Should You Remeasure Inharmonicity After a Pitch Raise?

You sampled inharmonicity on a piano sitting fifty cents flat, pulled it up to pitch, and the instrument in front of you now is under about six percent more tension than the one your device measured. The question is whether that stored number still earns its place.

Your stored B reads 5.95 percent high, not 5.6 percent low

The inharmonicity coefficient goes as B = π³Qd⁴ / (64L²T), with Q the Young's modulus of the wire, d its diameter, L the speaking length and T the tension. That's exact for a stiff string with pinned ends, and the useful part is where T sits: on the bottom. Raise the pitch and you raise tension, so B goes down.

Frequency goes as the square root of tension, so tension goes as frequency squared. A fifty-cent raise is a frequency ratio of 1.0293, a tension ratio of 1.0595, and B falls 5.61 percent.

That's the wrong number to carry around, though. You aren't asking how far B moved. You're asking how wrong the figure sitting in your device is, and those are reciprocals:

• 20 cents flat — stored B reads 2.34% high
• 50 cents flat — stored B reads 5.95% high
• 100 cents flat — stored B reads 12.25% high
• 200 cents flat — stored B reads 25.99% high

There's a tidy check hiding in that column. The percentage your stored B reads high is numerically the same as the percentage tension rose, both being 2^(2c/1200) − 1. That falls straight out of B going as 1/T, and it means you can work the staleness in your head from the pitch deficit alone.

Where 865.6 comes from, and which n you multiply

Partial frequencies go as f(n) = n·f₁·√(1 + Bn²), which puts the nth partial sharp of a true harmonic by roughly 865.6·B·n² cents for small B. The constant is 1200/(2·ln 2). Robert Young published the cgs version of this in the Journal of the Acoustical Society of America in 1952, and the Piano Technicians Guild hosts the paper, so it sits in your own library and not behind a paywall.

One trap before you compare any figure here against a source you already trust. Two conventions are in circulation. Young measures partial deviation against the ideal stiffness-free string, giving 865.6·B·n². Most technician material measures it against the note's own sounding first partial, giving 865.6·B·(n²−1). At the second partial those differ by exactly four thirds, 1.38 cents against 1.04. Everything below is referenced to the note's own first partial, which is what your device displays.

The error equals 5.95 percent of that note's own stretch

Here's the step that makes the whole question tractable. Every partial deviation is linear in B when B is small. Every beatless-octave condition is a difference of two partial deviations, so it's linear in B too. The tuning curve is a running sum of those octave conditions, so it's linear in B as well.

Scale every B on the piano by one common factor and the curve scales by that same factor. The shape is preserved and only the amplitude moves. So the error at any note is that note's own stretch multiplied by your fractional B error, which for a fifty-cent raise is just under 5.95 percent. Modelling an 88-note chain gives between 5.8 and 5.9 percent across the compass, the shortfall being second-order saturation in the logarithm.

Two cents at A0 and C8, a twentieth of a cent at A4

Anchor that to a published stretch envelope and you get real figures. Railsback-type curves run to roughly ±30 to ±35 cents at the extremes, so a fifty-cent raise costs you about 1.8 to 2.1 cents at the bottom and top of the keyboard, and about 3.7 to 4.3 cents after a hundred-cent raise.

Through the temperament, A3 to A5, the same error is 0.05 to 0.15 cents. Nobody is hearing that, and if your customer judges the tuning by the middle of the piano then the stale number was fine.

Two cents is small. It isn't invisible, and calling it unmeasurable would be wrong: it's roughly twenty times the 0.1-cent step your device displays. Small enough to ignore in the temperament, large enough to see on the screen at the ends.

B is a ratio, so a drifting piano doesn't spoil it

The obvious objection to measuring straight after a raise is that the piano is unstable and still settling. It's a weaker objection than it looks.

Your device fits B from the ratios between partials of a single note, f(n) over f₁. A uniform frequency drift multiplies every partial by the same factor and cancels out of the ratio entirely. What's left is the genuine change in B while you sample, and that's second order: five cents of drift during a sample moves B by 0.58 percent, against the 5.6 percent you're there to correct. Roughly a tenth of the problem.

This is why the Verituner can sample continuously through a coarse pass. Its manual is explicit that measuring happens automatically in the background while you tune, and that target recalculation is continuous in coarse tuning with targets left unlocked.

One thing not to overstate. Detensioning a piano isn't a uniform frequency scaling, because it changes B itself and so the partials don't all move together. On a plain wire at B = 4.5×10⁻⁴, dropping the first partial 50 cents drops the eighth partial by 48.6. The estimator is invariant to uniform scaling. Flatness isn't uniform scaling.

Chameleon won't sample a piano more than 50 cents flat

On a serious raise the choice was never fresh measurement against stale measurement. Reyburn's manuals state that Chameleon needs the piano within 50 cents to sample it, the stated reason being note identification: at a big enough deficit a very flat A4 starts to look like G♯4.

So on the pianos where a stale B costs the most, your device may have declined to measure at all and handed you a factory average instead. Worth knowing which of those two happened before you decide the reading can stand.

TuneLab says measure after. Illenberger says 10 to 20 cents.

Three vendors, three architectures, one direction of travel.

• TuneLab documents the mechanism and the instruction together: for larger pitch raises, pulling the string up changes the inharmonicity, so it isn't worth measuring first, and on the second pass you start a new tuning file with fresh readings. It also scopes the other side plainly, saying that for small pitch raises the normal inharmonicity measurements will be sufficient.
• Reyburn puts it as workflow: after pitch raising, go back to Chameleon, re-sample and calculate another tuning, for optimum accuracy. Note their well-known 8-cent figure is the trigger for doing a pitch raise at all, and isn't a remeasurement threshold.
• Frank Illenberger, who wrote pianoscope, puts the remeasure threshold at 10 to 20 cents.

Twenty cents is the round number that sits comfortably above where anyone argues it matters, and it leaves you a 2.3 percent stale-B error, which is below anything worth chasing. That's where the recommendation at the top comes from.

Overpull is a different animal

Worth a firewall, because the two get run together and an experienced tuner will notice immediately.

Overpull compensates for the pitch of notes you've already tuned drifting as total tension redistributes through the plate, the bridge and the non-speaking lengths. It has nothing to do with B. On a fifty-cent-flat piano a Sanderson Accu-Tuner offers overpull in the region of 25 to 34 percent, so twelve to seventeen cents of deliberate overshoot. That's about ten times the stale-B error, by a completely separate mechanism.

What we could not locate

We could not find any published measurement of B taken on the same piano string at two different tensions. The relation is derived, it cross-checks against Young's own published constant to about a tenth of a percent, and as far as we could establish nobody has tested it on an instrument. Every cents figure above is modelled, and it depends on the stretch of the particular piano in front of you.

We also could not find any shipping tuning device that applies a tension correction to a stored file, which it could: scaling stored B by the square of the frequency ratio would get most of the way there. Pitch deficits aren't uniform across a real piano, so a single scalar leaves residual error, and measuring again is faster than reasoning about it.

If you want the ground-level explanation of what inharmonicity and stretch are before any of this, Savvy Pianist has a piece on why pianos are never really in tune.