Piano stretch tuning and inharmonicity, explained

Updated October 31, 2026 · Fernhart Labs · sources at the end

Short answer

Piano strings are stiff, so their overtones (partials) sound slightly sharp of whole-number multiples of the fundamental. This is inharmonicity. Octaves tuned to sound clean therefore come out a little wider than 2:1, and over the whole keyboard the tuning stretches: flatter in the bass, sharper in the treble. Plotted against equal temperament, this is the Railsback curve.

The stretch is typically some tens of cents at the ends of the keyboard, and larger on smaller pianos, whose shorter, stiffer strings have more inharmonicity.

Inharmonicity in one formula

For a stiff string, partial *n* sounds at fₙ = n · f₀ · √(1 + B · n²) (Fletcher, 1964). B is the string's inharmonicity coefficient: zero for an ideal flexible string, about 10⁻⁴ in a grand's tenor, and up to around 10⁻² near the top of the keyboard. The higher the partial, the sharper it runs: with B = 0.001, the 10th partial is about 82 cents sharp of 10 × f₀.

Young (1952) measured B across real pianos: lowest in the tenor and bass, rising steeply in the treble, and higher on smaller pianos.

From inharmonicity to stretched octaves

A tuner sets octaves by listening for beats between partials: a 2:1 octave matches the upper note's fundamental with the lower note's 2nd partial; a 4:2 octave matches their 2nd and 4th partials; a 6:3 octave their 3rd and 6th. Because the lower note's partials are sharp, every one of these octaves is wider than an exact doubling, and wider octave types stretch more. Chained from the middle of the keyboard outwards, the small excess adds up to the Railsback curve.

How big the stretch is

These figures come from a typical-piano model of inharmonicity, with three common octave styles. They show the shape and the differences between pianos; a real piano's stretch is what you measure on it.

Stretch in cents from equal temperament, from a typical-piano model
Key	Grand, gentle (2:1)	Grand, standard	Grand, wide	Upright, standard	Spinet, standard
A0	-2.9	-12.0	-19.3	-16.7	-23.8
C1	-2.4	-8.9	-14.0	-12.5	-17.8
C2	-1.9	-3.7	-4.7	-5.2	-7.4
C3	-1.6	-2.2	-2.2	-3.1	-4.4
A3	-1.7	-1.7	-1.7	-2.4	-3.4
C4	-1.3	-1.3	-1.3	-1.8	-2.5
A4	+0.0	+0.0	+0.0	+0.0	+0.0
C5	-0.4	+0.9	+0.9	+1.2	+1.7
C6	+2.1	+6.5	+6.5	+9.0	+12.8
C7	+8.4	+12.9	+21.1	+18.0	+25.5
C8	+25.3	+29.7	+38.0	+41.3	+58.6

Computed with Pianowise's stretch-curve engine and its default inharmonicity model (a grand, and assumed levels for upright and spinet), A4 = 440 Hz. The model's levels and the octave styles are our documented choices, not a published standard.

A real piano for comparison

In our measurements of 45 sampled notes from a Yamaha C5 grand (Salamander Grand Piano) and the University of Iowa piano, the tuned instruments sat about −24 cents at A0 and C1 and rose smoothly to about +20 cents by B6, the same shape as the Railsback curve. Measured B values ran from about 10⁻⁴ in the tenor to 10⁻² in the top octave.

What this means when you tune

A tuner that targets plain equal temperament on every key will make a piano sound flat in the treble and muddy in the bass octaves.
Smaller pianos (spinets, small uprights) need more stretch than concert grands.
The right amount depends on the piano: measuring its inharmonicity and building the stretch from it gives octaves that sound clean on that instrument.

Questions

What is stretch tuning on a piano?

Tuning the piano's octaves slightly wider than an exact 2:1, so the bass sits a little flat and the treble a little sharp of equal temperament. It is needed because stiff strings have sharp overtones.

What is piano inharmonicity?

The amount by which a string's overtones run sharp of whole-number multiples of its fundamental, because the string is stiff. It is described by a coefficient B in fₙ = n·f₀·√(1 + B·n²).

How many cents is a piano stretched?

Typically some tens of cents at the ends of the keyboard: a few to around 25 cents flat in the low bass and 20 to 40 or more cents sharp at the top, more on smaller pianos.

What is the Railsback curve?

The plot of a well-tuned piano's deviation from equal temperament across the keyboard, first published by O. L. Railsback in 1938: flat in the bass, sharp in the treble.

Do uprights need more stretch than grands?

Usually yes: their shorter strings are stiffer relative to their length, so inharmonicity and stretch are higher.

Pianowise: Piano Tuner

Piano tuner app: every key free, measured stretch, checks and unisons.

Pianowise measures your piano's inharmonicity from the partials of six keys, builds the stretch curve for that piano with the octave style you choose (gentle, standard or wide), and shows every key against it, with each partial's offset.

Coming to Google Play

Sources

H. Fletcher, "Normal vibration frequencies of a stiff piano string", J. Acoust. Soc. Am. 36 (1964) 203–209 doi.org
R. W. Young, "Inharmonicity of plain wire piano strings", J. Acoust. Soc. Am. 24 (1952) 267–273 doi.org
O. L. Railsback, "Scale temperament as applied to piano tuning", J. Acoust. Soc. Am. 9 (1938) 274
N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd ed. (Springer, 1998), §2.18 and chapter 12
H. Hinrichsen, "Entropy-based tuning of musical instruments", Revista Brasileira de Ensino de Física 34 (2012) 2301 arXiv
Fernhart Labs, Pianowise real-audio evaluation (Salamander Grand Piano V3 samples of a Yamaha C5, CC-BY 3.0, and the University of Iowa MIS piano), October 2026

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