Semitone Interval Ratio Calculator
Calculate the frequency ratio for any musical interval in semitones. 12 semitones = 2.0 (one octave). Used in synthesis, tuning, and acoustics.
How to use this tool
- Enter semitones in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your frequency ratio and the full breakdown beneath it.
Find the frequency ratio for any interval expressed in semitones. One octave = 12 semitones = ratio 2.0.
⚠ This tool provides general estimates for education only and is not financial, tax or legal advice. Figures may not reflect your situation — verify with a qualified professional.
Formula
Frequency ratio = 2semitones ÷ 12
Cents = semitones × 100
Inverse ratio = 1 ÷ (2semitones ÷ 12)
How it works
This calculator derives the frequency ratio corresponding to any interval expressed in semitones under 12-tone equal temperament. Because an octave (12 semitones) doubles frequency, each semitone multiplies frequency by the twelfth root of 2 (≈ 1.05946), and n semitones multiply by 2n/12. The inverse ratio gives the factor by which the lower note's frequency must be multiplied to reach the upper note when the direction is reversed.
Worked example
- Semitones = 12 (one octave).
- Ratio = 2^(12 ÷ 12) = 2^1 = 2.0.
- Cents = 12 × 100 = 1 200 ¢.
- Inverse ratio = 1 ÷ 2.0 = 0.5.
12 semitones → frequency ratio 2.0 (one octave); 1 200 ¢.
Common mistakes to avoid
- Assuming 7 semitones (a perfect fifth) has an exact ratio of 3:2; in equal temperament the ratio is 2^(7/12) ≈ 1.4983, not 1.5.
- Forgetting that transposing down uses negative semitone values; -12 semitones gives a ratio of 0.5, not 2.
- Confusing the interval ratio with a percentage increase; a ratio of 1.5 means 50% higher in frequency, not 150%.
Key terms
- Semitone
- The smallest standard interval in 12-TET, one step on the chromatic scale. Each semitone is a frequency ratio of 2^(1/12) ≈ 1.05946.
- Frequency ratio
- The multiplicative factor between two pitches. A ratio of 2.0 means the upper note vibrates twice as fast as the lower one.
- Cents
- A logarithmic unit of musical interval: 100 cents per semitone, 1 200 cents per octave.
- Octave
- An interval of 12 semitones corresponding to a frequency ratio of exactly 2:1 in equal temperament.
- Inverse ratio
- The reciprocal of the frequency ratio (1 ÷ ratio). It represents the downward interval: a note at ratio 2.0 above corresponds to a note at 0.5 times the frequency below.
Frequently asked questions
- What is the frequency ratio of an octave?
- An octave is exactly 12 semitones, corresponding to a frequency ratio of 2:1.
- What is the perfect fifth ratio?
- A perfect fifth is 7 semitones; the equal-temperament ratio is 2^(7/12) ≈ 1.4983.