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Inharmonicity: Why Piano Strings Ring Sharp (Railsback & Stretch)
The ideal string vs. the real string
In theory, a perfectly flexible string under tension vibrates in a clean harmonic series: the fundamental f1, then exact integer multiples 2f1, 3f1, 4f1, and so on. Under that ideal model, tuning is simple - every overtone sits exactly where the maths predicts.
Real strings are not ideal. They have stiffness. A piano string is thick, high-tension steel (wrapped in copper in the bass for mass), and that stiffness resists bending. The stiffer the string, the more its higher overtones are pushed sharp of the pure harmonic series. Those real, slightly-sharp overtones are called partials, to distinguish them from ideal harmonics.
Why stiffness pushes the overtones sharp
The stiffness of a real string means its overtones do not sit at clean whole-number multiples of the fundamental. They get pushed progressively sharper the higher you go, and the deviation grows quickly - roughly with the square of the partial number. So while the first few partials are close to ideal, the upper ones land noticeably sharp of where the math says a pure harmonic would be.
That sharp drift high up is large enough to hear. It is why a piano tuned to pure equal temperament sounds flat and lifeless in the treble and muddy in the bass: the upper partials of each low note clash with the fundamentals above them. Every piano has its own stiffness curve, which is why per-instrument measurement matters. For the formula and the full maths (the B-coefficient, partial-frequency equations, typical ranges), see our inharmonicity blog article.
Why this forces stretch tuning (the Railsback curve)
Because the upper partials of a low note ring sharp, a piano only sounds in tune with itself if each octave is deliberately stretched: the higher note's fundamental is placed to align with the sharp upper partial of the note below it. Across the keyboard this stretching is summarised as the Railsback curve - bass progressively flatter, treble progressively sharper than pure equal temperament.
A tuner that ignores inharmonicity fights this: it shows the fundamental as "in tune" while the overtones your ear actually hears clash. That is why a piano needs a tuner that is aware of inharmonicity, not one that just targets equal-temperament frequencies.
How StroboPro handles inharmonicity
StroboPro's Piano Tuning Assistant (Beta) does not assume a textbook harmonic series. It models the inharmonicity of the string being measured - the B-coefficient - so the strobe targets shift to the real, stretched partial positions. The result is a strobe that reflects what the string actually does: the fundamental ring settles and the upper harmonic rings agree with it, rather than the tuner insisting on integer multiples the string cannot produce.
For the full keyboard, a real tuning still calls for setting temperament and tuning unisons by ear with mutes. StroboPro is a precise reading aid and inharmonicity-aware reference that speeds that work, not a replacement for a trained technician.
Related guides
Frequently asked questions
What is inharmonicity in music?
Inharmonicity is the tendency of a real, stiff string's overtones to ring slightly sharp of the pure whole-number multiples (harmonics) predicted by ideal physics. It is caused by string stiffness and grows stronger for higher overtones. It is the reason pianos are stretch-tuned rather than tuned to pure equal temperament.
Why are pianos tuned with stretched octaves?
Because of inharmonicity. A low piano string's upper partials ring sharp, so for the piano to sound in tune with itself, each higher note must be tuned slightly sharp to align with those sharp partials. This progressive stretching across the keyboard is described by the Railsback curve: bass flatter, treble sharper than pure equal temperament.
What is the inharmonicity coefficient B?
B is a small number describing how inharmonic a particular string is, depending on its length, tension, diameter, and stiffness. It sets how far each partial drifts sharp of a pure harmonic, with the drift growing quickly for higher partials. It varies per instrument, which is why a measured B matters for piano work. The formula and typical B ranges are covered in our [inharmonicity blog article](https://blog.strobopro.se/inharmonicity-in-stringed-instruments/).
Can a tuner app account for inharmonicity?
Yes. StroboPro's Piano Tuning Assistant (Beta) models the inharmonicity of the measured string - its B-coefficient - and shifts the strobe targets to the real, stretched partial positions instead of textbook integer multiples. This makes the reading match what the string actually does.
Does inharmonicity affect guitar as well as piano?
Yes, but far less. Guitar strings are thinner and under lower tension than piano strings, so their inharmonicity is small and rarely demands stretch tuning. Piano strings are thick, high-tension, and stiff, which is why inharmonicity is mainly a piano-tuning concern.
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Published by Applicaudia, the maker of StroboPro. Inharmonicity behaviour is described here in plain English; the formula, B-coefficient ranges, and worked examples live in our blog article. StroboPro is a tuning aid; a full concert piano tuning still requires a trained technician setting temperament and unisons aurally.
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