Pitch is multiplicative — double the frequency for an octave, double again for the next — so to lay intervals on an even ruler we take the LOGARITHM. The unit is the CENT: 1200 to an octave, 100 to a semitone. It lets you say a note is ‘5 cents flat’ and reveals that the pure fifth (3:2) is actually 702 cents, two cents sharp of the piano’s 700. Slide a ratio and read it in cents.
Because pitch perception is logarithmic, intervals are measured in CENTS: cents = 1200·log₂(ratio), so an octave (2:1) is exactly 1200 cents and an equal-tempered semitone is 100. This turns multiplicative frequency ratios into an additive ruler — you can add intervals by adding cents and quantify how ‘out of tune’ a note is. It exposes the small gaps temperament hides: the just perfect fifth (3:2) is 702 cents, 2 cents sharp of the tempered 700; the just major third (5:4) is 386, 14 cents flat of the tempered 400. A fail-loud self-check throws unless the octave is 1200 cents and the pure fifth is 702. ◆ real music theory, node-verified.
The exact cents definition (1200·log₂); the just-out-of-tune amounts are exact, and the ear’s just-noticeable-difference is roughly 5–10 cents — the log-ruler-of-pitch is exact and universal in tuning.