The piano is tuned wrong on purpose

Measure the pitches of a piano that a good tuner has just finished, against an electronic reference. The middle of the instrument will be close to equal temperament. The top will be sharp — by ten cents at the top of the treble staff and thirty or more at the extreme. The bottom will be flat, by a comparable amount.

That is not a badly tuned piano. It is what a well-tuned piano is, every one of them, and a piano tuned to match the reference exactly sounds wrong to everybody including people who cannot say why.

A real string is not the model string #

The harmonic series belongs to an ideal string: perfectly flexible, with tension as its only restoring force. Such a string vibrates in one, two, three and more equal parts, and the frequencies of those modes are exact whole-number multiples of the fundamental.

A steel wire is not perfectly flexible. It resists bending as well as stretching, and the bending stiffness adds a restoring force that grows with how sharply the wire has to curve — which, for a mode with more nodes, means a great deal more.

The result is the standard stiff-string relation:

$$f_n = n f_0 \sqrt{1 + B n^2}.$$

Every partial is sharp of where a harmonic series would put it, and the sharpness grows as the square of the partial number. $B$ is the inharmonicity coefficient, and everything about a particular piano follows from it.

Which computation produced the number #

$B$ is not a fudge factor. It comes out of the physics of a steel wire under tension:

$$B = \frac{\pi^2 E d^2}{64 \rho L^4 f_0^2},$$

where $E$ is Young’s modulus for steel, $\rho$ its density, $d$ the wire diameter, $L$ the speaking length and $f_0$ the pitch it is tuned to.

The strong dependence is on the length: $L^4$ in the denominator means halving the string quadruples-and-then-quadruples the inharmonicity. That is why the treble of a piano is far more inharmonic than the middle, and why a short piano is more inharmonic than a long one throughout.

A stated string design lets the whole compass be computed. The figures here use a small upright: a speaking length falling as $f^{-0.93}$ above the break, capped at 1.15 metres by the case, with a mild taper in wire gauge. That gives $B \approx 4\times10^{-4}$ at A3, $1.2\times10^{-3}$ at A4 and $9\times10^{-3}$ two octaves above that. Published measurements for small pianos are in that range.

Why the octave gets wider #

Here is the step that turns a property of wire into a tuning practice.

A tuner setting an octave does not measure a ratio. The tuner listens, and adjusts until the beating stops.

What beats, in an octave, is the lower note’s second partial against the upper note’s first. On an ideal string those coincide when the fundamentals are in the ratio 2:1. On a stiff string the lower note’s second partial is not at $2f_0$ — it is at $2 f_0 \sqrt{1 + 4B}$, sharp — and the upper note’s first partial is at $f_0'\sqrt{1+B}$.

Setting them equal:

$$\frac{f_0'}{f_0} = \frac{2\sqrt{1+4B}}{\sqrt{1+B}},$$

which is more than two. Expressed in cents, the octave is wider than 1200 by

$$600 \log_2 \frac{1+4B}{1+B}.$$

For the A4 string above, that is 3.0 cents. For a top-octave string it is tens of cents. And it accumulates: a tuner working outward from the middle by beatless octaves adds this stretch at every step, so the departure from equal temperament grows in both directions.

That accumulation is the curve in the first figure. The bass ends up about 18 cents flat and the top note 53 cents sharp, and both are what beatless octaves on stiff strings come to.

Which beatless interval, though #

The derivation above chose one thing without saying it was a choice: it matched the lower note’s second partial against the upper note’s first. A tuner has other pairs available in the same octave — the lower note’s fourth against the upper note’s second, or its sixth against the upper’s third — and those are different tunings of the same octave, known in the trade as the 2:1, the 4:2 and the 6:3. A tuner also has the twelfth and the double octave, and many use them for the extremes precisely because the octave becomes unreliable there.

Accumulating outward from A4 by each of them, on the same strings:

The curve is not a consequence of stiff strings. It is a consequence of stiff strings and one decision, and the decision moves the answer by a factor of eight. Per semitone at A4 the stretch is 0.099 cents on a 2:1 octave, 0.203 on a twelfth, 0.337 on a double octave and 0.389 on a 4:2 octave — four numbers for one instrument.

