When two tones with nearby frequencies f1 and f2=f1+Δf play together, the trig identity sin(2πf1t)+sin(2πf2t)=2cos(πΔft)sin(2πfˉt)
rewrites the sum as a fast tone at the average frequency fˉ=(f1+f2)/2 multiplied by a slow envelope at ∣Δf∣/2. We hear the envelope's full-wave-rectified rhythm — so the perceived **beat frequency equals ∣Δf∣** and the beat period is Tbeat=1/∣Δf∣. The top two panels show each tone alone, the bottom panel shows their sum (yellow) with the envelope drawn dashed and the envelope's zeros marked in green. Sweep Δf with the arrow keys: at unison the sum has constant amplitude; as you detune, slow pulsing appears and quickens. This is how a piano tuner zeroes a string — they listen for the beats between the string and a reference to slow to zero.