Cents, Equal Temperament, and Just Intonation

Alexander J. Ellis developed the cent scale in the nineteenth century. For two frequencies f₁ and f₂, the interval is:

cents = 1200 × log₂(f₂ / f₁)

• An octave, with a frequency ratio of 2:1, is 1,200 cents.
• An equal-tempered semitone is 100 cents.
• One cent is a ratio of 2^(1/1200) ≈ 1.0005778.

Because the scale is logarithmic, the same number of cents represents the same musical interval at any frequency, even though the absolute difference in hertz changes.

Most modern pianos, fretted guitars, and electronic tuners use 12-tone equal temperament. It divides the octave into twelve equal frequency ratios:

semitone ratio = 2^(1/12) ≈ 1.059463

Using A4 = 440 Hz, as specified by ISO 16, selected notes are:

The advantage is consistency: every key has the same interval pattern. The compromise is that most intervals differ slightly from simple whole-number frequency ratios.

Just intonation builds intervals from simple ratios such as 3:2, 4:3, and 5:4.

The difference in a major third is large enough to be audible in exposed harmony. Singers and flexible-pitch instruments can adjust toward simple ratios, reducing beats and producing a more settled chord.

A set of just ratios works around a particular tonal centre. Move to a distant key and the same fixed pitches no longer preserve all of those ratios. Keyboard and fretted instruments therefore need a compromise that works in every key.

Equal temperament makes each semitone identical and slightly narrows the pure 3:2 fifth. Earlier “well temperaments” also allowed performance in many keys but retained different interval colours by key; Bach's Well-Tempered Clavier should not be treated as proof that he used today's mathematically equal 12-tone system.

Pitch discrimination depends on frequency, sound level, duration, timbre, musical context, and training. Psychoacoustic literature commonly places fine discrimination in the single-digit-cent range under favourable conditions, while some listeners need a larger change.

A tuner reading is therefore not an absolute verdict on musical quality. A sustained unison exposes small offsets more clearly than a brief note in a complex mix, and performers may deliberately adjust intervals away from equal temperament.

• Compare a recorded pitch with a tuning reference.
• Detune doubled tracks by a few cents to widen a sound.
• Measure the difference between equal-tempered and just intervals.
• Design microtonal scales such as 22- or 31-tone equal divisions.
• Describe pitch bends and tuning offsets independently of octave.

NanToo's Chord Progression Player lets you hear chord progressions and transpose them directly in the browser.

• cents = 1200 × log₂(f₂/f₁).
• An octave is 1,200 cents; an equal-tempered semitone is 100 cents.
• Equal temperament supports every key with one consistent compromise.
• Just intonation uses simple ratios and can produce cleaner sustained harmony.
• Audibility depends on the listener, sound, and musical context.