When Two Notes Meet

At a football stadium, a "Mexican wave" can circle the whole ground in seconds, yet every fan stays in their seat. Each person just stands up and sits down. What travels is the pattern. That's what a wave is: a disturbance that moves through something while the something stays roughly where it was.

Sound is exactly that, in air. A guitar string or your vocal cords push on nearby air, which pushes on the next air, and a pattern of squeezed and stretched air races outward at about 343 metres per second (in dry air at 20 °C). The air itself just shuffles back and forth a tiny distance. When the pattern reaches your eardrum, it pushes it in and out in the same rhythm, and you hear it.

Pitch is how fast the pattern repeats

The number of squeezes per second is the frequency, measured in hertz (Hz). The note A above middle C is fixed by international standard (ISO 16) at 440 Hz. Divide the speed by the frequency and you get the wavelength, the distance between one squeeze and the next:

Sound	Frequency	Wavelength in air
Lowest pitch people hear	about 20 Hz	about 17 m
Concert A	440 Hz	78 cm
Highest pitch people hear	about 20,000 Hz	about 1.7 cm

That 20–20,000 Hz range is the usual textbook figure. The top end shrinks over a lifetime, starting in childhood, which is why some high-pitched sounds are audible to teenagers and not to their parents.

The one rule: waves add up

When two sound waves arrive at the same place, the air simply does both things at once. Its push is the sum of the two pushes. This is called superposition. Where two squeezes coincide, you get a bigger squeeze. Where a squeeze meets a stretch, they cancel. After they've passed, each wave carries on exactly as if the other had never been there. That is why, in a crowded room, everyone's voice reaches you intact.

Beats: hearing the difference

Set the waves to 440 and 443 Hz. Most of the time they drift in and out of step: in step, they add up to a loud tone; out of step, they nearly cancel. That swelling and fading happens 443 − 440 = 3 times a second. These are called beats, and the beat rate is always the difference between the two frequencies.

That gives musicians an exquisitely sensitive tuning tool. Two guitar strings that should match are played together; if you hear a slow wobble, they're slightly off, and you turn the peg until the wobble slows and disappears. A difference of a fraction of a hertz, impossible to hear in either note alone, becomes a clear, slow pulse.

Your ear can't tell 440 from 441 on its own. Play them together and it hears the difference once every second.

Piano tuners go further and count beats on purpose. On a modern piano the fifths (like A to E) are tuned very slightly narrow, so that every key works equally well. The E above concert A is set to about 659.26 Hz rather than a "pure" 660. Every piano note also contains higher harmonics, and the A's third harmonic (1,320 Hz) and the E's second harmonic (about 1,318.5 Hz) are then close but not equal. They beat about 1.5 times a second, a rate a tuner can listen for. (For why every note contains a stack of harmonics, see The Fourier Transform.)

Cancelling sound with sound

Now set both waves to 440 Hz and shift the second by 180°. Every squeeze of one meets a stretch of the other, and the sum is flat. Two sounds have made silence. In 1936 the German inventor Paul Lueg was granted a US patent, "Process of silencing sound oscillations", for exactly this idea: pick up the unwanted sound with a microphone and play it back, inverted, through a loudspeaker. The electronics of his day couldn't do it well. Modern noise-cancelling headphones can: tiny microphones listen to the outside noise and the earpiece adds its mirror image, continuously.

They work best on steady, low drones like engine hum. A low note's wave is long and predictable, so it is easy to match. Sudden, high-pitched sounds such as voices or a clatter change too quickly to mirror perfectly, which is why headphones also rely on simple padding.

Does cancelling sound destroy energy? No. When two speakers in a room play the same tone, there are quiet spots where the waves cancel and loud spots where they reinforce, and overall the energy is all accounted for. Interference doesn't delete energy. It moves it around.

Why this matters beyond music

Superposition is one of the most general rules in physics. Ripples on a pond, radio signals, light and earthquake waves all add in the same way. Light waves interfering produce the colours in soap bubbles. In quantum mechanics, the same rule applied to matter produces the famous double-slit pattern (see Quantum Mechanics: The Weirdness at the Bottom). And at a concert, a hundred instruments push on the same air at the same moment, the pushes simply add, and your ear and brain pull the sum apart again into a violin, an oboe and a cough in row G.