Convert frequency ratios into cents, compare musical intervals, reverse cents into ratios, transpose frequencies, view formulas, and export clear calculation results instantly online.
Ratios must be positive. Two-frequency mode accepts positive frequencies in the same selected unit.
The standard relationship between a frequency ratio and cents is logarithmic.
For two frequencies, use r = f₂ / f₁. The inverse relationship is:
A transposed frequency is f₂ = f₁ × 2^(c / 1200).
| Ratio | Approx. cents | Common description |
|---|---|---|
| 1:1 | 0.00 | Unison |
| 16:15 | 111.73 | Just minor second |
| 9:8 | 203.91 | Just major second |
| 5:4 | 386.31 | Just major third |
| 4:3 | 498.04 | Perfect fourth |
| 3:2 | 701.96 | Perfect fifth |
| 2:1 | 1200.00 | Octave |
Frequency ratios describe the relationship between two vibrations. A ratio of 2:1 means the second frequency is twice the first. In acoustics, that relationship is an octave. Cents provide a logarithmic way to express the interval. One octave equals 1200 cents, and one equal-tempered semitone equals 100 cents. This scale makes pitch differences easier to compare.
The calculator uses the formula c = 1200 × log₂(r). The symbol r represents a positive frequency ratio. When the ratio exceeds one, the cents value is positive. A ratio below one produces a negative value. A ratio of exactly one returns zero cents. The sign therefore indicates pitch direction, while the value shows interval size.
You can calculate the ratio from two frequencies. Dividing the second frequency by the first gives r. 660 Hz divided by 440 Hz equals 1.5. That ratio corresponds to about 701.955 cents, close to seven equal-tempered semitones. This comparison is useful when studying tuning systems, resonance, harmonics, instruments, and audio signals.
The inverse equation converts cents back into a ratio. It uses r = 2^(c/1200). This is helpful when you know a pitch shift. A starting frequency can then be multiplied by that ratio to find a transposed frequency. Positive cents raise pitch, while negative cents lower pitch.
Common simple ratios often appear in just intonation. The ratio 3:2 represents a perfect fifth near 701.955 cents. The ratio 5:4 represents a major third near 386.314 cents. Equal temperament adjusts these intervals slightly so every semitone has the same ratio.
Always use positive frequencies and ratios. Frequency units cancel when both frequencies share the same unit. More decimal precision helps in scientific work, while fewer decimals are often sufficient for practical tuning. Use the comparison details to see octave values, semitone values, and deviation from the nearest equal-tempered semitone.
| Concept | Value | Interpretation |
|---|---|---|
| Unison | 0 cents | Ratio 1:1 |
| Equal-tempered semitone | 100 cents | Ratio 2^(1/12) |
| Equal-tempered fifth | 700 cents | Seven semitones |
| Octave | 1200 cents | Ratio 2:1 |
A cent is one hundredth of an equal-tempered semitone. There are 1200 cents in one octave.
Yes. The calculator accepts fraction notation, colon notation, or a decimal ratio.
Negative cents indicate the second pitch is lower than the first, or the ratio is below one.
No, provided both frequencies use the same unit. Their units cancel when the ratio is formed.
Use r = 2^(c/1200). The calculator includes this conversion directly.
It shows how many cents the result differs from the closest equal-tempered semitone.
Accurate ratios make pitch comparisons clearer and easier today.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.