What is the difference between equal temperament and just intonation?

Just intonation tunes intervals to simple whole-number frequency ratios such as 3:2 for a perfect fifth, which sound pure and beatless. Equal temperament divides the octave into twelve identical steps so music can be played in any key, at the cost of detuning most intervals slightly.

A cent is one hundredth of an equal-temperament semitone, so an octave contains 1200 cents. It is a logarithmic unit, meaning equal cent differences sound like equal pitch steps regardless of the absolute frequency. Around 5 cents is the threshold most listeners can detect.

Which interval is most out of tune in equal temperament?

The major third is the worst, sitting about 14 cents sharp of the pure 5:4 ratio. The minor third is about 16 cents flat of 6:5. These tempered thirds are why barbershop and a cappella singers, who tune by ear, sound purer than a piano.

Why is the perfect fifth almost perfect in equal temperament?

The equal-temperament fifth is only about 2 cents narrower than the pure 3:2 ratio, far below the audible threshold. That tiny error is why fifths and octaves on a piano sound clean while thirds sound slightly rough.

How is the equal-temperament frequency computed?

Each semitone multiplies the frequency by the twelfth root of two, about 1.059463. So the frequency n semitones above the root equals root times two to the power n divided by twelve. Just-intonation frequencies instead multiply the root by a small whole-number ratio.

Equal Temperament vs Just Intonation Comparator

Western instruments are tuned in twelve-tone equal temperament, where every semitone is exactly the same size so you can play in any key. Pure or just intonation instead tunes each interval to a simple whole-number ratio that sounds beatless. This tool lays the two systems side by side so you can see precisely how much each interval is bent.

How it works

In equal temperament the octave is split into twelve identical steps. The frequency n semitones above a root of f hertz is:

f_et = f × 2^(n/12)

In just intonation each interval uses a small whole-number ratio derived from the harmonic series, for example 3:2 for the perfect fifth and 5:4 for the major third:

f_just = f × (ratio numerator / ratio denominator)

The difference is measured in cents, a logarithmic unit where an octave is 1200 cents and a semitone is 100 cents:

cents = 1200 × log2(f_just / f_et)

A positive value means the pure interval is sharper than the tempered one; a negative value means it is flatter.

Worked example

Take A4 = 440 Hz. The equal-temperament major third (4 semitones up) is 440 × 2^(4/12) ≈ 554.37 Hz. The pure 5:4 major third is 440 × 5/4 = 550.00 Hz. The difference is 1200 × log2(550 / 554.37) ≈ −13.7 cents: the pure third is nearly 14 cents flatter than the piano’s third, which is exactly why equal-temperament thirds shimmer with beats.

Tips and notes

Octaves and perfect fifths come out almost identical in both systems, while thirds and sixths show the biggest gaps — this is the central compromise of equal temperament. The ratios used here are the standard five-limit just intonation set. Choirs and string quartets drift toward these pure ratios by ear, which is why unaccompanied ensembles can sound sweeter than fixed-pitch keyboards. Everything runs locally in your browser.