Cents Between Frequencies

Cents and musical intervals: formula and example

To measure the interval between two frequencies in cents, use cents = 1200 · log₂(f₂/f₁). An octave comes out to 1200 cents. A tempered semitone is exactly 100. The just perfect fifth (3:2) lands around 702 cents, and the just perfect fourth (4:3) at roughly 498.045. Put 440 Hz next to 466.16 Hz and you get about 100 cents, which is one semitone. Trained ears pick up deviations as small as 5 cents or so.

Applications

When it comes to fine tuning, cents are the common language everyone speaks. Professional tuners like the Korg OT-120 show deviations in cents. Microtonal composers such as Harry Partch and Wendy Carlos build scales whose steps aren't 100 cents. Ethnomusicologists reach for cents to write down the quarter-tones of Arabic maqam (around 50 cents) or the Indian shruti. Violinists nudge their harmonics a few cents past equal temperament, and ear-training software grades pitch accuracy the same way.

FAQ

Why use a logarithm? Our ears hear pitch logarithmically. Double the frequency and it always sounds like one octave, no matter which note you started from.

How many cents can the human ear distinguish? Most people notice somewhere around 10-15 cents. Trained ears get down to about 5 cents when conditions are right.

Why is the tempered fifth 700 cents instead of 702? Equal temperament shaves about 2 cents off the just 3:2 ratio. That small compromise is what lets every key sound equally in tune.

Can cents be negative? Yes. A negative result just means f₂ sits below f₁.