The measurement is a chord, and the gap is a sagitta
Relief is measured by turning the string itself into a straightedge. Hold it down at a low fret and at a high fret and it becomes a straight line between those two points. The fretboard curves away from that line, and the gap you read with a feeler gauge somewhere in the middle is how far it has curved. In geometry that gap is a sagitta and the string is a chord.
The gap depends on where along the chord you read it. At either end it is zero, because that is where the string touches. It is largest at the middle. Anywhere else you read a fraction of the peak, and the fraction follows the shape of the curve. Treating the bow as a parabola, the fraction is four times the position along the chord times one minus that position: at the exact middle it is 1, at a quarter of the way along it is 0.75, at a tenth it is 0.36.
The trap is that the fret in the middle of the fret numbers is not the fret in the middle of the chord. Frets crowd together as they go up the neck. Held at the first and fourteenth frets on a 25.5 inch scale, the chord runs from 1.431 inches to 14.141 inches from the nut, so its midpoint is at 7.786 inches, which lands near fret 6.3. Fret 7 is at 8.481 inches, past the peak. Fret 8 is further past it still. So the seventh-fret reading and the eighth-fret reading on one neck differ, and both understate the peak.
How much this actually changes the answer
Less than you might fear near the middle, and that is worth knowing before you start correcting readings. Held at 1 and 14 with the reading at fret 7 on a 25.5 inch scale, the shape factor is 0.988, so an 8 thou reading is a peak of 8.1 thou. That is inside the resolution of most feeler gauge sets and nobody needs to care.
It stops being negligible when the reading point drifts. Hold at 1 and 17 and read at 5, and the factor is 0.900, so an 8 thou reading is a peak of 8.9. Hold at 1 and 24 and read at 4, and the factor is 0.678: you are reading about two thirds of the bow, and 8 thou at the gauge is nearly 12 at the peak. The correction matters most for people comparing a measurement to somebody else description of a method, because the method description is usually missing exactly the detail that changes the number.
Why the reading changes when nothing about the neck changed
Three things move a relief reading without the neck itself moving. Changing the fret you hold at the body end changes the chord and therefore the whole curve you are sampling. Changing the string you measure on changes the answer because the strings pull unevenly across the neck and the low side usually carries more bow. And measuring with the instrument on its back rather than in playing position changes it, because the neck is a beam and gravity is one of the loads on it.
None of that makes any single reading wrong. It makes an unlabelled reading useless. Write down which string, which two frets, which reading fret and what position the instrument was in, and the number becomes something you can repeat next month.
Fall-away past the joint
The frets over the body are a separate problem from relief. The fretboard extension sits on the body rather than on the neck, so it does not follow the neck when the rod moves, and when the neck takes up relief that extension tends to be left standing proud of the curve the rest of the board is making. The frets on it then sit in the way of a string that is trying to clear the frets below them.
Fall-away is the answer: the frets past the joint are levelled progressively lower, so the board drops away from the string as it approaches the body. The table above lays that out as a straight taper from the joint fret to the last fret, which is the easiest version to lay out and check. In practice fall-away goes in with a levelling beam, is checked with a straightedge and a feeler, and often comes out slightly curved rather than perfectly straight because the beam removes more where it bridges high spots.
Whether an instrument wants any fall-away, and how much, belongs to whoever is doing the work. Some builds get none by design. What the table gives you is the geometry once you have decided.
Where this connects
Relief is one of three things holding the string off the frets, and the other two have their own pages. The saddle sets the height at the bridge end, which is the string action and saddle height calculator, and the nut sets it at the other end, which is the nut slot depth calculator. The fret positions the measurements above are located from come from the fret position calculator. Relief moves with the load on the neck, and that load is the whole set of strings pulling at once, which the string tension calculator totals up.
Questions people ask
Which fret should I read the gap at?
Whichever one you can repeat, as long as you write it down along with the two frets you held. There is no single correct fret, because the correct answer is the peak of the bow and no fret sits exactly on it. What matters is consistency: if you read at the seventh fret holding at one and fourteen, keep doing exactly that, and your readings become comparable to each other over time. The page exists so that you can also translate a reading taken one way into a reading taken another way, which is what you need when comparing notes with somebody who measures differently.
Why is the peak not under the middle fret number?
Because fret spacing shrinks geometrically. Each fret sits closer to the last, so the physical middle of a stretch of neck is always at a lower fret number than the arithmetic middle of the fret numbers. Holding at the first and fourteenth frets, the arithmetic middle is 7.5 but the physical middle lands near fret 6.3. The higher the top fret you hold at, the further apart those two answers get, which is the main reason two people using slightly different methods arrive at different relief figures on the same instrument.
Should I measure relief on the low string or the high string?
Pick one and stay with it. The two sides of a neck rarely bow the same amount, partly because the strings do not pull evenly and partly because the neck itself is not symmetrical once a truss rod channel is cut into it. Measuring the low side usually shows more relief. Neither reading is the neck; both are samples of it, and the useful practice is to take the same sample every time so that a change in the number means a change in the neck.
What is fall-away and does every instrument need it?
Fall-away is a deliberate downward taper of the frets past the point where the neck meets the body, so the board drops away from the string in the last stretch rather than staying in its path. It is a fix for a specific geometry, where the fretboard extension is anchored to the body and cannot follow the neck when relief goes in. Plenty of instruments are set up without it and play cleanly. Whether a given instrument wants any is a judgement for whoever is working on it, and the amount is a judgement too. The table here only lays out the taper once both have been decided.
How much does a truss rod turn move the relief?
There is no general answer and this page deliberately does not offer one. The relationship depends on the rod design, the neck stiffness, the wood, the string tension currently on it and how much of the rod range is already used. The same eighth of a turn can be a thou on one neck and five on another. The workable method is to make a small change, let the neck settle, measure again, and learn the ratio for that particular instrument. The hazard is at the end of the range: a rod that has stopped moving is finished, and forcing it breaks the rod or the anchor, which is a repair that costs more than the setup did.