Clock Escapement Beat Rate Calculator
Calculate clock escapement beat rate in BPH from pendulum length or gear train ratios.
Returns seconds per beat and pendulum adjustment to correct timing.
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.