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Clock Gear Train and Beats Per Hour Calculator

A clock train is one long multiplication and the whole of a movement timekeeping follows from it. Each wheel drives the pinion of the next arbor, so every mesh multiplies the speed by wheel teeth over pinion leaves, and by the time you reach the escape wheel you know how many times a minute it turns. Multiply that by the teeth on it and by the beats each tooth gives, and you have the beats per hour the pendulum has to match. Count the teeth off the plates, put them in here, and the arithmetic that used to be done on the back of a workshop envelope comes out in one go.

The centre wheel carrying the minute hand turns once an hour on most movements, which is where the 1 comes from. If you are starting further back at the great wheel, put that wheel turns per hour in instead — time one turn if you do not know it.
Counted off the wheel itself. Mark a tooth with a scrap of tape and count round once.
Leave this pair at 0 and 0 if the train has only one mesh before the escape wheel.
For a longer train. Both boxes at 0 skips this mesh.
The wheel the pallets work on, on the arbor driven by the last pinion in the list.
How many ticks you hear while one tooth passes. Where the two pallets are each released once per tooth this is 2. Count the ticks against a marked tooth on your own movement rather than taking the number on trust.
Only used for the pendulum length line. Gravity varies with latitude and altitude by a few tenths of a percent, which is worth tens of seconds a day on a clock. The box holds the conventional figure used when nothing better is to hand.
Clock Gear Train Calculator — Train Count and BeatsBuildFigure

One multiplication, start to finish

A going train is a chain of meshes and each mesh does exactly one thing: it multiplies speed by the driving wheel teeth divided by the driven pinion leaves. Nothing else in the train changes the number. Start at a wheel whose speed you know — the centre wheel makes one turn an hour because it carries the minute hand — and multiply your way along to the escape wheel.

With the counts in the boxes: 96 teeth into an 8 leaf pinion is 12, so the third wheel turns 12 times an hour. 40 teeth into another 8 leaf pinion is 5, so the escape wheel arbor turns 60 times an hour, once a minute. Thirty teeth on the escape wheel at two beats a tooth is 60 beats a minute, 3,600 an hour, one beat a second.

Beat, swing, tick, period

These four words cause more arithmetic errors in clockwork than anything else, and the error is always a factor of two. A beat is one tick: the pendulum going from one extreme to the other, one pallet letting a tooth go. A full period is the pendulum going over and coming back, which is two beats. The pendulum formula uses the period. Beats per hour uses the beat.

So a seconds pendulum — the long one in a longcase clock — beats once a second and has a period of two seconds. Its length works out near 39.1 inches, or 994 mm. If you find yourself with a pendulum of about 9.8 inches when you wanted a seconds pendulum, you have used the beat where the period belonged, because a quarter of the length gives half the period.

What the escape wheel table is for

Changing the escape wheel count is often the cheapest way to move a train onto a beat you can actually make a pendulum for, because it is one wheel rather than a repinioned arbor. The table walks three teeth either side. On the default train, dropping from 30 teeth to 28 takes the beat from 1.0000 seconds to 1.0714, which wants a pendulum near 44.9 inches instead of 39.1 — the sort of change that decides whether a case will take the movement.

Where this stops being reliable

The train count is arithmetic and it is exact. Everything downstream of it depends on numbers this page cannot see. The pendulum length assumes a point mass on a weightless rod swinging through nothing; a real one swings wide and carries its mass over a span, and both push the effective length up. Beats per tooth assumes an escapement you have looked at. And no train count says anything about whether the movement will actually go — that is depthing, pivots, end shake, wear and power, and none of it is visible from tooth counts.

Questions people ask

How do you work out beats per hour from tooth counts?

Multiply wheel teeth over pinion leaves for every mesh from a wheel whose speed you know down to the escape wheel, multiply that by the turns per hour of the wheel you started at, then multiply by the escape wheel teeth and by the beats each tooth gives. A centre wheel at one turn an hour through 96 into 8 and 40 into 8 puts the escape wheel at 60 turns an hour; 30 teeth at two beats each gives 3,600 beats an hour.

Is a beat the same as a swing?

Yes. One beat is one tick, one swing of the pendulum in one direction, one pallet releasing one tooth. A full period, over and back, is two beats. The pendulum length formula takes the period, so a seconds pendulum beating once a second has a two second period. Confusing the two changes a pendulum length by a factor of four, which is the most common error in this whole subject.

What is a train count?

The ratio between the wheel you started measuring from and the escape wheel, worked out by multiplying every wheel over pinion in between. It is a pure number: on the default train it is 60 to 1, meaning the escape wheel turns 60 times for every turn of the centre wheel. It is the part of the calculation that is exact.

Why is my pendulum longer than the calculated length?

Because the formula describes a point mass on a weightless rod swinging through a vanishing arc, and yours is none of those. The mass is spread along the bob and the rod, so the length that governs the period runs to the centre of oscillation rather than to the middle of the bob, and a finite swing lengthens the period a little as well. Cut the rod long from the calculated figure and regulate down to rate.

Can I change the beat without changing the pendulum?

Only by changing the train, and the escape wheel is usually the least work — one wheel instead of a repinioned arbor. The table on the page walks the count three teeth either way and shows what each does to the beat and to the pendulum that beat would want. Whether the new wheel will actually work with your pallets is a bench question about span, drop and depthing, not one this page answers.

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