A Tool To Test Inharmonicity of Piano Strings

A piano-tuning physics demonstration

Stretched Octaves

Real strings resist bending, so their overtones ring sharp of a pure harmonic series — an effect called inharmonicity. Tune two such strings a mathematically exact octave apart and the octave beats. Set the string's stiffness below, then slide the octave sharp until the clash disappears.

String pitch

A2 · 110 Hz A3 · 220 Hz A4 · 440 Hz

Stiffness coefficient · B —

perfectly flexiblereal wireshort & stiff

Choose a real-world string… Nylon guitar string Steel-string guitar Piano — wound tenor string Piano — long treble string Piano — short upright treble Toy piano / stiff wire (extreme)

Match test

2:1 — lower's 2nd partial to octave's fundamental 4:2 — lower's 4th partial to octave's 2nd partial 6:3 — lower's 6th partial to octave's 3rd partial

which pair of partials should ring beat-free

Octave stretch —

ideal

flattheoretical 2:1widely stretched

Snap to theoretical (0¢) Snap to ideal stretch

lower string — partials octave string — partials

Partial 2 vs. octave fundamental — Hz

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▶ Play the octave Lower string alone Octave string alone Press play, then move the stretch slider — listen for the beating to settle.

Model: each string's n-th partial sits at fⁿ = n·f₀·√(1+B·n²), the standard stiff-string approximation (B is the inharmonicity coefficient). "Theoretical" isn't a single octave size — it depends which partials you're asking to line up. The 2:1, 4:2 and 6:3 tests all describe an octave, but each picks a different partial pair to beat-match, so each implies a slightly different ideal stretch. Both strings are modeled with the same B for clarity — real instruments pair strings of different lengths and gauges, which is part of why tuners stretch by ear across several tests rather than by a single formula.

Notice a few things happen even when you choose a “decent string” with low inharmonicity.

1. No matter where you place the octave, the wavy beating can never be perfectly eliminated. This is just basic physics. The partial chart shows why this is the case.

2. If you add no stretch to the octave (the theoretical tuning a digital tuner would give you), it sounds quite bad—much worse than a stretched octave.

3. A 4:2 octave sounds pretty good, but at A3 to A4, right in the middle of the piano, a stretch of 9 cents is needed!