Frequency Ratio Calculator

This frequency ratio calculator expresses one tone in terms of another. Musical intervals are perceived logarithmically: the brain hears a 440 → 880 Hz step (ratio 2:1) and a 220 → 440 Hz step as the "same" interval — both octaves. The two tuning systems below organize music's vocabulary of ratios in different ways. The same mathematics applies whenever you need to compare any two pitched sounds — from instrument tuning to acoustic resonance analysis to synthesizer oscillator detuning.

Just intonation — small integer ratios

Just intonation (JI) defines intervals as small whole-number ratios: 2:1 octave, 3:2 fifth, 4:3 fourth, 5:4 major third, 6:5 minor third, etc. These ratios maximize "consonance" — the overtones of one note line up perfectly with those of the other, producing a beat-free sound. Pythagoras built tuning systems from just 2:1 and 3:2 (Pythagorean tuning); 5-limit just intonation adds 5:1 (the harmonic). Extended just intonation (7-limit, 11-limit) introduces ratios like 7:4 (the harmonic seventh) and 11:8, beloved in microtonal and spectral music. Drawback: only one key sounds perfect; transposing requires re-tuning.

12-tone equal temperament — geometric spacing

12-TET divides the octave into 12 equal logarithmic steps. Each semitone is 21/12 ≈ 1.05946 (exactly 100 cents). All keys sound equally (slightly) out of tune. Drawback: the major third in 12-TET (400 cents) is 14 cents sharper than the just 5:4 (386 cents), giving piano thirds their characteristic beating. Benefit: any key sounds the same, transposition is free. Other equal temperaments — 19-TET, 31-TET, 53-TET — are also in use among microtonalists; this calculator handles any ratio regardless of the underlying scale system.

Cents — the universal interval unit

One cent is 1/100 of a 12-TET semitone, or 1/1200 of an octave. Cents convert any ratio to a linear measurement: cents = 1200 · log₂(ratio). Most listeners can detect pitch differences down to ~5 cents in isolation, ~10-20 cents in musical context. Beat rates also help: a 7 cent deviation at 440 Hz gives ~1.8 beats per second. To explore the full overtone structure that makes JI intervals so consonant, the Harmonic Series Calculator shows every partial of a fundamental and how they map onto the interval lattice.

Why are pianos not perfectly in tune? Equal temperament vs just intonation

Equal temperament (ET) divides the octave into 12 equal semitones, each with a frequency ratio of 2^1/12 ≈ 1.0595 (exactly 100 cents). Every key sounds identical, so a piano can play in any key without re-tuning. The trade-off is that most intervals are slightly impure: the ET major third (400 cents) is about 14 cents sharper than the just 5:4 major third (386 cents), and the ET perfect fifth (700 cents) is roughly 2 cents flatter than the just 3:2 fifth (702 cents). Those small deviations create the audible beating that gives a piano its characteristic shimmer.

Just intonation (JI) uses simple whole-number ratios — 3:2 for the perfect fifth, 5:4 for the major third — that produce beat-free, "locked-in" sound. The cost is key-dependency: a JI tuning perfect for C major sounds noticeably off in D major. Ensembles with flexible pitch (choirs, string quartets) naturally drift toward JI ratios mid-performance. To hear the difference concretely, use the Chromatic Tuner to check where a played note lands in cents, then use the Interval Calculator to see how far that interval sits from the nearest pure JI ratio.

Continued-fraction approximation

This tool's "Best integer ratio" finds the simplest p:q with q ≤ 1000 closest to your decimal ratio, using continued fractions. For an exact 3:2 input (660/440 = 1.5), it returns 3:2 exactly. For a 12-TET fifth (440 × 2^7/12 ≈ 659.255 Hz), it returns 1.4983:1 — close to but not 3:2, illustrating the tuning gap. Continued-fraction reduction is also the algorithm behind Stern–Brocot tree approaches to ratio simplification, which rank fractions by harmonic complexity rather than size.