Sound two close notes together and you hear a slow throb — the beat a piano tuner listens for. Two steady tones simply add, and the sum of f1 and f2 is one tone at the average frequency wrapped in an envelope that swells and dies |f1 − f2| times a second. Down the center, data flows: the two frequencies go in, superposition runs, the audible beat rate comes out. The blue team builds and defends it; the red team tries to break it.
source Joseph Sauveur, Systême général des intervalles des sons — Mémoires de l'Académie Royale des Sciences, année 1701 (first explanation of beats by coincidence of pulses). No stable open per-article link — cited by author/title/year. Rendered, not quoted.
Nothing new is created; the two waves only add. The trigonometric identity turns that sum into a single carrier inside a slow envelope:
cos(2π f1 t) + cos(2π f2 t) = 2·cos(2π favg t)·cos(2π (fdiff/2) t), with favg = (f1+f2)/2 and fdiff = |f1−f2|. The carrier rings at the average; the envelope |cos| hits a loudness peak twice per cosine cycle — so you HEAR the full |f1−f2|, not half of it.
For the current pair, live:
| quantity | value |
|---|
Two notes throb — beats. The audible rate |f1 − f2| is the loudness pulse of a product-to-sum envelope: zero at unison, and it grows as the notes detune.
These are two of the-speed-of-sound's waves interfering — the same c that carries a single tone carries the pair, and their difference is the throb a tuner nulls to zero. Each sphere is the next one's premise.
The blue team's live check: measure the loudness-pulse rate off the envelope waveform and confirm it equals the reported beat rate |f1−f2|. If red misreports the rate, this badge is where it shows.
The input is just two tones: f1 and f2, in hertz, each of unit amplitude. Around A4 (440 Hz) a piano tuner sets one string against a fork and detunes the other by a few hertz.
| symbol | meaning |
|---|---|
| f1 | reference tone (Hz) |
| f2 | the tone being tuned (Hz) |
| favg | (f1+f2)/2 — the pitch you hear |
| |f1−f2| | the throb you count |
Equal amplitudes are assumed AMBER — with unequal loudness the envelope never quite reaches silence, but its pulse rate is still |f1−f2|.
A tuner turns the peg until the throb slows to a stop — at zero beats the strings are locked in unison.
The band is the real sum of the two tones; it pinches to silence |f1−f2| times a second. Every number is computed live from superposition, never looked up.
What the machine proves: the sum is a carrier at favg inside an envelope whose loudness peaks |f1−f2| times a second (twice per envelope cosine cycle). Identical tones give a steady sound — zero beats; detuning speeds the throb.
The blue team's witness (left) measures this rate off the waveform live; the red team (right) tries to misreport it.
And equal amplitudes are an idealisation: unequal tones never reach true silence, so "the beats vanish" is exact only when the two are matched. The |f1−f2| law is the envelope's pulse rate — clean, but only part of what an ear does.
"The beat frequency is half the difference, |f1−f2|/2." Cut. That is the envelope carrier. Loudness peaks twice per cycle, so the counted rate is the full |f1−f2| — this is the exact tamper below.
"Beats are the two tones distorting each other." Cut. Superposition is linear — no new frequency is made. Remove one tone and the throb vanishes; the ear just tracks a slowly changing amplitude.
"A difference tone at |f1−f2| is the beat you hear." Kept, corrected. A difference tone is a separate nonlinear ear effect; the beat here is a linear amplitude throb at the very same rate.
The red team's move: report the beat rate as |f1−f2|/2 — the envelope carrier, not the loudness pulse. A tuner would then hear twice as many beats as the meter claims.
Halve the reported rate and the meter disagrees with the throbs measured off the waveform — the witness (window 7) recomputes, catches it, and turns red. Nothing is faked; the attack is real and it is caught.