Beat Frequency Calculator: Beats per Second, Beat Period and Tuning by Ear
Find the beat frequency |f₁ − f₂|, beat period, beats per minute and perceived pitch of two tones, or work out an unknown frequency from the beats you hear.
How it works
f_beat = |f₁ − f₂| · beat period T = 1 ÷ f_beat · perceived pitch = (f₁ + f₂) ÷ 2 · unknown tone = f_ref ± f_beat · cents = 1200 × log₂(f₂ ÷ f₁)
When two tones with slightly different frequencies sound together, their waves drift in and out of step. When the crests line up the sound gets louder, and half a cycle of the difference later they cancel and it gets quieter. That swell happens |f₁ − f₂| times per second, which is the beat frequency. The ear hears a single tone at the average of the two frequencies, pulsing at the beat rate. The sum of the two waves is 2A·cos(2π·Δf/2·t)·cos(2π·f_avg·t), and the slow cosine factor is the envelope plotted in the chart. Beats are how musicians and piano tuners tune by ear: as two strings get closer, the beats slow down and stop at unison. Because the formula uses an absolute value, one beat rate fits two answers, one tone below the reference and one above. To tell them apart, raise the unknown tone slightly. If the beats slow down it was flat, and if they speed up it was sharp. Choose "Unknown tone from beats" to do this, and use the mistuning table to see how many beats per second a given error in cents produces at your pitch. Beats are clearest below about 10–15 per second; faster than that the sound becomes rough and then splits into two notes. This calculator works with absolute frequencies; for converting cents to frequency ratios use the cents and frequency ratio calculator.
Worked example
A tuning fork at 440 Hz and a guitar string at 443 Hz beat |440 − 443| = 3 Hz, or 180 pulses a minute, one loud pulse every 1 ÷ 3 = 0.3333 s. You hear a pitch of (440 + 443) ÷ 2 = 441.5 Hz, and the string is 1200 × log₂(443 ÷ 440) = 11.8 cents sharp. In reverse, if you hear 3 beats per second against the 440 Hz fork and the beats speed up when you tighten the string, the string is at 440 + 3 = 443 Hz.
Sources: OpenStax University Physics Vol. 1, 17.6 Beats (Example 17.7: 256 Hz and 512 Hz forks) · HyperPhysics: Beats · Omni Calculator: Beat frequency (235 Hz and 335 Hz example)
Frequently asked questions
What is beat frequency?
It is the number of loud-soft pulses per second you hear when two tones of slightly different frequency play together. It equals the difference between the frequencies: 440 Hz and 444 Hz give 4 beats per second.
How do you find an unknown frequency from beats?
Add and subtract the beat rate from the known frequency. With a 256 Hz fork and 4 beats per second, the unknown tone is 252 Hz or 260 Hz. Raise the unknown tone slightly: if the beats slow down it was 252 Hz, if they speed up it was 260 Hz.
Why do beats slow down as you get in tune?
The beat rate is the frequency difference, so as the two tones approach each other the difference shrinks. At exact unison it is 0 and the pulsing stops.
What frequency do you actually hear?
For close tones you hear one pitch at the average frequency, (f₁ + f₂) ÷ 2, getting louder and softer at the beat rate. When the gap grows beyond about 15–20 Hz, the pulsing turns into roughness and then two separate notes.
How many beats per second does a few cents of mistuning give?
It depends on the pitch: 1 cent is about 0.058% of the frequency. At A4 = 440 Hz, 2 cents gives about 0.51 beats per second and 10 cents about 2.55. At A2 = 110 Hz the same errors beat four times slower, which is why low notes are tuned using their higher harmonics.
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This calculator is for educational purposes. Double-check important results before acting on them.