PlaygroundPerception · 08
The missing fundamental
Preset
Harmonics of 220 Hz
Hear
What's going on
The pitch you hear is not simply the lowest frequency in a sound. It is how often the whole sound repeats. Harmonics at 440, 660 and 880 Hz line up again every 1/220 of a second, so your brain hears 220 Hz, even with nothing playing there.
This is why a phone speaker, which can barely produce anything below a few hundred hertz, still lets you follow a bass line: your brain rebuilds the missing notes from their harmonics. Remove every other harmonic, though, and the pattern repeats twice as often: the pitch jumps up an octave.
Why it matters: pitch is computed by the brain, not simply measured by the ear. Pitch-detection algorithms in tuners, music software and speech processing have to do the same, finding the repetition rather than the lowest peak, and small speakers and headphones use it on purpose to suggest bass they cannot play.
The maths behind itFor the curious · formulas and details
A sum of harmonics n·f₀ for n in a set S is periodic, and its period is set by the largest common spacing of the frequencies present:
x(t) = Σ_(n∈S) aₙ·sin(2π·n·f₀·t) period T = 1 / (gcd(S) · f₀)
With S = 4 the gcd is 1, so the period is still 1/f₀: the waveform repeats at 220 Hz although no component sits there. With S = 8 the gcd is 2 and the pitch moves up an octave. The heard pitch is called the residue or virtual pitch (Schouten, 1940).
A simple pitch detector makes this concrete with the autocorrelation r(τ) = ∫ x(t)·x(t + τ) dt, which peaks at τ = T. Models of human pitch perception work along similar lines, combining the periodicity in each frequency band of the inner ear.