That is not a defect in the account so much as the missing half of it. Every one of those tunings is beatless in the interval the tuner listened to, and every one leaves beats in the others; a tuner who sets 2:1 octaves has a beating 4:2, and one who sets 4:2 has a beating 2:1. The stretch a real piano ends up with is a negotiated compromise between them, which is why published Railsback curves run to about thirty cents at each end and sit between the first two rows of this table rather than on either.

So what the physics fixes is the sign and the shape, and what a tuner fixes is the size. The curve turns up at the top and down at the bottom on every row, because that is what B does; how far it turns is a matter of which beat somebody chose to remove, and the range of defensible choices is wider than the effect the essay is describing.

What the tuner actually hears #

The stretch is small in cents and not small in beats, which is why it is impossible to miss at the instrument and easy to miss on a meter.

Take the A3–A4 octave on the piano modelled here. A3’s second partial sits at 440.36 hertz. If A4 were tuned to an exact 2:1 with A3’s own pitch, its first partial would be at 440.09. The two would beat against each other at 0.27 hertz — a slow pulse, about one every four seconds, and completely audible in a sustained octave.

Go up an octave and the same comparison gives 1.56 hertz. Another octave, 8.8. Another, 49 — which is no longer a beat at all but a rough buzz.

So the instruction “tune the octaves pure” is not a subtle refinement. Following it produces a stretch of a cent in the middle of the instrument and tens of cents at the top, and not following it produces beating that grows from a gentle undulation to something nobody would tolerate. A tuner is not choosing to stretch. A tuner is removing beats, and the stretch is the arithmetic consequence.

This also explains the shape of the curve rather than just its sign. The flat middle is where inharmonicity is lowest, because the strings there are as long as the case allows relative to their pitch. The curve turns up sharply in the treble because $B$ grows with $L^{-4}$ and the treble strings are very short, and it turns down in the bass because the wound strings are also short for their pitch and their stiffness is high again.

And then the temperament follows #

The octaves are not the end of it. Once they are stretched, everything inside them has to move too.

Equal temperament divides an octave into twelve equal parts. If the octave being divided is 1203 cents rather than 1200, the twelve parts are 100.25 cents rather than 100, and every interval within it is correspondingly wide. A fifth in the stretched treble is not 700 cents; it is 700 times whatever the local stretch factor is.

That is why a piano’s tuning curve is a curve and not a set of octave corrections. The tuner sets a temperament in the middle — one octave’s worth of fifths and thirds, checked by their beat rates — and then extends outward by octaves, and the extension carries the whole temperament with it, stretched.

It also means the piano is not, strictly speaking, in equal temperament anywhere except within the one octave where the temperament was set. Above and below, it is in equal temperament of a slightly different octave, and by the extremes those octaves differ from 1200 cents by enough to matter.

What it is not #

Three explanations circulate for the stretched piano and two of them are wrong.

It is not about temperament. Nothing above involves fifths, thirds or the comma. The stretch would exist on a piano tuned in just intonation, in meantone or in any other system, because it comes from the strings rather than from the arithmetic of intervals.

It is not a perceptual preference for stretched octaves. There is a real and separate phenomenon — listeners asked to tune an octave by ear to pure tones tend to set it slightly wide, by a few cents — and it is sometimes offered as the explanation. It is much too small to account for a thirty-cent departure at the top of the compass, and it does not explain why the effect is larger on short pianos.

It is a mechanical property of the strings, and the tuner is doing exactly the right thing. A piano tuned to exact 2:1 octaves has audible beating in every octave, because the partials that ought to coincide do not. Removing that beating is the tuner’s job, and stretching is what removing it requires.

Where the model stops #

The model over-predicts the treble, and by a lot. The computed curve reaches 53 cents at the top note, against 20 to 30 in published measurements of comparable instruments — it is out by a factor of two. The reason is stated in the model itself: it chains 2:1 octaves all the way up the compass, and real tuners in the top octave listen to double octaves and to other partial matches, which produce considerably less stretch than a chain of single ones. The over-prediction is a consequence of a simplification that can be named and whose direction is predictable, which is the best kind of wrong answer available.

And under-predicts the bass. The computed bass reaches −18 cents, against −30 or worse on real small pianos. The wound bass strings are modelled as plain steel of the core diameter, which understates their stiffness, and real tuners down there listen to the fourth and sixth partials rather than the second — and those are sharper still, so the octaves come out wider.

