Two layouts that both look right on paper
There are two ways to place strings across a span and they give different answers. Equal centre spacing divides the distance between the outer string centres by the number of gaps and marks each one at the same interval. Equal gap spacing places them so that the clear space between the strings is the same everywhere, which means the centres are not evenly spaced, because the strings are not the same thickness.
Divide a 1.375 inch span across six strings by five and you get 0.275 inch between centres. But a .046 next to a .036 leaves 0.234 inch of clear air, while a .013 next to a .010 leaves 0.2635, so the narrowest and widest gaps in that evenly-centred layout differ by about thirty thousandths of an inch. A finger landing between two strings is feeling the clear space, not the centres.
Working out the equal gap
The arithmetic is simpler than it looks. Take the span between the outer string centres, subtract everything the strings themselves occupy inside it, and divide what is left by the number of gaps. What they occupy is half of each outer string, because the span is measured to their centres, plus the whole of every string in between.
With a common set, that is half of .046 plus half of .010 plus .036 plus .026 plus .017 plus .013, which comes to 0.120 inch. Take that off 1.375 and you have 1.255 inch of clear space to share across five gaps, so each gap is 0.251 inch.
Then place them. The first string centre is at zero, and each following centre sits a gap plus half of each of the two neighbouring strings further along. Walk that all the way across the set and the last centre lands exactly on the span, which is the check that the arithmetic closed. If it does not land on the span, something in the gauges is wrong.
The differences against even centres are small but they are not nothing: on that set the second string moves 17 thousandths and the third moves 24, which is the largest shift in the set. That is well within the range a nut file cares about, and it is the difference between a nut that feels even and one that feels crowded at the bass end.
The margin is not symmetrical even when the layout is
Set the outer strings the same distance from each edge of the nut and the layout is symmetrical about the centreline, but the strings are not. On a 1.6875 inch nut with a 1.375 inch span, the outer centres sit 0.15625 inch from each edge. The low E is 0.046 across, so its outside edge is 0.1333 inch from the edge of the board. The high E is 0.010 across, so its outside edge is 0.1513 inch away.
That is 18 thousandths more clear board on the treble side than the bass side, from a layout that is symmetrical by construction. It is one of the reasons a low E can feel closer to sliding off the edge than a high E does. Whether to correct for it by moving the bass string in slightly is a preference, and a good many instruments do exactly that.
The bridge end and the taper between
The bridge span is wider than the nut span, and the same equal-gap arithmetic applies with the wider figure. The gaps come out larger and the differences against even centres come out the same, because what the strings occupy has not changed.
Between the two ends the paths run straight, so the span at any point along the neck is a straight interpolation. On a 25.5 inch scale with a 1.375 nut and a 2.0625 bridge, each outer path moves outward by about thirteen and a half thousandths of an inch for every inch of neck, and the span at the twelfth fret is 1.71875 inches. That taper is what the fingerboard edges have to accommodate, and it is why the margin outside the outer strings is a different number at every fret.
What this does not settle
Slot positions are one of three things a nut has to get right and they are the easiest. Slot width has to suit the gauge that goes in it: too narrow and the string binds instead of sliding, so it goes out of tune every time it is bent or a tuner is turned, and too wide and it rattles sideways in the slot. Slot depth is a separate geometry and has its own page. Slot angle matters too, because the string should leave the nut at a defined point on the fingerboard side rather than sitting on a flat.
Nor does this page have an opinion about the spans. What span an instrument should have depends on the design, on what it is played with and on the hand playing it, and the wide range of spans in use on instruments that all work is the evidence that there is no single answer.
Where this connects
How deep those slots go is the nut slot depth calculator, and the width they span across is on the fretboard radius calculator along with the arc across it. String heights over the frets are the string action and saddle height calculator, and the gauges that set the occupied width are also the input to the string tension calculator. Fret positions along the same neck come from the fret position calculator.
Questions people ask
What is the difference between equal gap and equal centre string spacing?
Equal centre spacing puts the string centres at even intervals across the span, which is what dividers and a rule naturally give you. Equal gap spacing puts the same amount of clear air between every neighbouring pair, which means the centres are not evenly spaced because the strings differ in diameter. On a common six-string set the two layouts differ by up to about twenty-four thousandths of an inch on individual strings. A hand feels the clear space rather than the centres, which is the argument for the equal-gap layout, but both are in use and neither is a correction of the other.
How is the equal gap actually calculated?
Subtract what the strings themselves occupy from the outer-centre span, then divide by the number of gaps. What they occupy is half of each outer string, because the span is measured centre to centre, plus the full diameter of every string in between. For a set running .046 down to .010 that comes to 0.120 inch, so a 1.375 inch span leaves 1.255 to share across five gaps, giving 0.251 each. Placing the centres from there is a walk across the set, adding a gap plus half of each adjoining string, and the last centre should land exactly on the span.
Why does the low string feel closer to the edge of the neck?
Because it is, even when the layout is symmetrical. The outer string centres sit the same distance from each edge, but the strings do not have the same diameter, so the outside edge of a .046 is 18 thousandths closer to the edge of the board than the outside edge of a .010 is. Some layouts deliberately pull the bass string in a little to even that up, which trades a symmetrical centre layout for a symmetrical clear-space layout. Which of those an instrument wants is a preference.
Should nut and bridge spacing be worked out the same way?
The arithmetic is the same and only the span changes, since what the strings occupy is fixed by their gauges. The gaps come out wider at the bridge because the span is wider. The two ends are not independent, though: the paths between them run straight, so the span at any fret is a straight interpolation between them, and the taper decides how much board is left outside the outer strings at every point along the neck. That is worth checking at the last fret as well as at the nut.
Does string spacing change if I change gauges?
The equal-gap positions do, because the strings occupy different amounts of the span. Going from a light set to a heavy one takes more of the span for the strings and leaves less for the gaps, so every gap narrows and the interior centres shift. The movement is small, in the low thousandths for a single gauge step, and it is well below what most people would notice. It becomes real if you change several gauges at once, and it is one of the arguments for cutting a nut for the strings an instrument is actually going to wear.