Just Intonation Cents Calculator

Cents and Frequency Ratios

Music intervals are perceived logarithmically. Each octave doubles the frequency, yet our ears hear every octave as the same size of step, so the useful unit has to be logarithmic too. Cents are that unit, and they make intervals comparable across tuning systems.

Formula

cents = 1200 × log₂(f₂ / f₁)

Inversely:

f₂ / f₁ = 2^(cents / 1200)

One octave = 1200 cents. One equal-tempered semitone = 100 cents. A skilled musician can hear differences as small as 5 to 10 cents.

Just Intonation vs Equal Temperament

Just intonation tunes intervals to small whole-number frequency ratios, the ratios that produce the cleanest, beatless harmonies. Equal temperament divides the octave into 12 equal steps, slightly detuning every interval except the octave so that any key sounds equally good.

Look down the Difference column and one thing stands out: the fifth and the fourth are within 2 cents, which nobody can hear, while the thirds and sixths are off by 14 to 16 cents, which everybody can. That is the whole bargain of equal temperament in one column. The intervals that hold a chord together were left almost alone, and the intervals that give it its color took the damage.

Why Equal Temperament Won

A keyboard tuned in just intonation sounds beautiful in one key and progressively worse as you modulate away from it. Bach’s Well-Tempered Clavier showcased the early compromises that led to today’s 12-tone equal temperament, one slightly detuned scale that works in every key.

Worked Examples

• A 3:2 perfect 5th (440 Hz to 660 Hz): cents = 1200 × log₂(660/440) = 1200 × log₂(1.5) = 701.96 cents.
• A 5:4 major 3rd (C to E in just intonation): 386.3 cents, about 14 cents flatter than the equal-tempered E. Play both against a sustained middle C and the just third goes still, while the tempered one beats around ten times a second. That is the fastest beating anywhere in a tempered triad, and it is why a piano tuner sets thirds last.

Cents in Practice

Limitations

Cents describe the size of an interval and nothing else. Two intervals with the same cent value have the same frequency ratio, by definition, so cents cannot tell you how an interval will actually sound on a given instrument.
What they leave out is timbre. A 7:4 septimal seventh at 968.8 cents locks in against a harmonically rich tone like a bassoon, while the 12-TET minor seventh at 1000 cents beats audibly against the same tone, and they are only 31 cents apart. On a pure sine wave, with no upper harmonics to beat against, that difference nearly disappears.