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1096; Piano octave stretching grows with pitch as strings shorten (+1 more)

Broader Connections

• Finite wire thickness raises higher piano partials, producing stretched harmonics #corroborates (0.90) — Eric J. Heller, Why You Hear What You Hear- An…, 1820
• Octave stretching on pianos emerges directly from string inharmonicity #exemplifies (0.85) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1096
• Piano octave stretching grows with pitch as strings shorten #expands (0.80)
• Pianos are stretch-tuned so low notes sit flat and high notes sharp #expands (0.80) — Eric J. Heller, Why You Hear What You Hear- An…, 1849