Cents and Musical Intervals Explained

How far apart are two notes, really? A tuner answers in cents, and one octave always spans exactly 1200 of them. That single fact lets you compare a bass note and a whistle with the same ruler.

This guide shows what a cent is and how to find the cents between two frequencies. It also shows why some “perfect” intervals sit a few cents away from the piano. You will also learn how to turn cents back into hertz for tuning and sound design.

What a Cent Measures

A cent is one hundredth of an equal-tempered semitone. Stack 100 cents and you get one piano key step. Stack 1200 cents and you get a full octave, where the frequency doubles.

Cents measure a ratio, not a gap in hertz. This matches how we hear pitch. The step from 110 Hz to 220 Hz sounds the same size as the step from 440 Hz to 880 Hz, because both are 2:1.

That is why cents beat hertz for tuning work. A tuner that says “12 cents flat” means the same thing on a low E string and on a high flute note. A hertz reading would need a different meaning at every pitch.

Each semitone multiplies the frequency by the twelfth root of 2, about 1.0595. Twelve of those steps in a row multiply it by exactly 2. So cents are simply a finer ruler laid over that same curve, with 100 marks inside every semitone.

How Is the Interval in Cents Calculated?

Divide the higher frequency by the lower one, take the base-two log, and multiply by 1200. The formula is cents = 1200 x log2(f2 / f1).

1. Pick a reference. Call the first frequency f1. For a tuning check, f1 is the target pitch, such as 440 Hz.
2. Find the ratio. Divide f2 by f1. For 440 Hz and 466.16 Hz, the ratio is about 1.0595.
3. Take the base-two log. log2(1.0595) is about 0.0833, which is one twelfth of an octave.
4. Multiply by 1200. The result is 99.99 cents, so the two notes are one semitone apart.
5. Read the sign. A positive answer means f2 is higher. A negative answer means f2 is lower.

Now try a small gap. From 440 Hz to 445 Hz is 19.56 cents. That is about a fifth of a semitone, and a trained ear notices it. For the reverse direction, 440 Hz down to 415.30 Hz gives about -100 cents.

Calculators without a log2 key still work. Use log(ratio) divided by log(2), then multiply by 1200.

The same number converts to other units in one step. Divide cents by 100 for semitones, so 386.31 cents is 3.86 semitones. Divide cents by 1200 for octaves, so 700 cents is 0.583 of an octave.

Why Do Just Intervals Differ From Equal Temperament?

Just intervals use whole-number ratios, while equal temperament splits the octave into 12 identical steps. The two systems agree on the octave and almost nothing else.

Any just ratio turns into cents with the same formula. For the fifth, 1200 x log2(3/2) gives 701.96. For the major third, 1200 x log2(5/4) gives 386.31. The table below lists the common ones.

The equal-tempered fifth is exactly 700 cents, only 1.96 cents narrower than the just fifth. Above A440, the just fifth is 660 Hz, and the piano fifth is 659.26 Hz.

Thirds show a bigger gap. The piano major third is 400 cents, which is 13.69 cents wider than the just 5:4 third.

Equal temperament gives up pure thirds so that every key sounds the same. That trade lets music move freely between keys on one piano.

HyperPhysics notes that the piano fifths and fourths stay within the 5-cent margin of their just versions. The thirds and sixths miss that margin by a wide stretch, at 13 to 16 cents.

Turning Cents Back Into Hertz

To reverse the math, raise 2 to the power of cents divided by 1200. That gives the frequency ratio. Multiply the reference frequency by that ratio to get the new pitch.

For example, 50 cents is a ratio of 1.0293. Adding 20 cents to 440 Hz gives 445.11 Hz, and subtracting 20 cents gives 434.95 Hz. Our guide to note frequencies and the A440 standard shows how each named note gets its hertz value.

How Many Cents Can a Listener Hear?

HyperPhysics gives about 5 cents as a reasonable estimate of the just noticeable pitch difference. It also cites measurements near 0.5 percent, about 8 cents, for pure tones between 1 and 4 kHz. The threshold shifts with loudness, tone length, and musical training.

Five cents covers very different hertz widths across the range. At 110 Hz it is only 0.32 Hz, but at 1760 Hz it is 5.09 Hz. A band tuned to A442 instead of A440 sits 7.85 cents sharp.

Use this when reading a tuner. A needle 3 cents off sits under the 5-cent estimate. A needle 15 cents off is three times that estimate, so most listeners will catch it.

Cents Traps and Better Habits

Most cents errors come from treating pitch as a straight line in hertz. The table pairs each trap with a better habit.

Cents also appear in sound design, where oscillator detune is set in cents. Timing effects use milliseconds instead, as our guide to syncing delay time to tempo explains.

Try it with your own notes. Enter a target pitch and a measured pitch in the cents calculator to see how sharp or flat you are.

Cents and Intervals: Frequently Asked Questions

Sources and Further Reading

This article explains the math of musical intervals for learning and tuning practice. Hearing thresholds vary by listener and by sound. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.