How Far Is That Wall
Clap, hear the echo, click the distance. You will be within about five per cent in a minute, and completely lost outside a band three and a half doublings wide.
What it is
Headphones, then click. A clap goes off, an echo comes back, and you click how far away you think the wall is. The scale runs from 0.6 to 40 metres and it’s logarithmic, because being two metres out at thirty isn’t the same mistake as being two metres out at three.
Walls are drawn from a wider range than the one you can actually do, on purpose. Both edges of what works are things you find by failing at them.
no colour takes the dullness out of the echo and leaves only the timing, which turns out to change almost nothing, and that’s the measurement.
How it works
The delay is the round trip, so twice the distance over the speed of sound. Three seconds a kilometre out and back, which is the rule of thumb, and 55 ms for a wall nine metres away. Reading it’s a skill you pick up in about a minute, which is the part that feels like a trick.
The interesting thing is where it stops, and it stops at both ends.
The near edge is fusion. Below the echo threshold the ear reports one event at the leading direction and suppresses the second, so there’s no delay to read because there’s no second sound. For clicks that threshold is around 8 ms, and 8 ms of round trip is 1.37 metres. Nearer than that and the task isn’t hard, it’s impossible.
The far edge is level. The echo has travelled twice the distance and come off a wall that absorbs some of it, so it gets quiet fast. Taking -34 dB against the clap as the point where you would lose it inside your own clap, that’s 17 metres.
So the whole usable band is 1.4 to 17 metres: 3.63 doublings. Every bit of echolocation anybody has ever done happens inside that, and it’s narrower than it feels like it should be.
For the resolution, the threshold on a time interval is roughly a fixed fraction of the interval with an absolute floor underneath. Six per cent with a floor of 1.2 ms, which predicts a flat percentage error across the window and a worse one close in where the floor takes over. The simulated listener does exactly that: 4.7 to 4.9 per cent from 2 metres out to 32, and 469 per cent at 1 metre, which is inside the fusion region and isn’t an error bar so much as a shrug.
| wall | echo level | predicted | simulated listener |
|---|---|---|---|
| 1 m | -9.0 dB | 20.6% | 469% |
| 2 m | -15.1 dB | 10.3% | 8.4% |
| 4 m | -21.1 dB | 6.0% | 4.9% |
| 8 m | -27.3 dB | 6.0% | 4.8% |
| 16 m | -33.5 dB | 6.0% | 4.9% |
| 32 m | -39.8 dB | 6.0% | 4.7% |
What surprised me
I built a second cue and then measured that it doesn’t exist. The plan was a cue ablation like day 58’s: delay against colour. Air absorbs high frequencies far faster than low ones, so a distant echo should arrive not just later and quieter but duller, and that dullness is a second, independent measure of the same distance.
It does, in principle. Then I measured how much of it there is inside the window. The frequency that has lost 6 dB to the air is 16 kHz at two metres and 13 kHz at seventeen.
That’s 0.3 octaves of change across 3.6 octaves of distance. Absorption is steep with frequency and gentle with distance, and the audible band simply doesn’t get far enough for the second to matter. The top of the range doesn’t lose 6 dB until the wall is 12 metres away, by which point there’s only half an octave of usable window left.
So there’s no second cue to ablate. There’s one cue and a rounding error. The no colour button stays because the two conditions lying on top of each other is a better way to show that than a paragraph is, but I set out to compare two cues and finished with an argument for why one of them can’t work.
The dead zone is bigger than the live one. The page draws walls from 0.6 to 30 metres and the usable band is 1.4 to 17, so a good fraction of trials are in a region where nobody can do anything. That felt like a design mistake until I looked at what it does: you discover both edges yourself, by being confidently wrong in a way that doesn’t improve with practice. Telling somebody the window is 3.6 doublings wide isn’t the same as running into both walls of it.
The pose had its own listener. The chart draws the simulated listener from idealGuess,
and the capture pose was generating its points with a second implementation I’d written
alongside it. They agreed everywhere except inside the fusion region, where one said 27 per
cent and the other 380. Nothing was wrong with either, they were different models, and only
one of them was the one on the chart. The pose now calls the same function.
What I would do next
The echo threshold is the number the near edge is built on, and I took 8 ms from the literature rather than measuring it. It’s the one quantity here that a page could actually find on a person: present a clap and an echo at a controlled delay and ask whether that was one sound or two, run a staircase on the delay, and the near edge stops being a constant and becomes yours.
That’s a better day than this one, and it’s the same machinery with the answer moved.
The other gap is that a real echo has a direction. Everything here is diotic, the same signal in both ears, so the precedence effect is doing its suppression on the basis of timing alone with no spatial disagreement to work with. Day 58 has the spatialiser that would fix that, and a wall off to one side should behave differently from a wall straight ahead.