What a drop cannot be smaller than

Two thirds of an escape wheel’s motion is drop: the interval between one tooth leaving a pallet and the next arriving at the other, during which nothing is touching anything and the train does no work at all. On the escapement this site draws, 3.95° out of every 6°.

That is a large fraction of a mechanism to be spending on nothing, and the obvious response is to spend less of it. The reason nobody does is a tolerance stack.

Why a small drop stops the clock #

The tooth has to clear. It leaves the end of one pallet’s impulse face and the wheel turns freely until the next tooth reaches the other pallet’s locking face; if the escapement’s dimensions are exactly nominal, the first tooth is well out of the way by then.

Give every dimension a tolerance and it may not be. A wheel cut with its teeth slightly unevenly spaced, an arbor a few hundredths out of position, a pallet face a fraction of a degree off its drawing — each of those moves the instant at which the next tooth lands, and if the accumulated movement exceeds the drop, the next tooth arrives before the last one has gone. The mechanism jams, and a clock that jams once a day is not a clock.

So the drop is not a waste to be minimised. It is a clearance, and clearances are sized by tolerance stacks.

Measuring the stack by re-solving #

The sensitivities are taken the way this site takes every sensitivity: by rebuilding the mechanism with one dimension perturbed and re-solving, rather than by differentiating a formula that might not be the formula the mechanism obeys.

Five dimensions, each with a plausible tolerance, on the thirty-tooth escapement:

The first row is the one to look at. A tooth cut a minute out of place takes a minute off the drop — the sensitivity is 1.00000, exactly, which is the sort of number that should be checked rather than admired, and it is: the assertion behind it requires that coefficient to be 1 within $10^{-9}$.

It is exactly one because the landing tooth’s angular position enters the drop directly and nothing modifies it on the way. Every other dimension reaches the drop through the geometry and is attenuated: the lift angle at about a half, the arbor distance at a fifteenth, the draw at a fiftieth.

Worst case against the square root #

The two totals are the two this site always reports together, and their ratio is 1.44.

The worst case adds the absolute contributions and asks what happens if every dimension is at the wrong end of its tolerance simultaneously and in the direction that hurts. It is 0.392°, and a design that carries that much drop cannot be assembled wrong.

The root sum square treats the errors as independent and asks what a typical build looks like. It is 0.273°, and a design that carries that much will be right for the great majority of specimens and will jam for the tail.

The gap between them is not large here — 1.44, against ratios above 2 for a four-bar with more contributing lengths — and the reason is instructive. The stack is dominated by one term. When one contribution is two and a half times the next largest, the worst case and the statistical estimate converge, because both are mostly that one term. A stack with five equal contributors has a ratio of $\sqrt{5} = 2.24$; a stack with one contributor has a ratio of 1.

So an escapement’s drop is a single-source clearance, and the source is the wheel-cutting. That is why the accuracy of the dividing engine mattered so much to clockmakers and why a wheel is a bought-in part rather than a made-in-the-workshop one.

The trap in a finer wheel #

Now the finding, and it is the reason this essay exists.

More teeth means a smaller half pitch, and the impulse does not shrink with it, so the drop shrinks faster than the pitch does. Meanwhile the tolerance stack does not shrink at all — a dividing engine that is out by a quarter of a degree is out by a quarter of a degree whether the wheel has fifteen teeth or fifty.

The middle column barely moves — 0.382, 0.392, 0.388, 0.387, 0.386 — and the left column falls by a factor of twenty. The margin collapses with it, and by seventy-two teeth it is 1.4: a wheel at the wrong end of its tolerance has almost no drop left, and a wheel slightly past it has none.

Beyond about ninety teeth the arithmetic returns a negative nominal drop, which is not a small drop but a mechanism in which the arriving tooth has already reached the pallet before the departing one has left. There is no tolerance in that; it is the design itself that does not work.

Where the sensitivities come from #

The five coefficients are worth reading rather than just tabulating, because four of them can be got at by hand and the fifth cannot.

