Definition
The 12-tone equal temperament (12-TET) divides the octave into 12 equal semitones, each with a frequency ratio of 2^(1/12) ≈ 1.059463. All keys have the same structure: you can modulate anywhere, and fixed-pitch instruments (piano, guitar, synth) play in every key. The price is that no interval except the octave is acoustically pure.
Cents
The cent is the logarithmic unit of intervals: 1 octave = 1200 cents, 1 tempered semitone = 100 cents.
Formula: cents = 1200 · log2(f2 / f1). Examples: 3/2 → 701.96 cents; 5/4 → 386.31; 2^(1/12) → 100.
Perception threshold: two successive sounds can typically be told apart from about 5–10 cents upward (it depends on frequency, timbre and training); played together, even smaller differences are heard as beating.
Comparison with natural intervals
| Interval | Pure ratio | Pure cents | 12-TET cents | Difference |
|---|---|---|---|---|
| Octave | 2/1 | 1200.00 | 1200 | 0 |
| Perfect fifth | 3/2 | 701.96 | 700 | −1.96 |
| Perfect fourth | 4/3 | 498.04 | 500 | +1.96 |
| Major third | 5/4 | 386.31 | 400 | +13.69 |
| Minor third | 6/5 | 315.64 | 300 | −15.64 |
| Major sixth | 5/3 | 884.36 | 900 | +15.64 |
| Minor sixth | 8/5 | 813.69 | 800 | −13.69 |
| Harmonic minor seventh | 7/4 | 968.83 | 1000 | +31.17 |
| Major tone | 9/8 | 203.91 | 200 | −3.91 |
Tempered fifths are almost pure (2 cents off), while major thirds are sharp by almost 14 cents: this is why a major triad on the piano “beats” slightly, while an a cappella choir, which naturally tends toward just intonation, sounds steadier and sweeter.
Beating
If two close frequencies sound together, the amplitude oscillates at a rate equal to their difference. Example: with tempered C4–E4, the 5th harmonic of C4 (1308.1 Hz) and the 4th harmonic of E4 (1318.5 Hz) produce a beat of about 10.4 Hz: this is the shimmer of piano thirds. Piano tuners count beats to temper by ear.
A brief history
The idea of dividing the octave into equal parts is old (Zhu Zaiyu in China, 1584; Simon Stevin in Europe, around 1585), but the widespread use of 12-TET on the piano only took hold in the 19th century. Bach's The Well-Tempered Clavier (Book I, 1722) was probably written for a non-equal well temperament, in which all keys were usable but with different colors (see the card on just intonation and historical temperaments).
Other equal systems
- 24-TET (quarter tones, 50 cents): an approximation used for Arabic music.
- 19-TET, 31-TET: much purer thirds; 31-TET approximates meantone.
- 53-TET: almost Pythagorean and almost just, the theoretical basis of Turkish music.
- Stretch tuning: real pianos have slightly widened octaves (sharpening treble, flattening bass) to compensate for the inharmonicity of the strings.
The cent in production work
- Synth detune: a supersaw uses voices detuned by 5–30 cents. Below 5 cents you get a slow chorus; above 25 you get a “honky-tonk” effect.
- Pitch correction: Melodyne and Auto-Tune think in cents; a voice within ±10–15 cents sounds natural, beyond ±25 it is perceived as out of tune.
- Sample tuning: an 808 that is 30 cents flat relative to the root creates beating with the bass; fix it with the channel's fine tune parameter in FL Studio.
- Microtonal scales: many synths (Pigments, Serum 2, Surge XT) import .scl/.tun files to play in just intonation or in 24-TET.
Example in a song
A cappella groups (e.g. The King's Singers) naturally tend toward pure thirds; the same harmonies played on the piano have the tempered color. Jacob Collier, in his arrangement of “In the Bleak Midwinter”, uses just intonation to let the tuning “drift” to a key a quarter tone away from 12-TET.
Exercise
- On a synth with two sine oscillators, play tempered C4 + E4, then lower the E by 13.7 cents: listen to the beating disappear.
- Create a pad with three oscillators at 0, +7 and −7 cents and compare it with ±20 cents.
- Use a chromatic tuner to measure how many cents off your voice is on a note held for 4 seconds.
Related topics
Pythagorean tuning and comma; Just intonation and historical temperaments; The harmonic series and timbre; Consonance and dissonance.