What is a cent in music?

The definition

In equal temperament, the tuning system of pianos and nearly all Western music since the nineteenth century, the octave is divided into twelve equal semitones. Each semitone is divided into one hundred cents. So a semitone is 100 cents, a whole tone is 200, a perfect fifth is 700, and an octave is 1,200. The unit was proposed by Alexander Ellis in the 1880s so that tuning systems could be compared with ordinary arithmetic.

"Equal" here means equal in ratio, not in Hertz. Each semitone multiplies the frequency by the same factor — the twelfth root of two, about 1.0595 — so each cent multiplies it by the 1,200th root of two, about 1.000578. Going up a cent always means a frequency about 0.058 per cent higher than where you were, whatever that was.

Why the unit is logarithmic

The ear hears pitch by ratio. A note an octave above another has twice its frequency, whether that is 110 to 220 Hz or 1,760 to 3,520 Hz, and both jumps sound like the same interval. A unit that measured pitch difference in Hertz would call the first jump 110 and the second 1,760, which matches nothing anyone hears. Cents measure the ratio, so the two octaves are both 1,200 cents.

The formula, in words: take the higher frequency divided by the lower, take the base-two logarithm of that ratio, and multiply by 1,200. For a ratio of exactly two, the logarithm is one and the result is 1,200 cents — an octave. For the 1.0595 ratio of a semitone, it is 100.

The two twenty-Hertz rows are the point of the whole unit. The same Hertz error is a gross mistake at the bottom of the range and a modest one at the top. Scored in Hertz, a pitch test would punish low notes and flatter high ones. Scored in cents, a miss is a miss of the same size wherever it happens.

How small a difference can you hear?

The just-noticeable difference for pitch depends on frequency, loudness, duration and whether the two tones are heard together or one after another. For sustained tones in the middle of the range, trained listeners under good conditions are commonly cited as detecting differences of roughly five to ten cents; figures as low as three or four cents appear in some laboratory conditions. Untrained listeners typically need more, and everybody needs more when the tones are short, quiet or separated by silence.

That last case — separated by silence — is the one that matters for pitch memory. Hearing that two simultaneous tones beat against each other is easy. Hearing a tone, waiting, and then judging whether a second tone matches it is a different task, and the error grows with the gap. Most people are surprised by how far they drift in a few seconds; it is normal to be fifty cents or more off on a first attempt and to improve quickly with feedback. How to improve pitch recognition covers what to do about it.

Cents in practice

• Tuners show the deviation of a played note from the nearest equal-tempered pitch, usually on a scale of plus or minus 50 cents. A string player aims for the needle at zero; many consider within five cents in tune.
• Just intonation is described in cents relative to equal temperament. A pure major third is about 14 cents narrower than the equal-tempered one, which is why a well-tuned barbershop chord sounds different from a piano chord.
• Intonation trainers and singing apps report how many cents sharp or flat your voice was, which is the only feedback precise enough to improve on.
• Pitch-memory tests score the distance in cents between the tone you heard and the one you reproduced. Percore's Pitch session does exactly this: a sine tone plays once, you find it again by dragging a waveform, and the accuracy comes from the cents distance — being about an octave off scores zero. How Percore scores has the detail.