Tooth spacing, 1.000. The drop is measured from where one tooth leaves to where the next lands, and the next tooth’s angular position is literally the quantity being perturbed. Nothing modifies it. This is the only sensitivity in the table with an obvious closed form, and having it come out at exactly one is the check that the perturbation is being applied where the model says it is.

Lift angle, −0.534. Adding lift lengthens the impulse, and the impulse and the drop share a fixed half pitch, so the drop must shorten. The coefficient is not −1 because the lift is measured at the pallet and the impulse arc at the wheel, and the two are related by the geometry of the impulse face: a degree at the pallet is about half a degree at the wheel on this escapement.

Arbor distance, −0.069 per wheel radius. Moving the arbor out lengthens the pallet arms and steepens the impulse, which lengthens the impulse arc, which shortens the drop. The sign is the same as the lift’s for the same reason.

Wheel radius, +0.111. A larger tip circle at a fixed arbor position is the arbor moving in relative to the wheel, so the sign is opposite to the arbor’s and the magnitude is comparable.

Draw, +0.022. Nearly nothing. Tilting the locking face changes where the arriving tooth settles, which shifts the landing by a fraction of the tilt, and the fraction is small. A designer choosing a draw angle is not choosing a drop.

None of the four derived coefficients is quoted from a formula here. Each is a difference of two re-solves, which is the same route the practice field takes to a linkage’s sensitivities, and it is the route that survives the model being changed underneath it.

What halving the dividing engine’s error buys #

The dominant term is the spacing, so it is the term worth improving, and the improvement is worth quantifying rather than gesturing at.

Each halving of the spacing tolerance moves the whole stack by less than a factor of two, because the other four terms do not move and eventually dominate: from 0.25° to 0.125° the stack falls by only a third, and halving again would gain almost nothing. The floor is 0.142° — the four remaining contributions — and no dividing engine gets below it.

That floor is what decides how fine an escape wheel can usefully be. With a perfect dividing engine and everything else as tabulated, the margin at forty-eight teeth would be 12.3 and the mechanism would run out at about a hundred and twenty teeth instead of ninety. A better engine moves the limit by a third, and no further, because the limit stops being about the engine.

The margin is a ratio, and only one of its terms moves #

The two numbers in this essay are a nominal drop and a stack, and neither of them is the quantity that decides anything. What decides it is their ratio — how many times the accumulated error fits inside the clearance that has to absorb it — and reading the two tables together says something neither says alone.

The nominal drop falls by a factor of twenty as the tooth count triples. The stack, over the same range, moves from 0.382 to 0.386, which is to say it does not move at all. So the ratio collapses at very nearly the rate the drop does, and it collapses because the numerator is geometry and the denominator is workshop practice, and the tooth count is in only one of them.

That is the sharp form of the finding, and it changes where the ceiling is. The arithmetic returns a negative nominal drop somewhere beyond ninety teeth, and it is tempting to read ninety as the limit. It is not: ninety is where the margin reaches zero, and a mechanism whose clearance exactly equals its expected error fails about half the time it is built. The usable limit is wherever the ratio falls to whatever a maker will accept — three, or five, or ten — and every one of those thresholds sits at a tooth count well below ninety, because the ratio is falling steeply through that whole region.

So the answer to how fine may an escape wheel be is not a tooth count at all. It is a tooth count per workshop, obtained by dividing a nominal drop that anybody can compute by a stack that depends entirely on whose dividing engine cut the wheel. Two makers with the same design and different machines have different ceilings, and neither ceiling appears anywhere in the design.

That also explains the historical shape of the subject better than a limit would. Escape wheel counts did not sit still and then jump; they crept upward alongside dividing engines, because every improvement in spacing tolerance moved one term of a ratio and thereby moved a ceiling that was never written down. A maker with a better engine could cut a finer wheel and get a clock that ran, and a maker copying the design without the engine got one that stopped — which is the same design, the same drawing, and two different mechanisms.

