A normal duplex watch escapement uses two tooth sets on one escape wheel: long rim teeth lock against a notched roller, while shorter teeth standing on the wheel face give impulse to a pallet on the balance shaft. It gives one impulse per complete balance oscillation. The return swing has no impulse and includes a small advance and recoil as the roller notch passes the resting tooth.
Duplex Escapement · Balance Rate and Wheel Gearing
Normal duplex: one impulse every two beats. Compare balance rate, tooth count and explicit seconds-hand gearing. The two tooth sets work at different axial levels.
Equation Used
- Two beats make one complete balance oscillation.
- Normal duplex advances one tooth per full oscillation.
- Both tooth sets have the entered count.
- G is escape-wheel rpm divided by seconds-hand rpm.
- The target seconds-hand speed is 1 rpm.
- No missing releases or extra releases are modelled.
Phase timing, notch profile and recoil are explanatory. Calculation does not predict balance dynamics, repair clearances or real-watch accuracy.
Separate locking and impulse contacts
The long pointed rim tooth rests on the cylindrical roller fixed to the balance shaft. On the working swing, the notch admits that tooth so the wheel can advance. A raised face tooth then drives the separate impulse pallet. The next long tooth reaches the roller and holds the wheel again.
On the opposite swing, the pallet passes between face teeth without receiving another impulse. The resting tooth can enter the notch slightly and then recoil; it does not advance another complete tooth. The two tooth sets share a shaft but work at different axial levels.
The animation enlarges these levels and slows the sequence for inspection. The impulse contact point follows the shown radial face of a raised tooth. Locking-notch outline, balance motion, recoil amount and transition timing are explanatory, not dimensions for making an escapement. No spring, friction or impact dynamics are solved.
Compare balance rate and train gearing
Enter the balance rate in beats per hour, the number of teeth in each set, and the speed reduction from escape wheel to seconds hand. One beat means one swing; two beats make a complete oscillation. In the normal duplex shown here, one complete oscillation advances one escape-wheel tooth.
The tooth-circle diameter supplies an arc-pitch measurement. The drawing automatically fits the scene, so changing diameter changes the dimensional readout rather than shrinking the complete mechanism on screen.
The separate Duplex-Jacot or double-duplex form can have three lost beats and one impulse every four swings. Its timing should not be substituted into this normal-duplex calculator.
One tooth per full oscillation
For balance rate B in beats/hour and N teeth per set, the impulse frequency is f=B/7200 impulses/s, and the mean escape-wheel speed is nₑ=B/(120N) rpm. Tooth pitch is 360/N degrees. For tip-circle diameter D, the corresponding arc pitch is πD/N.
If G is escape-wheel speed divided by seconds-hand speed, nₕ=nₑ/G. A seconds hand intended to turn once per minute has an indicated rate error of 86400(nₕ−1) seconds per actual day. Positive means gain; negative means loss.
This last result assumes fixed train gearing and continuous operation without missing or extra releases. It is not a diagnosis of a real watch. Changing tooth count changes mean wheel speed; it does not change the number of impulses per second at a fixed balance rate.
18,000 beats/hour with 15 teeth
The balance makes 2.5 complete oscillations per second and receives 2.5 impulses per second. Each cycle lasts 0.4 s. A 15-tooth wheel advances 24° per impulse and turns at 10 rpm on average.
A 10:1 speed reduction produces a 1 rpm seconds hand, so the ideal indicated daily error is zero. That hand advances 2.4° at each of the 2.5 releases per second. Gearing changes the angular size of the steps; it does not turn 2.5 release events per second into one event per second.
At a 12 mm locking-tip diameter, the arc pitch is π×12/15≈2.513 mm. This is spacing measured along the tip circle, not tooth thickness or a recommended clearance.
What this model does and does not establish
The normal duplex is a frictional-rest escapement: a long tooth bears against the roller during the resting portion of the balance swing. One impulse per full oscillation does not imply that friction is exactly halved, and it does not guarantee a particular daily accuracy.
Rate and pitch calculations do not specify roller-slot width, tooth profiles, balance amplitude, spring stiffness, unlocking margin or impact force. Those require a complete movement design and physical measurements. The illustration is intended to distinguish this mechanism from an anchor or lever escapement.
Duplex timing questions
Does 18,000 beats/hour mean one impulse per second?
No. A normal duplex at that rate has 2.5 impulses per second.
Is the escape wheel itself the seconds hand?
Not generally. The gearing between the wheel and hand must be included explicitly.
Does the wheel remain perfectly still throughout the return?
It remains on the same tooth but can make a small forward-and-back movement as the notch passes.
Does the calculator predict restoration clearances?
No. Tip-circle pitch is a geometric spacing; the drawing is not a manufacturing or repair specification.
Mechanism references
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, device 1180, printed page 290: balance-shaft stop, impulse pallet and raised wheel studs.
Horopedia, The Duplex Escapement, description of the two tooth sets, four operating phases and return-swing recoil. Clock and Watch Escapement Mechanics, chapter 19, geometric construction showing separate locking and impulse levels. The present proportions and phase timing are illustrative.
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