Boneyard Tools

Cents, ratios and equal temperament

How the cent divides the octave, why just intervals have simple ratios, and how equal temperament spreads the tuning error evenly.

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What a cent actually measures

A cent is a logarithmic unit of pitch: one twelve-hundredth of an octave. Because our sense of pitch is logarithmic, doubling a frequency always sounds like the same distance, an octave, whether you start at 55 Hz or 5000 Hz. The cent captures that even spacing. Splitting the octave into 1200 equal steps means a single semitone is 100 cents and a quarter tone is 50, which gives a fine ruler for describing intervals that a raw hertz difference cannot provide.

Just intervals and their simple ratios

The intervals that ring cleanest are built from small whole-number frequency ratios. An octave is 2 to 1, a perfect fifth is 3 to 2, a perfect fourth is 4 to 3, and a major third is 5 to 4. When two notes lock into one of these ratios their overtones align and the beating between them disappears. Converting those ratios to cents gives 1200, about 701.96, about 498.04 and about 386.31 respectively, which is why just intervals rarely land on round cent values.

Why equal temperament exists

Pure ratios do not stack neatly. Twelve just fifths overshoot seven octaves by about 23.5 cents, a gap called the Pythagorean comma, so an instrument tuned in pure fifths cannot play in every key. Equal temperament solves this by making every semitone identical at exactly 100 cents, which pulls the fifth down to 700 cents and pushes the major third up to 400. Each interval is slightly impure, but the error is shared evenly and music sounds acceptable in all keys.

Using cents to check intonation

Enter a measured pitch as Frequency 2 and the target pitch as Frequency 1 to see how many cents sharp or flat you are. A guitar string reading plus 8 cents is noticeably sharp and wants loosening, while a choir landing within about 3 cents of pure will sound locked in. Comparing the ratio the tool reports against a known just ratio also tells you whether a synth patch or sample is tuned to just intonation or to standard equal temperament.

Frequently asked questions

How many cents off is actually audible?

On sustained, harmonically rich tones many listeners notice errors of about 5 to 10 cents, and trained ears do better. On short or percussive sounds the threshold is far more forgiving, so a few cents rarely matters.

Why do some results end in .96 or .04?

Those come from pure just ratios, which do not divide the octave into whole cents. A 3 to 2 fifth is 701.96 cents and a 4 to 3 fourth is 498.04, so the fractional part is expected, not a rounding bug.

Is 440 Hz the only possible reference?

No. A4 at 440 Hz is the common concert standard, but you can enter any reference, such as 432 Hz or a baroque 415 Hz, and the cents result reflects the interval between whatever two frequencies you provide.