The Second Space · Mind · the science of the felt world

A Candle in Daylight

Light one candle in a dark room and the room changes. Light the same candle at noon and nothing happens — the flame is still there, pouring out the same light, and you cannot see it at all. Your senses do not report how much. They report how much more, as a fraction of what was already there.

This is the oldest quantitative law in psychology, and you can measure it on yourself in the next two minutes. The smallest change you can just barely notice — the just-noticeable difference — is not a fixed amount. It is a roughly constant percentage of the starting level. A gram added to a letter, you feel; a gram added to a suitcase, you don't. The rule that a candle in a bright room is worth less than a candle in a dark one is called Weber's law, and it is why almost every scale humans built for their own senses — loudness, star brightness, musical pitch — turns out to be logarithmic. Below, you'll make your own senses draw the line.

First, the candle

Here is a room. Drag the background up from dark to bright, then press + one candle — a fixed amount of extra light, the same every time. Watch what the same candle is worth against different backgrounds. The number that matters is the ratio — the extra light as a fraction of what was there.

Now measure your own threshold

Two grey squares will appear, almost the same. Click the brighter one (or press ← / →). You'll be right, then right, and the squares will creep closer together until you're guessing; then they'll ease apart again. This is a staircase — it hunts for the exact difference you can just detect. Run it at a dim, a medium, and a bright background, and your own answers will plot Weber's law: the threshold rises in step with the background.

Each finished run drops one point below: the background you ran at, against the threshold difference your eyes needed there. Weber's law predicts they fall on a straight line through the origin — ΔI = k · I — whose slope k is your Weber fraction.

Why this makes every scale logarithmic

Fechner saw the consequence in bed on the morning of 22 October 1850. If every just-noticeable step is a constant fraction of the stimulus, then to climb by equal felt steps you must multiply the stimulus by equal ratios. Stack the steps up and sensation grows as the logarithm of the stimulus: S = k · ln(I / I₀). Doubling is doubling whether you start quiet or loud; equal ratios, equal steps.

So when humans needed to write their sensations down as numbers, they kept reinventing the logarithm without meaning to. Move the slider through the physical world and watch three of those invented scales move in lockstep — because all three are the same compression your eye and ear were already doing.

The star scale is the eerie one. Pogson fixed it in 1856 so that five magnitudes mean exactly a hundredfold in brightness — one magnitude is a factor of 100^1/5 = 2.512. That was chosen to match a scale the eye had used for two millennia. The Sun and the full Moon differ by about 14 magnitudes; in raw light that is a ratio near 400,000 : 1, which your eye quietly folds into fourteen even-feeling steps. The compression isn't a convention astronomers imposed. It's the one your retina was already doing.

The twist: the sense that exaggerates

Fechner's law reigned for a century, and then S. S. Stevens broke it — gently. He simply asked people to put a number on how intense a thing felt: this light is "10," now here's another, give it a number. The answers didn't fit a logarithm. They fit a power law, S = k · Iⁿ, with an exponent that changes from sense to sense — and that exponent tells you the personality of each sense.

Below 1, a sense compresses: double the light and it looks far less than twice as bright (brightness n ≈ 0.33). At 1, it's honest (a line's length, n ≈ 1.0). Above 1, a sense expands — and one sense expands violently. Electric shock has n ≈ 3.5: a small rise in current feels like an enormous jump in pain. That isn't a flaw. A sense that exaggerates rising danger is a sense that keeps you alive. Pick a sense and double the world:

So which law is true — Fechner's logarithm or Stevens' power? The honest answer is that after a century and a half the question is still open. Both fit the middle of the range; both fail at the edges; the number you get depends on how you ask (magnitude estimation has real, documented biases). The reconciliations have their own literature. What no one disputes is the thing your own staircase showed you at the top of this page: the senses trade in ratios, and that single fact reshapes the world before you ever perceive it.

The two tables, laid open

Weber fractions — the fraction of the base you can just detect. Smaller means keener. These are averaged, condition-dependent teaching values; sources disagree (especially taste), and some senses only obey Weber's law over a middle range.

Stevens exponents — the personality of each sense. Below 1 compresses, above 1 expands.