Thought Toys · Waves & rhythm · Exhibit 105

The same difference. Obvious here, invisible there.

Two pairs of grey patches. In each pair, the right half has had exactly the same amount of light added. You will see one of them and not the other — and nothing about the light is different. What changed is what it was added to.

Top: the same difference on two backgrounds. Bottom: two grey ramps

dim pair, difference— bright pair, difference— bright pair would need— steps per doubling—

—

↳ drag the first slider left and watch the right pair's difference reappear — without adding any light

What you're seeing

Each pair is one patch cut in half. The right half of each has had the same amount of light added to it — the second slider sets that amount, and it is the same number on both sides of the stage. It is an amount of light, not an RGB number: the patches are encoded for the screen so the physics is honest. The only thing that differs is the grey underneath.

On the dim pair you see a step. On the bright pair you probably do not. Drag the first slider down and the bright pair's step walks back into view. Nothing was added. The background moved.

Your eye is not measuring the amount. It is measuring the fraction. A difference becomes visible at roughly 2% of the light already there, so a bright background needs a bigger absolute step to clear the same bar. That 2% is the Weber fraction, and it is a model number — your own eye, screen and room will move the line a little.

The two ramps below show what that does to a whole scale. The upper one steps in equal perceptual amounts; the lower one steps in equal amounts of light. Watch the doubling marks. They are evenly spaced on the upper ramp and piled against the dark end on the lower one.

That even spacing is the surprise worth keeping. Every doubling of light costs the same perceptual distance, wherever you start. Doubling a candle and doubling the sun are the same journey to you. That is why the octave, the decibel, the star magnitude and the Richter number are all ratios rather than amounts — each of them is a ruler built to match this eye and this ear.

The rule, exactly. Weber's law (1834) says the just-noticeable difference at intensity I is a fixed fraction of it, ΔI  =  k · I, so climbing n just-noticeable steps from I0 lands on I0(1+k)n — a geometric staircase. Counting steps therefore counts logarithms: n(I1→I2)  =  ln(I2/I1) / ln(1+k),    and  S = (1/k) ln(I/I0) in the small-k limit — Fechner's law, which is Weber's law integrated. The patches are painted through the real sRGB transfer function (1.055 L^(1/2.4) − 0.055), so the light on your screen is the light the model claims; at the default settings the dim pair is 8 code values apart and the bright pair only 3, and both pairs are still genuinely two different greys — what you fail to see up high is your eye, not the screen rounding them together. The companion proof (improve/verify/105-weber-fechner.js, 61 checks) climbs the staircase one step at a time against the closed form, confirms every doubling costs the same 35.0 steps at k = 2% whether you double 0.01 or 500, shows Fechner's continuous law is the limit of the discrete count, and runs four negative controls — chief among them a ratio-blind eye with a constant just-noticeable difference, whose staircase is arithmetic and whose bright-end doublings cost four times as much. Decibels and stellar magnitudes are checked against their published constants rather than gestured at.

Also in Waves & rhythm: Fourier epicycles →

All 13 in Waves & rhythm
  1. 105The same difference. Obvious here, invisible there. — you are here
  2. 12Fourier epicycles
  3. 19Standing waves & normal modes
  4. 21Lissajous figures
  5. 62Ask which slit, lose the wave
  6. 64Outrun your own sound
  7. 66Move, and your clock falls behind
  8. 70The angle past which light refuses to leave
  9. 72Push at the right rhythm and it tears itself apart
  10. 79Twelve perfect fifths overshoot the octave. Every piano pays for it.
  11. 89Every path leads to the same loop.
  12. 91Push one pendulum. It falls still — completely.
  13. 93Every crest is born at the back and dies at the front.

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