Temperament Cents Calculator
A piano is out of tune with itself on purpose. Equal temperament divides the octave into twelve identical steps so that every key works, and the price is that no interval except the octave is pure. This puts numbers on that compromise: how many cents each interval is from its just ratio, and how many beats a second you would hear between the two on your chosen root.
What this generator does
Computes each just interval from its whole-number ratio, computes the equal-tempered interval from the twelfth root of two, and reports the difference between them both as cents and as the beat rate between the two pitches.
How to use this tool
- Enter the root frequency you are tuning from.
- Press Work it out.
- Read the cents deviation for each interval.
- Read the beat rate — that is what you would actually hear.
Understanding the controls
- Root frequency
- The pitch everything is measured from, in hertz. 440 is concert A; 432 and 415 are common alternatives and change the beat rates without changing the cents.
Common use cases
- Seeing exactly how far a tempered third is from a pure one
- Working out the beat rate for tuning an interval by ear
- Explaining why barbershop and choral thirds sound different from piano ones
- Comparing tuning at 440 against 432 or 415
- Teaching cents as a logarithmic unit with real frequencies
How this generator works
A cent is a twelve-hundredth of an octave on a logarithmic scale, so the deviation of any interval is 1200 times the base-two logarithm of the ratio between the two frequencies. A just fifth is 3:2 and lands 1.96 cents above the tempered one; a just major third is 5:4 and lands 13.69 cents below, which is why thirds are the audible compromise. Beats are simpler: two pitches a few hertz apart produce a pulsing at exactly that difference, and that is the number a tuner counts. Before display every cent figure is recomputed from the two frequencies rather than from the ratio, the equal-tempered pitches are confirmed to land on whole semitones, and every beat rate is recomputed from the pitches it belongs to.
Randomness and fairness
Nothing here is random. A root frequency gives exactly one table, computed from ratios and logarithms rather than measured or sampled.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Only twelve-tone equal temperament and five-limit just intonation are compared. Meantone, well temperaments and other divisions of the octave are different questions.
- The just ratios chosen are the common five-limit ones; several intervals have more than one reasonable just form and the tritone especially is debatable.
- Beat rates assume two steady sine-like tones. Real instruments have overtones that beat too, which is what a tuner is usually listening to.
- This computes tuning; it does not produce sound, and nothing here can tune an instrument for you.
Privacy and your data
The calculation runs in your browser and the frequency you enter is never transmitted, stored or included in analytics.
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