Piano octaves are stretched because lower notes’ overtones are inharmonic

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

Category: Causal Relationships

• DEFINE: Explains the mechanism of octave stretching in piano tuning — a concrete, non-obvious psychoacoustic phenomenon.
• XREF: Extends the inharmonicity concept from the previous passage into practical tuning; connects to equal temperament discussion in chapter 3.

Related

• Piano octave stretching grows with pitch as strings shorten (similarity 0.91)
• Real piano strings force a stretched tuning to avoid audible beats (similarity 0.90)
• Octave stretching on pianos emerges directly from string inharmonicity (similarity 0.90)
• Pianos are stretch-tuned so low notes sit flat and high notes sharp (similarity 0.87)
• Even the perfect octave is routinely violated in piano tuning (similarity 0.86)
• Hammer impact speed and mass trade off to shape a piano’s tone (similarity 0.85)
• Inconsistent octave stretching that leaves partials linear is jarringly unpleasant (similarity 0.83)
• Piano tuning must sacrifice perfect octaves to avoid beating against sharp harmonics (similarity 0.83)
• Piano tuning stretched so far it has become a required sound (similarity 0.82)
• Stretched partials give the piano its characteristic musical color (similarity 0.82)
• Physically correct octaves actually cause audible beats in a real piano (similarity 0.81)
• Slightly mistuned piano strings produce richer, longer-lasting sound (similarity 0.81)
• Instrument tuning relies on high harmonics because pitch changes there are easier to hear (similarity 0.81)
• Moisture, cracking, and tension drift make every grand piano uniquely voiced over its lifetime (similarity 0.81)
• Finite wire thickness raises higher piano partials, producing stretched harmonics (similarity 0.81)