Why a general figure is no use
The chain from humidity to pitch runs through the soundboard. Damp air swells the panel across the grain, the crown rises, the bearing on the bridges increases, the strings are pushed harder and the pitch goes up. Dry air reverses all of it. How much pitch you get per point of humidity depends on the thickness and species of the panel, the rib scale, how much crown was there to begin with, how much the case restrains the panel, and how old and how tired the whole assembly is.
Two instruments in the same room genuinely respond differently, and the same instrument responds differently after a rebuild. So there is no general number worth quoting, and this page does not quote one. What it does is take two readings you took yourself and turn them into the slope for that instrument in that room, which is a number that means something.
Getting the two readings
Read the pitch at one named note with a device, and read the humidity in the room at the same moment, and write both down. Then leave the instrument alone and do it again when the room has moved a long way — in most heated houses that means the dry end of winter and the damp end of summer, so it is a year-long measurement rather than an afternoon one.
The two conditions have to be far apart, because the slope is a difference divided by a difference and a small denominator amplifies every error in the numerator. Readings ten points apart with a hygrometer that is three points out give a slope that could be wrong by half. Readings twenty five points apart with the same hygrometer are much more forgiving. And the instrument must not be tuned in between, or the second reading is measuring the tuning rather than the room.
Reading the calendar
With the defaults — 30 percent at 4 cents flat, 55 percent at 3 cents sharp — the slope is 0.28 cents per point. A year running from 28 to 58 percent is 30 points, so the whole swing is 8.4 cents from end to end. That is well over a twelfth of a semitone and easily audible against a fixed reference.
Tune in January at 28 percent and every other month is above you, so the worst case in July is the full 8.4 cents sharp. Tune in April at 42 percent, near the middle of the range, and the worst case is 4.5 cents, because the furthest reading in either direction is only 16 points away. Same instrument, same room, same tuning, half the worst-case drift.
The page does not assume the middle is best; it tries every reading in the list as the tuning point and keeps whichever has the smallest largest excursion. It happens to land near the middle every time, and the reason is straightforward — from any starting point the worst you can do is the distance to whichever end of the range is further away, and standing in the middle makes both ends as close as they can be.
What the slope is hiding
Hysteresis. Wood absorbs moisture faster than it releases it, so a panel arriving at 45 percent from a damp summer is not in the same state as the same panel arriving at 45 percent from a dry winter, and the pitch will differ. A straight line through two points averages over this and cannot describe it.
Lag is the other one. The room changes in hours and the soundboard changes in weeks, so a spot humidity reading taken today is not what the panel has been living in. If you read the room on an unusual day you will get a slope that is too shallow, because the pitch reflects the fortnight and the hygrometer reflects the morning. Where the readings can be taken after the room has been steady for a while, they are worth much more.
And the compass does not move as one. The middle of the board carries the most crown and moves furthest; the extremes move less. Reading at one note gives you one sample of a curve, which is why the note has to be the same one every time and why a single number cannot describe what the whole instrument did.
Questions people ask
How many cents does a piano move per percent of humidity?
There is no figure that holds across instruments, which is why this page asks for two readings instead of supplying one. The slope depends on the soundboard panel, the ribs, the crown, the case and the age of the assembly, and two pianos side by side in the same room will give different answers. Take a pitch reading and a humidity reading at the dry end of the year and again at the damp end, and the ratio between them is the number for that instrument.
Why does the calculator say to tune in the middle of the range?
It does not assume it — it tries every reading in the list as the tuning point and reports whichever gives the smallest worst-case drift. That always turns out to be a reading near the middle, for a simple reason: the worst you can drift from any starting humidity is the distance to whichever end of the year is further away, and starting in the middle makes the two ends equally far. Tuning at the driest point of the year guarantees the worst case is the entire annual range.
Does the pitch go sharp or flat when the air gets damp?
On most instruments damp air raises the pitch, because the soundboard swells, the crown increases, the bearing on the bridges goes up and the strings are stretched a little further. But the sign is something your own two readings establish rather than something to assume — the arithmetic here carries whatever sign your measurements give it, and reports the drift signed in both directions, because a piano four cents sharp and a piano four cents flat are different situations calling for different work.
My hygrometer might not be accurate. Does that matter?
It matters twice over. An offset that is the same at both readings cancels out of the slope, because the slope uses the difference between them — so a hygrometer reading uniformly five points low still gives the right cents per point. What it does wrong is the calendar, since every month is shifted by the same five points and the drift figures are unaffected but the humidity numbers printed beside them are not. A hygrometer whose error changes with humidity, which is common in cheap dial types, corrupts the slope itself and there is no way to recover from it in arithmetic.
Can I use this to decide how often to tune?
It gives you one input to that decision and not the decision. The calendar shows how many months sit inside whatever drift you are prepared to accept, and the size of the worst excursion, and those are real numbers. What it cannot see is how much the instrument goes out with itself rather than simply moving as a whole, which is what is actually heard, or how it is being played, or what the person listening to it wants. Those settle the schedule, and the drift figure informs it.