Musical Frequency Ratio Calculator
Convert semitone intervals into target frequencies and explore how tuning systems affect pitch relationships. Use it when transposing instruments, tuning synthesisers, or comparing equal temperament against just intonation.
Last updated: September 2026
Formula below · 2 sources (aes.org, Wikipedia) · Updated Sep 2026
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About this calculator
Every musical interval is a ratio between two frequencies. A perfect octave is 2:1 (exactly one frequency is double the other), a perfect fifth is 3:2, a major third is 5:4. This calculator takes a base frequency, an interval in equal-tempered semitones and an octave multiplier, and returns the target frequency: f = base × 2^(semitones/12) × octave multiplier.
Frequency to pitch — how it maps
Standard concert pitch fixes A4 at 440 Hz. From there every semitone up multiplies frequency by the twelfth root of 2 (≈ 1.05946), so A#4 is 466.16 Hz, B4 is 493.88 Hz, and C5 (one octave + minor third above A3) is 523.25 Hz. To go the other way, from a frequency to the nearest note and its cent deviation, use the frequency-to-note calculator.
Frequency ratio between two notes
The ratio between any two frequencies is f₂/f₁. For 12-tone equal temperament the ratio is 2^(n/12) where n is the semitone distance. For just intonation the ratios are small integer fractions: octave 2/1, fifth 3/2, fourth 4/3, major third 5/4, minor third 6/5. This calculator uses the equal-tempered ratio; compare its result with base × the just ratio to see how far the two systems differ.
Cents — measuring fine pitch difference
One semitone equals 100 cents. So a note 5 cents flat sits 5/100 of a semitone below perfect pitch — audible to a trained ear on sustained tones but usually inaudible on plucked or percussive notes. Tuning drift within ±5 cents is considered "in tune" in almost every context; classical string ensembles push for ±2 cents on sustained chords.
Just intonation vs equal temperament
Just intonation uses integer-ratio intervals that create beat-free chords but drift out of tune when you change key. Equal temperament divides the octave into 12 equal steps — every chord is slightly out of tune (major thirds are 14 cents sharp, fifths are 2 cents flat) but every key is equally usable. Piano and fretted-instrument tuning is always equal-tempered; vocal and string ensembles adjust toward just intonation in the moment.
How to use
Start with a base frequency of 440 Hz (A4), a target of 7 semitones (E5 in equal temperament), and the octave adjustment set to Same Octave (×1). Apply the formula: f = 440 × 2^(7 / 12) × 1 = 440 × 1.4983 ≈ 659.3 Hz. This is E5, the perfect fifth above A4. In just intonation, the pure fifth ratio is 3:2, giving 440 × 1.5 = 660 Hz — a difference of about 2 cents, audible to trained ears. Set the octave adjustment to Down 1 Octave (×0.5) to get the E below A4 instead (329.6 Hz).
Frequently asked questions
What is the difference between equal temperament and just intonation frequency ratios?
Equal temperament divides the octave into 12 mathematically equal semitones, each with a ratio of 2^(1/12), ensuring every key sounds equally in tune. Just intonation uses simple whole-number ratios (3:2 for a fifth, 5:4 for a major third) derived from the natural harmonic series, producing purer-sounding intervals in a single key. The trade-off is that just intonation sounds out of tune when you modulate to distant keys, which is why equal temperament became the standard for Western music.
How do I calculate the frequency of any note from A440?
Start with A4 = 440 Hz and count the semitones from A4 to your target note — positive for higher pitches, negative for lower. Plug those values into f = 440 × 2^(n/12), where n is the semitone count. For example, middle C (C4) is 9 semitones below A4, so f = 440 × 2^(−9/12) ≈ 261.6 Hz. This formula works for any reference pitch, not just 440 Hz.
Why do different instruments need different tuning system adjustments?
Fretted instruments like guitars are built around equal temperament because fixed frets must serve all keys equally. However, vocalists, violinists, and trombonists can adjust intonation in real time and naturally gravitate toward just ratios when playing sustained harmonies, creating richer resonance. Keyboard instruments like pianos and organs are permanently tempered, so microtuning adjustments are sometimes applied in electronic music production to match the natural harmonics of acoustic instruments or to create specific tonal colours.
How do I convert a frequency to a musical pitch?
The formula is n = 12 · log₂(f / 440) semitones from A4. Round to the nearest integer for the note, and the fractional part × 100 gives the cent deviation from perfect tuning. For 523.25 Hz that's +3 semitones from A4 — C5 in scientific pitch notation.
What is a frequency ratio?
The ratio f₂/f₁ between two frequencies. Musical intervals have specific ratios: octave 2/1, fifth 3/2, fourth 4/3, major third 5/4, minor third 6/5. In equal temperament the ratios are irrational (2^(n/12)) and only approximate the integer ratios of just intonation.