Tuning Temperament Comparison
Equal temperament divides the octave into twelve identical steps so that every key sounds the same, and it pays for that with every interval except the octave. The pure major third is a 5:4 ratio; the tempered one is fourteen cents sharp of it, which is enough to hear beating. This computes both sets of frequencies from a root you choose and shows the deviation in cents for each interval.
What this generator does
Computes each interval twice from your root: once as a pure whole-number ratio and once as a whole number of equal semitones. Both are converted to cents, and the difference between them is the compromise equal temperament makes for that interval.
How to use this tool
- Enter a root frequency; 440 Hz is the usual concert A.
- Compare the pure and tempered frequencies for each interval.
- Read the cents column — that is the compromise in each case.
- Notice the fifth is barely off and the third is not.
Understanding the controls
- Root frequency
- The note everything is measured from. Only the ratios matter for the cents figures, so changing this scales the frequencies without changing the deviations.
Common use cases
- Seeing exactly how far equal temperament sits from pure intervals
- Teaching cents as a logarithmic measure of pitch difference
- Comparing frequencies for a tuning or intonation exercise
- Explaining why thirds beat on a piano and fifths do not
- Producing reference frequencies for just intonation
How this generator works
Cents are a logarithmic measure: twelve hundred to the octave, so an interval's size in cents is twelve hundred times the base-two logarithm of its frequency ratio. A pure fifth is 3:2, which comes to just over 701 cents; equal temperament gives it exactly 700. A pure major third is 5:4, about 386 cents against the tempered 400. Before display every cents figure is recomputed from its own ratio, every frequency is checked against the root, and the octave is confirmed to be exact in both systems — which it is, and it is the only interval that is.
Randomness and fairness
Nothing here is random. A root frequency determines every figure in the table.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Only twelve-tone equal temperament is compared; historical temperaments like meantone and well temperaments distribute the compromise quite differently.
- The just intervals shown are the simplest ratios for each interval, and other tunings choose different ones — particularly for the seventh.
- Real instruments are not perfectly in tune with themselves, and stretched tuning on pianos deliberately widens octaves.
- Cents describe pitch difference, not perceived consonance, which also depends on timbre and register.
Privacy and your data
Every figure is computed in your browser. The frequency you enter is never transmitted, stored or included in analytics.
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