Why a Violin Is Not a Flute

Pluck a string and you do not hear one frequency. You hear a crowd.

The string vibrates along its whole length, which gives the pitch you name. It also vibrates in halves, which sounds an octave above. And in thirds, giving the octave plus a fifth. And in quarters, fifths, sixths, and onwards, all simultaneously. The frequencies are whole-number multiples of the lowest one, and the whole stack is called the harmonic series.

You can hear the members individually. Touch a guitar string lightly at its midpoint and pluck: the fundamental is suppressed and the octave rings out alone. Touch at a third of the length and you get the note a twelfth above. Guitarists and string players call these harmonics and use them as a tuning check, which is a musician's way of interrogating the physics directly.

Two consequences follow, and between them they account for most of what an ear does.

The first is timbre. Every instrument produces the same series and emphasises different members of it, and that recipe is what tells you what you are listening to. A flute is nearly pure, with weak upper harmonics. A violin is rich in them. A clarinet is missing its even-numbered ones almost entirely, because a cylindrical tube closed at one end suppresses them, which is why it sounds hollow, why it overblows at the twelfth rather than the octave, and why its bottom register has a completely different character from its top. One geometric fact about a pipe produces most of the instrument's personality.

But a static recipe is not enough, and here is the surprising part: most of what identifies an instrument is in the first few hundredths of a second.

Take a recorded piano note, cut off the first fifty milliseconds and play the rest. Listeners struggle to say what it is. The hammer strike, the fret noise, the breath, the scrape of rosin. The attack is where the identifying information lives, and the sustained tone that follows is comparatively anonymous. This is why synthesised instruments were unconvincing for years while the engineers worked on the steady state, and why a sampled piano that models the attack properly fools people immediately.

The second consequence is the octave.

Double a frequency and every harmonic of the upper note is already inside the lower note's series. The two overlap almost completely, which is why they fuse in a way no other interval does, and why musical systems around the world treat pitches an octave apart as versions of the same thing. Octave equivalence looks like a convention and behaves like acoustics.

The series also explains a trick your hearing performs without telling you.

Play the harmonics of a low note (the second, third, fourth and so on) without the fundamental itself, and listeners hear the fundamental. It is not there. The ear works out which fundamental would produce that pattern of spacing and supplies the pitch.

This is why a bass line survives on a phone speaker that cannot reproduce anything below a few hundred hertz. The bottom note is physically absent and the harmonics above it specify it unambiguously, so you hear it anyway. Cinema sound, telephone speech and every small speaker ever sold depend on the effect.

One last thing, which matters for the next chapter. The series is a set of whole-number ratios: 2:1, 3:2, 4:3, 5:4. Those ratios are where scales come from, because notes whose harmonics coincide sound related.

The awkward fact is that those ratios do not fit together. Stack enough of them and they fail to close, by a small and stubborn amount, and the whole history of tuning is the history of what to do about it.