And it gives the halving table its proper reading. Each halving of the spacing tolerance moves the stack by less than a factor of two because the other four terms do not halve with it, so the ratio improves by less than the effort put in and eventually stops improving at all. The floor that produces is a floor on the stack and therefore a ceiling on the tooth count, and it is reached while the other four terms are still what they are. Improving the dividing engine past that point buys nothing, and the next gain has to come from the arbor, the pallets or the wheel radius — which is a different workshop problem entirely.

The three things a designer can do #

Only three, and the table above rules out the first.

Fewer teeth. More drop absolutely, more margin, and more of the wheel’s travel wasted. This is what a coarse turret-clock escape wheel is, and it is why they run for a century on indifferent maintenance.

A shorter impulse. Less lift at the pallet leaves more of the half pitch for the drop, and costs the pendulum: a shorter impulse over a shorter arc, which is a smaller share of the swing spent being driven. The relation is exact and one-for-one — every degree taken off the impulse arrives in the drop.

A better dividing engine. The dominant term is the tooth spacing, so halving the spacing tolerance takes the worst-case stack from 0.392° to 0.267° and the margin at forty-eight teeth from 4.5 to 6.6. Historically this is the one that actually happened: the accuracy of wheel-cutting engines is most of the reason clocks got better between 1700 and 1850, and it shows up here as the only lever that improves the mechanism without costing it anything.

What loose tolerances do #

Worth one line, because it inverts the argument.

Take the tolerances to what a rough workshop would produce — a degree of spacing error, a hundredth of a radius on the arbor, most of a degree on the faces — and the worst-case stack goes to 1.36° against a nominal drop of 3.95°. The margin is 2.9, which is still buildable, and it is buildable only because the escapement is a coarse one. The same tolerances on the forty-eight-tooth wheel give a margin of 1.3, and on the sixty-tooth wheel the mechanism does not assemble.

Which is the whole relationship between this field and the practice field in one paragraph: the tolerance does not decide whether the mechanism works, it decides which design works. A coarse escapement made badly runs. A fine escapement made badly does not run at all, and the difference between the two is entirely in a clearance that exists to be wasted.

The same clearance in three other mechanisms #

Drop is not a horological peculiarity. It is what every mechanism in this field has instead of an overlap, and naming the equivalents makes the pattern visible.

A Geneva drive’s equivalent is the gap in its locking arc: the disc has to be cut away wide enough for the wheel to turn through its index, and too wide leaves the wheel free for a moment at each end. Too narrow and the wheel is still locked when the pin starts to drive it, which is a mechanism that jams — the same failure as an escapement whose drop has gone negative, arrived at from the other side.

A ratchet’s equivalent is the clearance between the pawl’s tip and the tooth back it has just ridden over. Too little and the pawl catches on the way past; too much and the pawl falls further into each root, which is lost motion.

A mutilated gear’s equivalent is the gap between the last engaged tooth and the locking arc. Every one of the four is a clearance that does nothing useful, is sized by a tolerance stack, and cannot be taken to zero.

The general statement is that a mechanism which switches between contact states needs slack at every switch, because the switch happens at a nominal instant and the parts arrive at a real one. A mechanism whose joints are all pins has no switches and needs no such slack — which is why the whole of this argument is absent from the first eleven fields of this site, and why every mechanism in this one has a number like the drop somewhere in it.

What is not in the stack #

Three things, named because leaving them out is a choice.

Wear. The stack above is an as-built stack, not an as-worn one. Every hour of running takes material off the pallet faces and the tooth tips, and taking material off changes the drop — in the direction that increases it, since a shorter tooth lands later. So an escapement’s drop grows with age, which is one of the few things in a clock that improves with use until it suddenly does not.

Temperature. A brass wheel in a steel frame changes the arbor distance with the seasons, and the arbor’s sensitivity is −0.069, so a distance change of a thousandth of a radius moves the drop by 0.004°. That is a hundredth of the stack, and it is the reason this term is not chased.

Anything about how hard the tooth arrives. The mechanism jams if the drop is negative and does not jam if it is positive, and how violently it lands in between is a question about torque and inertia. What is computed here is whether the tooth clears, which is a length, and nothing about what happens to it while it does.