Beat Frequency Calculator
Calculate the beat frequency between two close tones.
Used for instrument tuning, ultrasonic detection, and acoustics.
Find beat period and pitch midpoint.
Beat Frequency
When two sound waves of slightly different frequencies are played simultaneously, they alternately reinforce and cancel each other. The result is a slow pulsation in loudness called a beat. Beats are how piano tuners, guitarists, and orchestra musicians fine-tune instruments by ear.
Formula
f_beat = |f₁ − f₂|
The beat frequency is simply the absolute difference between the two source frequencies. Each beat is one full cycle of the loudness modulation — louder, quieter, louder.
Beat Period
T_beat = 1 / f_beat
If two tones differ by 2 Hz, you hear a 2 Hz beat — one full pulse every 0.5 seconds.
Worked Example — Guitar Tuning
You play a reference 440 Hz A and your guitar’s open A string at 437 Hz:
- f_beat = |440 − 437| = 3 Hz
You hear three beats per second. Tighten the string until the beats slow and disappear — at zero beats, the strings are in tune.
Perceived Pitch — The Carrier
When you superpose two close frequencies, the ear perceives a tone at the average frequency:
f_carrier = (f₁ + f₂) / 2
This is the actual pitch a listener hears, modulated by the slow beat envelope at f_beat.
Useful Beat Ranges
| Beat Frequency | Use |
|---|---|
| 0 Hz | Perfect unison (tuned) |
| 1–7 Hz | Audible beats — used in tuning |
| 7–20 Hz | Roughness, fast pulsing |
| > 20 Hz | Two distinct tones (not heard as beats) |
Beats Beyond Music
| Field | Application |
|---|---|
| Ultrasonics | Doppler velocity from beat with reference |
| Lasers | Heterodyne detection in interferometry |
| Radio | AM and FM detection circuits use beat principles |
| Bat / sonar | Pulse-echo frequency comparison |
| Bridge engineering | Detect resonance close to driving frequency |
Caveats
The beat formula assumes pure sinusoidal tones. Real instruments contain harmonics, so you can hear beats from harmonic pairs even if the fundamentals seem in tune. For piano, octave stretching introduces small beat patterns that are intentionally tuned for richness.
Mathematical Origin
cos(2πf₁t) + cos(2πf₂t) = 2 × cos(2π × f_avg × t) × cos(2π × f_beat/2 × t)
The slow envelope (cosine of half the beat frequency) modulates the fast carrier — that envelope is what your ear perceives as the beat pattern.