Cents Calculator: Frequency to Note, Cents Between Frequencies, Ratios and Tuning
Convert between cents, frequency ratios and semitones: the nearest note and cents offset for a measured frequency, the cents between two frequencies, a frequency shifted by cents, the frequency of any note at A = 440, 442, 432, 415 or a custom pitch, the closest simple ratio, and just vs equal-tempered intervals.
How it works
cents = 1,200 × log₂(f₂ ÷ f₁) · ratio = 2^(cents ÷ 1,200) · new frequency = f × 2^(cents ÷ 1,200) · note number n = 69 + 12 × log₂(f ÷ A4) · f = A4 × 2^((n − 69) ÷ 12) · offset = (n − round(n)) × 100 cents
Pitch is logarithmic. Every octave doubles the frequency and every equal-tempered semitone multiplies it by the twelfth root of two (1.05946), so musicians measure intervals and tuning errors in cents: hundredths of a semitone, 1,200 to the octave. Cents describe a ratio, not a frequency: the same 10 Hz error is about a third of a semitone at 100 Hz but only a few cents at 1,000 Hz. To find the nearest note, the calculator converts the frequency to a MIDI note number, n = 69 + 12 × log₂(f ÷ A4), rounds it to the nearest note and reports the remainder × 100 as the offset in cents. It also gives the difference in hertz, which equals the beat rate you hear against a reference tone. Most listeners notice about 5–6 cents on a sustained note, and singers and string players routinely adjust by that much to make intervals pure. Pure (just) intervals use small whole-number ratios: a fifth of 3:2 is 701.96 cents, 2 cents wider than the tempered 700, and a major third of 5:4 is 386.31 cents, 14 cents narrower than 400. That compromise lets a piano play in every key. Concert pitch sets the whole grid: A4 = 440 Hz is the ISO 16 standard, 442–443 Hz is common in orchestras, and 415 Hz is the Baroque convention about a semitone lower.
Worked example
A tuner hears 446 Hz on an A string with A4 = 440 Hz. n = 69 + 12 × log₂(446 ÷ 440) = 69.2345, so the nearest note is A4 (MIDI 69, exactly 440.00 Hz) and the offset is 0.2345 × 100 = +23.4 cents: noticeably sharp, about 6 beats per second against a 440 Hz reference. Against an orchestra at A4 = 442 Hz the same string would be only +15.6 cents. In the other direction, a 3:2 fifth is 1,200 × log₂(1.5) = 701.96 cents, 1.96 cents wider than the tempered 700, and 702 cents is a ratio of 1.50004.
Sources: sengpielaudio: interval, cents and frequency ratio calculators · Wikipedia: Cent (music)
Frequently asked questions
How many cents are in a semitone and an octave?
An equal-tempered semitone is 100 cents and an octave is 1,200 cents. A tempered whole tone is 200, a major third 400 and a perfect fifth 700 cents.
How do I convert Hz to cents?
You cannot convert a single frequency to cents, because cents measure the ratio between two frequencies. Use cents = 1,200 × log₂(f₂ ÷ f₁). For example, 446 Hz against 440 Hz is 1,200 × log₂(1.01364) = +23.4 cents.
How many cents out of tune is noticeable?
About 5–6 cents is the usual threshold for a sustained tone heard alone; in a unison with another instrument the beating makes even 2–3 cents audible. Most tuners show ±50 cents either side of the note.
What is the difference between A = 440 and A = 432?
A4 = 432 Hz is 31.8 cents below 440 Hz, about a third of a semitone flat. ISO 16 sets 440 Hz as the standard; many orchestras tune to 442 or 443 Hz, and period-instrument groups often use 415 Hz.
What is a comma?
A small gap between stacks of pure intervals. The syntonic comma (81:80, 21.51 cents) separates four pure fifths from a pure major third two octaves up; the Pythagorean comma (about 23.46 cents) separates twelve pure fifths from seven octaves.
Why do just and equal-tempered intervals differ?
Just intervals use whole-number ratios (3:2, 5:4) that sound beatless, but they do not add up evenly around the circle of fifths. Equal temperament makes every semitone exactly 100 cents so that every key sounds the same, at the cost of thirds about 14 cents off pure.
Related calculators
Music-theory and studio arithmetic; use your ears for the final call. Genre conventions and the material itself override any formula.