Cents & Musical Interval Calculator
Enter two frequencies to get the interval between them in cents, or convert cents and frequency ratios back and forth. There are 1200 cents in an octave and 100 in an equal-tempered semitone.
How this differs from the note-name converter
The Note Frequency Converter converts between note names or MIDI numbers and frequencies, and shows cents only as the offset from the nearest note. This calculator expresses the interval between any two frequencies in cents. Use it for a tuner reading, a measured frequency, or the interval of a tuning ratio such as 3:2.
What is a cent?
A cent is a unit for an interval (the ratio of two frequencies). One octave (twice the frequency) is 1200 cents, and an equal-tempered semitone is 100 cents. The interval from frequency f₁ to f₂ is
cents = 1200 × log₂(f₂ ÷ f₁)
and conversely f₂ = f₁ × 2^(cents ÷ 1200). These definitions were checked against explanatory references (Wikipedia, "Cent (music)" and "Interval (music)").
The sign
If f₂ is higher than f₁ the result is positive, and if it is lower the result is negative. This is this calculator's convention, set by the formula. The same frequency gives 0 cents. A value that rounds to 0 is shown without a sign.
The four calculations
- Frequencies → cents: the difference in cents from f₁ and f₂.
- Cents → frequency: f₂ from the reference frequency f₁ and the cents.
- Ratio → cents: a frequency ratio (such as 3/2) in cents.
- Cents → ratio: cents as a frequency ratio.
Precision and "exact" values
Logarithms and powers are computed in floating point and then rounded to 12 significant digits. When the frequency ratio is a whole-number power of 2 (2, 1/2, 4 and so on), the result is exact: a whole multiple of 1200 cents. Everything else is marked as a rounded value.
The nearest equal-tempered interval
The interval is shown as a number of semitones plus an offset in cents. The offset is from −50 up to (but not including) 50, and exactly 50 cents counts as the higher one (the same rule as in the Note Frequency Converter). For 0 to 12 semitones, the name of the interval is added too (perfect unison, minor second, major second, minor third, major third, perfect fourth, tritone, perfect fifth, minor sixth, major sixth, minor seventh, major seventh, perfect octave).
Examples
| Input | Result |
|---|---|
| 440 Hz → 880 Hz | +1200 cents (one octave, exact) |
| 440 Hz → 220 Hz | −1200 cents |
| 440 Hz → 440 Hz | 0 cents |
| 440 Hz → 450 Hz | about +38.91 cents |
| Frequency ratio 3/2 | about 701.96 cents (calculated from the formula) |
| Frequency ratio 2^(1/12) | 100 cents |
The just perfect fifth (3:2) is about 701.955 cents, which is about 1.955 cents from the equal-tempered perfect fifth (700 cents). These figures are calculated from the formula.
Things to know
- How large a difference people can hear varies a lot with the person, the sound and the volume. This page gives no figure for it.
- The frequency must be from 0.000001 to 999,999,999 Hz, the cents from −24,000 to 24,000, and the frequency ratio from 0.000001 to 1,000,000.
- Beat frequencies, overtones and lists of tunings are not covered.
Frequently asked questions
How big is 100 cents?
One equal-tempered semitone.
Does a larger number of cents mean a larger difference in frequency?
Cents depend on the ratio of the frequencies, so the same number of cents is a larger difference in hertz at a higher frequency.
Can I turn a frequency into a wavelength?
Use the Sound Wavelength Calculator. To convert between frequency and period, use the Frequency ⇄ Period Converter.