Clock Escapement Beat Rate Calculator
Calculate the beat rate of a clock escapement from pendulum length or gear train ratio.
Verify timing accuracy.
What is beat rate?
The beat rate (or vibration rate) of a clock is the number of ticks per hour (abbreviated BPH or vph). Each “tick” is one swing of the pendulum in one direction — so a full back-and-forth swing produces two beats. The beat rate determines how finely a clock can divide time and is critical for accurate timekeeping.
Pendulum period formula: The period (time for one full swing) of an ideal simple pendulum is:
T = 2π × √(L / g)
Where:
- T = period in seconds (one complete cycle)
- L = effective pendulum length in meters (center of suspension to center of bob)
- g = gravitational acceleration ≈ 9.81 m/s²
From period to beat rate: Beats per hour = 3600 / (T / 2) = 7200 / T
Each full period T contains 2 beats (tick and tock).
Common clock beat rates:
| Clock Type | Beat Rate (BPH) | Pendulum Length | Period (s) |
|---|---|---|---|
| Seconds pendulum (grandfather) | 3,600 | 994 mm (39.1 in) | 2.000 |
| Vienna regulator | 7,200 | 248 mm (9.8 in) | 1.000 |
| Mantel clock (typical) | 9,600 | 140 mm (5.5 in) | 0.750 |
| Cuckoo clock (1-day) | 5,400 | 332 mm (13.1 in) | 1.333 |
| French carriage clock | 14,400 | 62 mm (2.4 in) | 0.500 |
| 400-day anniversary clock | 900 | 4,380 mm (172 in) | 8.000 |
Gear train verification: The escapement beat rate can also be calculated from the gear train. Count the teeth on each wheel and pinion from the center wheel to the escape wheel:
BPH = (Teeth product / Pinion product) × 2 × Escape wheel teeth / 60
If the pendulum-based calculation and the gear-train calculation disagree, the pendulum length is wrong — adjust it.
Worked example: A grandfather clock with a 994 mm pendulum:
- T = 2π × √(0.994 / 9.81) = 2π × 0.3183 = 2.000 seconds
- BPH = 7200 / 2.000 = 3,600 beats per hour
- That is 60 beats per minute, or one tick per second — the classic grandfather clock rhythm.
Adjusting beat rate:
- To speed up (increase BPH): shorten the pendulum by raising the bob
- To slow down (decrease BPH): lower the bob
- One turn of the rating nut typically changes rate by 10–30 seconds per day
- A 1 mm change in a seconds pendulum changes the rate by about 86 seconds per day
Temperature compensation: Metal pendulum rods expand with heat, making the clock run slower in summer. A 1°C rise expands a steel rod by ~11 parts per million, slowing a seconds pendulum by ~0.5 seconds/day. Compensated pendulums (mercury, wood-and-metal, invar) minimize this drift.