What Are Cents in Music?

Q: How do you calculate cents between two frequencies?

Use cents = 1200 x log2(f2 / f1). For example, 440 Hz to 660 Hz is a ratio of 1.5, and 1200 x log2(1.5) is about 702 cents, a pure perfect fifth.

The tiny logarithmic unit that tuners, ear-trainers, and intonation obsessives all speak in.

Short answer: a cent is a unit of musical pitch equal to 1/100 of a semitone and 1/1200 of an octave, so an octave spans exactly 1200 cents. It is a logarithmic unit, which means the same number of cents represents the same perceived interval anywhere on the instrument, whether you are near the bottom of the piano or the top.

Lower (Hz) Upper (Hz)

702 cents · ratio 3:2 · perfect fifth

The distance between two pitches in cents (1200 = one octave), the approximate frequency ratio, and the nearest just interval.

The definition, exactly

By definition, one octave equals 1200 cents, one equal-tempered semitone equals 100 cents, and one cent is therefore 1/1200 of an octave. These are not approximations. They are how the unit is built. Split the octave into twelve equal semitones, then split each semitone into a hundred equal parts, and each of those parts is one cent. Because an octave is a frequency ratio of 2:1, dividing it into 1200 equal steps makes each cent a fixed frequency ratio, the 1200th root of 2, or about 1.0005777895. Multiply a frequency by that number and you have raised it by exactly one cent; do it 1200 times and you have doubled the frequency, one full octave up.

The formula, with a worked example

To find the distance in cents between any two frequencies, the formula is cents = 1200 × log₂(f2 / f1). Take concert A at 440 Hz and the A-sharp just above it at about 466.16 Hz. Their ratio is 1.0595, and 1200 × log₂(1.0595) comes to almost exactly 100 cents, one semitone, as expected. Try a wider gap: a pure perfect fifth is a 3:2 ratio, so from 440 Hz to 660 Hz gives 1200 × log₂(1.5), which is about 702 cents. That 702 is quietly informative, because the equal-tempered fifth on a piano is set to exactly 700 cents. The two-cent gap is the small compromise equal temperament makes so that every key sounds equally usable.

Interval	Semitones	Cents (12-TET)
Unison	0	0
Semitone (half step)	1	100
Whole tone	2	200
Minor third	3	300
Major third	4	400
Perfect fourth	5	500
Perfect fifth	7	700
Octave	12	1200

The cents in that table are the equal-tempered values. Their pure, whole-number counterparts sit a little to one side: a just major third (5:4) is about 386 cents, roughly 14 cents flatter than the tempered 400, which is one reason barbershop and a cappella singers can sound sweeter than a keyboard on the same chord.

Why cents are logarithmic, and why that matters

Pitch perception follows ratios, not differences. Going up an octave always means doubling the frequency, but in absolute hertz that doubling is a small jump low on the keyboard and a huge one high up. A unit measured in raw hertz would describe the same musical interval with wildly different numbers depending on register. Cents fix this by being multiplicative rather than additive. One hundred cents is a semitone whether you are at 100 Hz or 4000 Hz. That register independence is exactly why tuners, intonation charts, and ear-training tools all report deviations in cents: a reading of "+7 cents sharp" means the same amount out of tune no matter which note you played.

Where the unit came from

The cent was introduced by the English phonetician and mathematician Alexander J. Ellis in 1885, in his influential study of the history of musical pitch and in his translation of Helmholtz's On the Sensations of Tone. He needed a practical way to compare tunings and temperaments from many cultures on one common scale. More than a century later the cent remains the standard fine-grained unit for pitch, precisely because its logarithmic design matches how we actually hear intervals.

How small is a cent, really?

A single cent is far finer than ordinary musical judgment. Trained listeners can detect differences as small as about 5 to 10 cents under focused conditions, while in everyday listening a gap nearer 20 to 25 cents is a fairer guide to what most people reliably notice. So a lone cent is imperceptible in isolation, yet cents are still the right ruler for the small deviations that separate "in tune" from "not quite," which is where careful listening earns its keep. If you want to see how fine your own threshold is, that is precisely what a discrimination test measures.

Train your ear in cents

Now put the unit to work. In Hearfork, two tones sit a small, measured distance apart, and your job is to decide whether they match or how they differ. It is a direct way to build a feel for what a handful of cents actually sounds like, the same skill a tuner leans on every time it flags a note a few cents sharp or flat.

Play Hearfork →

Frequently asked questions

How many cents are in a semitone?

Exactly 100 cents in one equal-tempered semitone, or half step. The cent was defined this way so that each of the twelve semitones in an octave divides into a convenient hundred parts.

How many cents are in an octave?

Exactly 1200 cents, twelve semitones of 100 cents each. Because a cent is 1/1200 of an octave, this holds in every register of the instrument.

How do you calculate cents between two frequencies?

Use cents = 1200 × log2(f2 / f1). For example, 440 Hz to 660 Hz is a ratio of 1.5, and 1200 × log2(1.5) is about 702 cents, a pure perfect fifth.

Why measure pitch in cents instead of hertz?

Because pitch perception is logarithmic. The same musical interval spans very different numbers of hertz in low versus high registers but always the same number of cents. Cents give one register-independent yardstick for how far apart two pitches sound.

How many cents can a person hear?

It depends on training and conditions. Trained listeners can catch differences of about 5 to 10 cents when concentrating, while in ordinary listening a difference closer to 20 to 25 cents is a more realistic threshold.