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 with 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 why equal-temperament thirds shimmer with audible beats.
The compromise interval by interval
Not all intervals are equally bent in equal temperament. The table below shows the typical cent deviations using five-limit just intonation ratios:
| Interval | ET semitones | Just ratio | Deviation |
|---|---|---|---|
| Unison | 0 | 1:1 | 0 ¢ |
| Minor third | 3 | 6:5 | −15.6 ¢ |
| Major third | 4 | 5:4 | −13.7 ¢ |
| Perfect fourth | 5 | 4:3 | +2.0 ¢ |
| Perfect fifth | 7 | 3:2 | −2.0 ¢ |
| Major sixth | 9 | 5:3 | −15.6 ¢ |
| Octave | 12 | 2:1 | 0 ¢ |
The perfect fifth is only 2 cents narrow — far below the typical audibility threshold of about 5 cents. Thirds and sixths carry a much larger gap, which is the fundamental trade-off of equal temperament.
Why equal temperament won
Earlier historical tuning systems — meantone, Pythagorean, various well temperaments — also tried to balance pure intervals against practical key-modulation. Equal temperament does not optimise any single key but instead distributes the error equally across all twelve. This means every key sounds equally in tune (or equally out of tune), which became essential as composers explored distant key relationships and keyboard instruments became standard.
Who cares about the difference today
- A cappella ensembles and choirs tune by ear and naturally drift toward just ratios — particularly for thirds and sixths — producing the characteristic purity of unaccompanied choral sound.
- String quartets playing without piano can subtly adjust intonation, and experienced ensembles actively match the harmonic series in sustained chords.
- Barbershop harmony is explicitly trained on locking just-intonation chords, and practitioners learn to hear the “ringing” of a locked chord versus a slightly tempered one.
- Synthesiser designers and microtonalists use this comparison to design or evaluate alternative tuning systems.
Everything runs locally in your browser — enter any root frequency to explore the relationships in a key you are working in.