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.
About this calculator
In equal temperament, the octave is divided into 12 equal semitones, each with a frequency ratio of 2^(1/12) ≈ 1.05946. The target frequency is calculated as: f = baseFrequency × 2^((targetNote / 12) + octaveAdjustment). Here, baseFrequency is the known reference pitch in Hz, targetNote is the number of semitones above or below that reference, and octaveAdjustment shifts the result up or down by whole octaves. For example, A4 = 440 Hz is the universal tuning standard, and every other note can be derived from it using this formula. Cents deviation — the difference between equal temperament and just intonation — can be measured as 1200 × log₂(f_just / f_equal), where 100 cents equals exactly one semitone. Understanding these ratios is essential for intonation work, microtuning synthesisers, and instrument building.
How to use
Start with a base frequency of 440 Hz (A4), a target of 7 semitones (E5 in equal temperament), and an octave adjustment of 0. Apply the formula: f = 440 × 2^((7 / 12) + 0) = 440 × 2^(0.5833) = 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. Toggle the octave adjustment to −1 to get the E below A4 instead.
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.