Two notes close in pitch beat at a rate equal to the difference of their frequencies: 440 Hz and 444 Hz beat four times a second.
Adding two sine waves of frequencies f1 and f2 gives a wave at the average frequency whose height swells and fades at |f1 − f2|, as the waves drift in and out of step. Tuning by ear works by slowing the beats until they stop.
It needs the two notes to be close: when they differ by more than about 15 to 20 Hz the beats blur into a rough buzz and then into two separate pitches. The loudness swells and fades twice for every cycle of the envelope, so the rate you hear is the difference, not twice it. It is for steady tones; real instruments have vibrato and decay.
Estimate with the rule, then check it against the calculator that models it properly.
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