Beats, and Tuning a Guitar by Ear
The problem
A guitar's A string should sound 440 Hz. Against a 440 Hz reference you hear the loudness throb three times per second. What is the string's frequency, and by how much must you change the tension?
The wrinkle: beats tell you the size of the error but not its sign, and the fix for that is a nice piece of practical physics.
Where the throb comes from
Two tones of nearly equal frequency add by superposition:
Read the two factors:
- The fast one oscillates at the average frequency \((f_1+f_2)/2 \approx 441.5\) Hz — the pitch you perceive.
- The slow one, at \((f_1-f_2)/2 = 1.5\) Hz, modulates the amplitude — the envelope.
The ear hears loudness, and loudness peaks twice per envelope cycle (once at each extreme of the envelope, positive and negative). So:
That factor of two is the whole subtlety. The envelope oscillates at 1.5 Hz; you hear 3 throbs per second. Beat period = 1/3 s.
Step 1: the string is 437 or 443 Hz
\(|f_1 - 440| = 3\), so the string is at 437 Hz or 443 Hz and beats cannot distinguish them. Nothing about the throb tells you which.
The resolution: deliberately make it worse. Slacken the string slightly and listen.
- If the beat rate increases, you were flat (437) and you have gone further flat.
- If it decreases, you were sharp (443) and are heading toward 440.
This "detune and listen" trick is what every guitarist does without thinking about it, and it is a genuinely useful idea: when a measurement is degenerate, perturb the system in a known direction and watch which way the degeneracy breaks.
Step 2: how much tension?
For a string of length \(L\) and mass per length \(\mu\) under tension \(T\), the fundamental is
Take logarithms and differentiate:
For \(\Delta f = 3\) Hz out of 440:
A typical A string carries about 70 N, so this is roughly 1 newton — about a 100 g weight, and a very small turn of the peg. Which is exactly why guitars go out of tune so readily: a 1.4% tension change is well within what temperature, humidity and a few minutes of playing will do.
Why this method is so sensitive
Suppose you tried to judge 3 Hz out of 440 by pitch alone — that is 12 cents, at the edge of what a trained ear can detect in isolation. But as a beat it is a throb every third of a second, completely unmistakable.
Better still, the sensitivity improves as you approach the target: at 1 Hz off you get a throb every second, at 0.2 Hz off one every five seconds. A null method — listening for the absence of something — beats a direct measurement, and the same idea underlies Wheatstone bridges, interferometers and lock-in amplifiers.
Where else beats appear
- Tuning any instrument, and the "wolf tones" of unequal temperament.
- Two aircraft propellers slightly out of sync — the characteristic slow throb of a twin.
- Heterodyne detection. Mix an unknown signal with a known local oscillator and read the beat: this is how every radio receiver works, and how laser frequencies are measured against a comb.
- Neutrino oscillations, where mass eigenstates of slightly different frequency beat against each other over hundreds of kilometres.
Common mistakes
- Using \(f_{\rm beat} = |f_1-f_2|/2\). The envelope is at half the difference, but loudness peaks twice per envelope period, so the audible beat is the full difference.
- Assuming beats give the sign. They give \(|{\Delta f}|\) only. Perturb to break the degeneracy.
- Forgetting \(f \propto \sqrt{T}\). Using \(\Delta T/T = \Delta f/f\) gives half the right answer.
- Confusing beats with interference in space. Beats are interference in time, from a frequency difference; standing waves are interference in space, from counter-propagating waves.
Related
Beats · Superposition & interference · Sound waves · Traveling waves · Simple harmonic motion · Wave studio — set two nearby frequencies and watch the envelope