Hong Life CalculatorsUnderstanding the beat frequency calculator — concept and principle
When two tones with slightly different frequencies sound together, you hear a single tone whose loudness swells and fades regularly. These pulsations are called beats, and their rate equals the difference between the two frequencies. Musicians and piano technicians use beats to tune by ear.
How it is calculated
- Beat frequency: f_beat = |f₁ − f₂| (Hz, beats per second)
- Perceived tone: f_avg = (f₁ + f₂) / 2 (Hz)
Enter both frequencies in hertz. The order does not matter because the absolute value is taken. The beat period, the time between two loudness peaks, is 1 / f_beat seconds.
The principle
Adding two equal-amplitude sine waves and applying the sum-to-product identity gives
sin(2πf₁t) + sin(2πf₂t) = 2 cos(2π · (f₁ − f₂)/2 · t) · sin(2π · (f₁ + f₂)/2 · t).
The fast factor oscillates at the average frequency, which is the pitch you hear. The slow cosine factor is an envelope at (f₁ − f₂)/2. The ear responds to the size of the envelope, not its sign, and the envelope's magnitude peaks twice per cosine cycle, so loudness peaks occur at |f₁ − f₂| per second.
Worked example
- Inputs: f₁ = 440 Hz (concert A4), f₂ = 444 Hz.
- Beat frequency = |440 − 444| = 4 Hz, i.e. four swells of loudness per second, one every 0.25 s.
- Perceived tone = (440 + 444) / 2 = 442 Hz.
| Frequency 2 (Hz) | Beat frequency (Hz) |
|---|---|
| 111 | 329 |
| 222 | 218 |
| 333 | 107 |
| 444 | 4 |
| 555 | 115 |
| 666 | 226 |
| 888 | 448 |
How to read the result
Slow beats (below about 1 Hz) mean the two tones are almost in tune; tuning is finished when the beats disappear. A few beats per second sounds like a vibrato-like wobble. As the difference grows past roughly 15–20 Hz the beats blur into a rough, dissonant sound, and with larger differences the ear separates two distinct pitches. The exact thresholds depend on the frequency range and the listener, so treat these as rough guides.
Common mistakes
- Expecting the beat rate to be half the difference because the envelope formula contains (f₁ − f₂)/2; the audible loudness peaks occur at the full difference.
- Assuming beats only happen between pure tones; harmonics of musical notes also beat, which is how intervals such as fifths are tuned.
Frequently asked questions
How do I tune a guitar string using beats?
Play the reference note and the string together and adjust the tuning peg so the beats slow down; when they stop, the frequencies match.
Can beats be used to measure an unknown frequency?
Yes. Sound it with a known reference, count the beats, then change the reference slightly; if the beats speed up or slow down you learn whether the unknown is above or below.
Sources: D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, Wiley (chapter on sound waves, beats) · H. von Helmholtz, On the Sensations of Tone (1863; English translation by A. J. Ellis)