This is a piano, not the piano. The string design is stated rather than measured, and it describes a small upright. A concert grand’s longer strings give it perhaps half the inharmonicity and correspondingly less stretch, which is a prediction the model makes and which matches what is observed.

Inharmonicity is not all bad. A perfectly harmonic piano would sound thin and organ-like; a modest amount of inharmonicity is part of what a piano sounds like, and instrument designers do not attempt to eliminate it. The figures here treat it as something to be compensated for, which is the tuner’s view and not the builder’s.

The stiff-string formula is itself an approximation. It assumes small amplitude, uniform wire and simply supported ends, none of which is exactly true. Real terminations at the bridge and agraffe are somewhere between simply supported and clamped, which changes $B$ by a modest factor.

Whose music, and when #

The curve is named after O. L. Railsback, who published measurements of tuned pianos in 1938 and found the same shape on every instrument he examined. The finding was initially treated as evidence that tuners were systematically inaccurate, and the explanation in terms of inharmonicity followed.

The practical consequences reach further than they look.

Electronic tuning devices have to implement the stretch, and the good ones do it by measuring each string’s actual inharmonicity and computing the tuning curve for that individual piano rather than applying a stored average. A device that tuned to exact equal temperament would produce an instrument every tuner would reject.

Digital pianos that sample real instruments inherit the stretch in their samples, and those that synthesise have to add it deliberately. An early digital piano with exactly harmonic partials and exact octaves is instantly identifiable, and the reason is this.

Ensemble intonation, which already has a comma problem, acquires a second one. A piano’s top octave is sharp of equal temperament and a flute’s is not, so a flute playing in octaves with a piano’s top register is playing against a stretched instrument. Orchestral players adjust to the piano, which is the standard practice and is rarely discussed as what it is.

Harpsichords and clavichords have much thinner strings at lower tension and are correspondingly far less inharmonic, so they are tuned with much less stretch. That is one of several reasons a historical keyboard sounds different from a modern one in a way that has nothing to do with which temperament it is in.

And the deepest consequence is a theoretical one. Every argument on this site about consonance and ratios — why a fifth is smooth, why the comma is where it is, what a pure third is — assumes harmonic partials. On a real piano they are not harmonic, so those arguments hold approximately in the middle of the compass and progressively less well towards either end. The theory describes an instrument nobody owns.

What it does to the rest of the theory #

The last consequence is the one that reaches furthest, and it is uncomfortable.

Every argument about consonance on this site runs on partials that coincide. A fifth is smooth because the third partial of the lower note and the second of the upper are the same frequency. A pure major third is smooth because the fifth partial of one meets the fourth of the other. The whole edifice of small-whole-number ratios is a claim about where partials fall.

On a real piano string they do not fall there. The third partial of a middle-register note is a couple of cents sharp of three times the fundamental; the fifth is five or six cents sharp. So the ratios that would make partials coincide are not 3:2 and 5:4 — they are 3:2 and 5:4 adjusted by the inharmonicity, and the adjustment is different for every note on the instrument.

The practical size of this is modest in the middle of the compass and considerable at the ends. What it means is that a piano is never quite the instrument the theory describes, and that the agreement between the theory and the sound is closest where the strings are longest.

A more interesting version of the same point is what happens when the partials are not merely stretched but genuinely unrelated to any fundamental — bells, gongs, tuned metallophones. There the ratios that minimise roughness are not the small integers at all, and the scales built around such instruments are not approximations of a chromatic scale. Whether a gamelan’s tuning sits where its own dissonance minima are is a question that can be asked precisely, and the answer is at least partly yes.

The ladder from here #

Later rungs on this anchor: the partial-matching strategies tuners actually use, and what a 4:2 or 6:3 octave is. Individual inharmonicity measurement, and how a tuning curve is computed for one instrument. Bells and gongs, whose partials are not merely stretched but bear no simple relation to a fundamental at all, and what tuning means for them. Dissonance curves computed for inharmonic spectra, which have their minima somewhere other than the small-integer ratios — and the gamelan tunings that appear to sit where those minima are. And the question this raises about the whole subject: what a theory of consonance built on real spectra rather than ideal ones would say.

A tuner spends a career making an instrument measurably wrong, on purpose, because the wire will not do what the arithmetic assumes.