Equal temperament
Equal temperament divides the octave into twelve equally large semitones. Each semitone has the same frequency ratio, the twelfth root of two, about 1.0595. In return no interval except the octave is quite pure.
- Semitone
- 100 cents, factor 1.0595 (twelfth root of 2)
- Octave
- 1200 cents, factor 2
- Fifth
- 700 cents, about 2 cents smaller than pure
- Major third
- 400 cents, about 14 cents larger than pure
- Counterparts
- just intonation, meantone, well-tempered
The principle: twelve equal steps
An octave means double the frequency. If you divide it into twelve equal semitones, each semitone must multiply the frequency by the same factor f, and f to the twelfth power must give exactly two. From this follows f = 2^(1/12), the twelfth root of two, about 1.059463. Starting from the standard pitch A4 = 440 Hz you get, for example, C5 = 440 · 2^(3/12) = 523.25 Hz, because the C lies three semitones above the A. The page Standard pitch calculates the same for C4 and E4.
C major triad in equal temperament (distances in cents)
The numbers under the notes are the distances in cents as the tuning fixes them. The marked major third C–E has 400 cents, the pure major third (frequency ratio 5:4) about 386 cents. On the piano it therefore sounds about 14 cents too high.
The price: no pure third
The pure intervals arise from simple number ratios, as they occur in the overtone series. Twelve pure fifths, however, exceed seven octaves by the Pythagorean comma of about 23.46 cents. Equal temperament distributes this error over all twelve fifths, just under 2 cents each. That is how the circle of fifths closes, and all keys sound the same. The table shows the result.
| Interval | Pure ratio | Pure in cents | Equal-tempered | Deviation |
|---|---|---|---|---|
| Octave | 2:1 | 1200.00 | 1200 | 0 |
| Fifth | 3:2 | 701.96 | 700 | −1.96 |
| 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 |
That the fifth deviates from pure by only just under 2 cents is hardly heard by anyone. The major third, by contrast, lies about 14 cents too high and the minor about 16 cents too low. These are the values that the article Equal, meantone, just — why your piano is slightly out of tune names too. The ear gets used to it, but thirds held for a long time audibly beat.
What “tempered” means
In pure tuning a chord sounds calm and steady; in equal temperament the intervals beat a little. The deviation is equal in cents: every fifth is too small by the same amount, every major third too large by the same amount. Counted in beats per second the beating becomes faster with rising pitch. The word “temperament” here means the procedure of “tempering”, that is moderating, the intervals, not warmth.
The advantage that every key can be played and that enharmonic equivalents such as C♯ and D♭ give the same note has to be set against the loss of the character of individual keys. How that went in the history of tuning is in the glossary entry Well-tempered: Bach's time did not necessarily mean equal temperament by it, and equal temperament became the standard only in the 19th century.
Common mistakes
- Equating equal temperament and well-tempered: the two do not mean the same.
- Assuming the third is “wrong”: it is a compromise that makes every key usable.
- Confusing cents with hertz: cents describe a ratio, not a frequency.
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Related terms
Well-temperedJust intonationPythagorean commaCentSemitoneStandard pitchSchismaEnharmonic equivalenceCircle of fifths
Editor: Maik Krohm · reviewed on 3 October 2026
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