what we hear
hearing is three mechanisms stacked. a wave either bends around your head or is blocked by it, which decides whether you can point at it. a coiled tube then sorts it by pitch, mechanically. and your sensitivity is wildly uneven — in a way that changes shape depending on how loud things are.
a wave meeting your head, solved live
where the pitch lands inside
sound arrives from the left. the amber disc is a head, to scale, at 18 cm across — 1.68 m of air on a 280×190 grid.
frequency
500 Hz
wavelength
68.6 cm
nearest note
B4
ear-to-ear, simulated
0.0 dB
the wave is far larger than your head, so it diffracts straight around — both ears hear almost the same thing and the level difference collapses toward zero.
that figure is measured off the running field, not looked up — the simulation averages the pressure over an arc on each side of the head and takes the ratio. two caveats worth knowing: this is a 2D slice, so sound can't pass over or under the head the way it really does, and the shadow comes out deeper than a real head's by a few dB. and below about 314Hz the wave is longer than the whole simulated box, so the number stops meaning anything — the real answer down there is under a decibel, which is exactly why you can't point at a bass note.
level isn't the only cue. sound reaching the far ear arrives about 525 microseconds later, and below roughly 1500 Hz your brain switches to using that delay instead — precisely because the level difference has nothing left to say.
inside the cochlea this note peaks 9.9 mm from the apex of about 35. the map is logarithmic, so every octave gets roughly the same length of tissue — your ear is built for ratios, which is why an octave sounds like an octave wherever you play it.
headphones, low volume. the sweep climbs about an octave a second from 2 kHz.
equal loudness — and why quiet music loses its bass
every curve joins the tones that sound equally loud. these are the published ISO 226 contours, not an impression of them. the bottom one, 0 phon, is the threshold of hearing.
drag it and watch the curves flatten as they rise. at 60 phon, a tone at 500 Hz needs 62 dB SPL to sound as loud as a 1 kHz tone at 60 dB — 2 dB more. compare 20 Hz at 110 dB against the same 1 kHz reference: that is the gap that matters, and it widens the quieter you listen. it is why turning music down seems to strip the bass out first — not the speakers, but you. old stereos had a “loudness” button that tilted the bass back up to compensate.
why the ear canal is 2.5 cm long
it's a tube closed at one end by the eardrum — the same object as the closed pipe elsewhere in the lab. a 2.5 cm closed pipe resonates near 3.4 kHz and boosts everything around there by roughly 10 dB before the sound even reaches the drum. that free amplification is exactly where the contours dip lowest, and it's where consonants live — the difference between s and f, which is the first thing to go when hearing fades. losing the top octave doesn't make speech quieter, it makes it blurry.
past the drum, three of the smallest bones you own lever the vibration onto a membrane about twenty times smaller, concentrating the pressure enough to move fluid instead of air. without that gearing most of the sound would bounce straight off the water in your head, the way shouting at a lake doesn't disturb the fish.