About

When two sound waves of slightly different frequencies f₁ and f₂ overlap, their superposition produces a periodic amplitude modulation perceived as "beats." The beat frequency f_beat = |f₁ − f₂| determines how many volume oscillations occur per second. Piano tuners rely on this phenomenon: they adjust a string until the beats slow to zero, confirming unison. Misidentifying beat rate by even 0.5 Hz compounds across an instrument, producing audible dissonance in chords. This calculator computes f_beat and the perceived average pitch, then synthesizes both tones through the Web Audio API so you hear the actual interference - not a recording or approximation.

The tool assumes ideal sinusoidal waves in a linear medium. Real instruments produce harmonics that create additional combination tones beyond the fundamental beat. For frequencies differing by more than roughly 15 Hz, the human ear resolves two distinct pitches rather than perceiving beats - a psychoacoustic boundary known as the roughness limit. Results are most musically relevant when f_beat < 15 Hz.

Formulas

The resultant displacement of two superimposed sinusoidal waves at a fixed point is:

y=sin(2πf₁t)+sin(2πf₂t)=2cos(2π⋅f₁ − f₂2t)⋅sin(2π⋅f₁ + f₂2t)

The cosine term modulates amplitude at half the difference frequency. Because amplitude peaks occur twice per full cosine cycle (positive and negative peaks both produce loudness maxima), the perceived beat frequency is:

f_beat=|f₁−f₂|

The perceived average pitch (the carrier) is:

f_avg=f₁ + f₂2

The beat period (time between successive loudness maxima):

T_beat=1f_beat

Where f₁ = frequency of first wave Hz, f₂ = frequency of second wave Hz, t = time s, T_beat = beat period s.

Reference Data

Musical Interval	f₁ Hz	f₂ Hz	f_beat Hz	Context
A4 Unison (in tune)	440.00	440.00	0.00	Perfect tuning reference
A4 slightly sharp	440.00	442.00	2.00	Common orchestral A pitch
A4 vs B♭4	440.00	466.16	26.16	Minor second - dissonant
C4 vs C♯4	261.63	277.18	15.55	Semitone roughness boundary
Guitar E2 strings	82.41	83.00	0.59	Slow beat → nearly tuned
Tuning fork A4 + piano	440.00	439.00	1.00	1 beat/sec audible wobble
Equal temperament 5th (C4 - G4)	261.63	392.00	130.37	Two distinct pitches heard
Just intonation 5th (C4 - G4)	261.63	392.44	130.81	Pure ratio 3:2
Binaural 10 Hz alpha	200.00	210.00	10.00	Alpha brainwave entrainment claim
Binaural 4 Hz theta	150.00	154.00	4.00	Theta brainwave entrainment claim
Violin A-string detuned	440.00	437.00	3.00	Noticeable wobble during performance
Oboe vs Clarinet A4	440.00	441.50	1.50	Ensemble tuning discrepancy
Two tuning forks (demo)	256.00	260.00	4.00	Classic physics lab experiment
Sub-bass interference	30.00	31.00	1.00	Felt more than heard
Ultrasonic near-limit	19000	19005	5.00	Carriers inaudible, beat may be heard

Frequently Asked Questions

The human auditory system can only track amplitude fluctuations up to roughly 15 Hz as a single fused percept. Beyond that threshold, the cochlea resolves two separate frequency channels, and the listener perceives two distinct pitches with a sensation of "roughness" rather than a smooth wobble. This is related to the critical bandwidth of the basilar membrane, which varies from about 100 Hz wide at low frequencies to over 1000 Hz at high frequencies.

A tuner strikes a reference source (tuning fork at 440 Hz or electronic tone) simultaneously with the piano string. If beats are heard, the string is sharp or flat. The tuner adjusts the pin until f_beat = 0. For temperament stretching, tuners intentionally leave specific intervals with controlled beat rates (e.g., a tempered fifth beats at approximately 0.89 Hz for A3 - E4).

The formula f_beat = |f₁ − f₂| strictly applies to pure sinusoidal tones. Complex waveforms contain harmonics, and each harmonic pair can produce its own beat pattern. A violin and oboe playing near-unison will exhibit multiple simultaneous beat rates from their overtone series, making the perceived beating more complex than a single-frequency prediction.

Acoustic beats occur when two tones physically mix in the air (or in one ear). Binaural beats require headphones: one frequency in the left ear, another in the right. The "beat" is generated neurologically in the brainstem, not by physical wave superposition. This calculator demonstrates acoustic beats. For binaural beats, use headphones and note that the perceptual effect is subjective and contested in peer-reviewed literature.

No. Beat frequency is defined as the absolute value of the difference, so it is always ≥ 0. The sign of f₁ − f₂ only indicates which source is higher in pitch, not a physical direction of the beat.

The sum of two close-frequency sinusoids produces a carrier wave at the average frequency modulated by an envelope at half the difference frequency. The envelope's shape is a cosine function. Because both the positive and negative lobes of the cosine produce amplitude maxima, you see |f₁ − f₂| loudness peaks per second, which is the beat frequency.