Two tones that almost agree don't just sound out of tune — they beat: the
sound swells and fades at exactly the rate their coinciding harmonics disagree. Beats are how tuners
tuned pianos long before electronics, why a tempered third shimmers, and — in Balinese gamelan —
a feature: paired instruments are tuned a few Hz apart on purpose so the ensemble breathes.
That shimmer is called ombak ("wave"). Detune the pair and hear it.
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the two coinciding harmonics, summed — the slow envelope IS the beat
The rate is simple arithmetic: for an interval a:b the b-th harmonic of the upper tone and the
a-th of the lower land almost together, and the beat is their difference in Hz. A piano tuner
counting "one beat a second" on a fifth is measuring that arithmetic by ear. In just
intonation the difference is zero — that's what "beatless" means on the
Just Intonation explorer, and it's why the gamelan pages'
stretched octaves shimmer on the tunings page.