Musical Interval Calculator
Map interval names to semitones, cents and frequency ratios.
Overview
The musical interval calculator maps interval names (minor third, perfect fifth, tritone, major seventh) to their semitone counts, frequency ratios, and cent values. Pick "minor third" and the tool returns 3 semitones, ratio 1.1892, 300 cents in equal temperament — and the just-intonation ratio 6/5 for comparison.
It's used by ear-training students learning to hear intervals, composers picking colours for voicings, microtonal practitioners comparing tempered versus pure ratios, and acoustics enthusiasts exploring why some intervals sound consonant and others tense. Interval is the basic building block of harmony, so understanding the math behind each one shortens the path from theory to practice.
How it works
Every interval has a name (number and quality), a semitone count, and one or more frequency ratios. The semitone count comes from counting half-steps between the two notes: C to E is 4 semitones (major third). In 12-tone equal temperament, the ratio for n semitones is 2^(n/12), so a major third is about 1.2599 — close to but not identical to the just-intonation ratio 5/4 (1.25).
Cents quantify any pitch distance: 100 per semitone, 1,200 per octave. A perfect fifth is 700 cents in equal temperament but 702 cents in just intonation (ratio 3/2), a deviation of -2 cents that's barely audible. The major third deviates more: just is 386 cents, tempered is 400 — a 14-cent stretch that gives equal-tempered thirds their slightly "buzzy" character compared to barbershop or string-quartet thirds.
Examples
Perfect fifth → 7 semitones, ratio 3/2 (1.5), 702 cents (just)
Major third → 4 semitones, ratio 5/4 (1.25), 386 cents (just)
Tritone → 6 semitones, ratio ~1.4142, 600 cents (12-TET)
Octave → 12 semitones, ratio 2/1, 1200 cents (exact)
FAQ
Why is the tritone called "the devil's interval"?
Medieval theorists labelled it "diabolus in musica" because its strong tension was avoided in chant. It became a defining sound of dominant seventh chords and blues much later.
What's the difference between just and tempered intervals?
Just intervals use small whole-number ratios (3/2, 5/4) that sound very pure in one key. Equal temperament approximates each ratio so all keys sound equally usable but no interval is exactly pure.
Why do harmonics sound consonant?
Frequencies with simple ratios (2/1, 3/2, 4/3, 5/4) share many overtones, so they reinforce rather than beat against each other. The ear perceives that pattern as "blending."
Is a perfect fifth always 3/2?
In just intonation, yes. In equal temperament it's 2^(7/12), which is slightly narrower than 3/2 by about 2 cents.
How small an interval can the ear detect?
About 5-10 cents on sustained tones, and beating between two near-unison pitches reveals even smaller differences as a slow wavering.