The distance is the easy part
Two gears of the same pitch mesh at the sum of their pitch radii, and the pitch radius is the tooth count times the module over two. Sixty teeth and eight leaves at 0.8 module gives radii of 24 mm and 3.2 mm, so the centres are 27.2 mm apart. That much is exact, holds for any tooth form, and is the number to drill to when you have nothing else.
Everything after it is the interesting part, because no plate is drilled exactly and no pair of wheels is cut exactly, and the difference is what you feel when you turn the train by hand.
Where backlash comes from
Two places. Opening the centres beyond theoretical lets the teeth slide out of each other, and cutting the teeth thinner than theoretical leaves a gap at the pitch line to begin with. The two add. Fifty microns of extra centre distance at a 20 degree pressure angle contributes 36 microns of backlash, and twenty microns of thinning on each member contributes forty more, for 76 microns in total — three thou.
Note that the thinning term does not depend on the centres at all, so a pair cut thin has backlash even when the plate is perfect. That is usually deliberate. A pair with no backlash anywhere is a pair that binds somewhere.
The involute caveat
The two times tangent relationship is involute geometry. Clock wheels have traditionally been cut to cycloidal forms, which are a different curve and do not obey it. On a cycloidal pair the number this page gives is a rough guide to the order of magnitude and nothing more, and the real answer comes off a depthing tool: set the pair up, run them together through a full turn, and find the depth where they run freely at the worst place.
That is not a failure of the arithmetic so much as a description of why depthing tools exist. The pressure angle field is a field precisely because the answer depends on a tooth form this page cannot see.
Three thou at the hands
Seventy-six microns of backlash on a wheel of 24 mm pitch radius is 0.18 degrees of play. If that wheel turns once an hour, 0.18 degrees is 1.8 seconds of dial, and at the tip of a four inch minute hand it is about 0.3 mm of movement before the train takes up. That is the whole reason backlash matters on a clock: not efficiency, but the fact that the hand does not point at anything definite.
Compounded down a train it is worse, because every mesh adds its own and the play at the hands is the sum seen through the ratios. It is also why a movement that rattles at the escapement will not hold a beat however well the pendulum is regulated.
Measured at the worst place
The figure on the page is what the geometry gives at a nominal position. Real backlash varies round a turn with pitch error and with how truly each wheel runs on its arbor. A pair that measures nicely where you happened to stop can be tight a quarter turn on. Turn both members through a complete mesh cycle and take the tightest and the loosest, and those two numbers are the ones that decide whether the pair will do.
Questions people ask
How do I calculate the centre distance between two gears?
Add the two pitch radii. Each pitch radius is the tooth count times the module divided by two, so sixty teeth and eight leaves at 0.8 module gives 24 mm and 3.2 mm and centres 27.2 mm apart. In diametral pitch it is the two counts added and divided by twice the pitch. Both members must be the same pitch for any of it to hold.
How much does centre distance affect backlash?
For involute teeth, backlash increases by twice the distance error times the tangent of the pressure angle. At 20 degrees that is about 0.73 times the error doubled, so fifty microns of extra centre distance adds 36 microns of backlash. Opening the centres by 0.035 mm adds about one thou at the pitch line.
Does this work for clock wheels?
The centre distance and the ratio do, and they hold for any tooth form. The backlash figure does not, because it is involute geometry and clock wheels are traditionally cycloidal. Treat it as an order of magnitude on a cycloidal pair and set the real depth on a depthing tool by running the pair through a full turn, which is the job that tool exists for.
What does gear backlash look like at the hands?
Divide the backlash by the pitch radius to get the play in radians. Seventy-six microns on a 24 mm radius is 0.18 degrees, and on a wheel turning once an hour that is 1.8 seconds of dial and about 0.3 mm at the tip of a four inch minute hand. Every mesh in the train adds its own, which is why hands on a worn movement feel vague.
Should a gear pair have any backlash at all?
This page has no view on how much yours should have, only on what your numbers produce. What the arithmetic does show is that the thinning term is independent of the centre distance, so a pair cut thin carries backlash even on perfect centres. A pair with zero clearance everywhere has nowhere to put pitch error, eccentricity or dirt, and it binds somewhere.