Why do your ears measure sound with an invisible ruler?

Because your senses tend to notice changes in proportion rather than in absolute amounts, and a ruler built on proportions is a logarithmic one. In the 19th century, Ernst Weber found that the smallest difference between two weights a person can notice is roughly proportional to the weights themselves: if 105 grams can only just be told apart from 100 grams, then at 200 grams it takes 210. His student Gustav Fechner went further and argued that sensation grows as the logarithm of the stimulus, so that equal ratios feel like equal steps.

Hearing shows this most clearly with pitch. Our ears perceive equal ratios of frequency as equal differences in pitch, so going up an octave, a doubling of frequency, always sounds like the same-sized step. That is how a logarithmic scale works: each step multiplies the previous value by the same amount, so 1, 2, 3, 4, 5 are not evenly spaced on it, while 1, 10, 100, 1000 are. Sound level is measured this way too, in decibels, a logarithmic unit, and so are earthquake magnitudes, pH and the brightness of stars.

The ruler isn't perfect. Weber's law does not quite hold for loudness: it is a fair approximation at higher intensities but not at lower ones, and since the 1960s many researchers have considered Stevens's power law a more general rule than Fechner's logarithm. Still, logarithms may come naturally to people: studies of young children in an isolated tribe found logarithmic scales to be the most natural way for them to display numbers.