Cents & Interval Calculator

Frequently Asked Questions

What is a cent in music?

A cent is 1/100 of an equal-tempered semitone, so an octave is 1,200 cents. It measures pitch differences finely: humans typically notice mistuning of about 5–10 cents on sustained notes.

How do I calculate cents between two frequencies?

Cents = 1200 × log2(f2 ÷ f1). A perfect fifth (3:2 ratio) is about 702 cents, while the equal-tempered fifth is exactly 700 cents - a 2-cent difference.

How do I find the frequency after shifting by N cents?

f2 = f1 × 2^(cents ÷ 1200). Raising 440 Hz by 50 cents gives 440 × 2^(50/1200) ≈ 452.9 Hz (a quarter-tone up).

Why are cents better than Hz for comparing tuning?

A fixed Hz difference is a bigger pitch change at low frequencies than at high ones. Cents are logarithmic and constant per musical interval, so a 10-cent error sounds the same in any octave.

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How This Calculator Works

A cent is the standard fine unit of musical pitch: there are exactly 100 cents in an equal-tempered semitone and 1200 cents in an octave. This calculator measures the interval in cents between any two frequencies, reports the underlying frequency ratio, and in reverse computes the frequency you get by detuning a base frequency by a given number of cents. Tuners, intonation analysis, microtonal scales, and comparisons between tuning systems (equal temperament versus just intonation) all live in cents because the unit is logarithmic and therefore matches how we perceive pitch - a fixed cent difference sounds like the same amount of detuning whether the notes are low or high. Hertz alone is a poor way to talk about tuning errors: a few hertz is a huge interval in the bass and inaudible in the treble, whereas a cents figure means the same musical amount everywhere. That single property is why tuner displays, pitch-correction plugins, and microtonal scale designers all express deviation in cents rather than in raw frequency. It is also why string players, who can bend pitch continuously, talk about playing a note "a few cents sharp" for expressive effect, and why discussions of historical temperaments are framed entirely in cents of deviation from equal temperament.

Formula / Method

The cents between two frequencies is c = 1200 × log₂(f₂ ÷ f₁). The 1200 comes from dividing the octave (a 2:1 ratio) into 1200 equal logarithmic parts. To go the other way - apply N cents to a base frequency - invert it: f = f_base × 2^{(N ÷ 1200)}. The pure frequency ratio is just 2^(c÷1200). Because the relationship is logarithmic, intervals add: stacking a 700-cent fifth on a 400-cent third gives an 1100-cent interval, and the ratios multiply correspondingly.

Worked Example

Compare A4 = 440 Hz with A#4 in 12-TET = 466.164 Hz. The ratio is 466.164 ÷ 440 ≈ 1.05946, and 1200 × log₂(1.05946) ≈ 100 cents - exactly one equal-tempered semitone, as expected. A pure (just) perfect fifth has ratio 3:2; 1200 × log₂(1.5) ≈ 701.96 cents, which is about 2 cents wider than the 700-cent equal-tempered fifth - the small but audible difference that defines temperament. In reverse, detuning 440 Hz by +700 cents gives 440 × 2^700÷1200 ≈ 659.3 Hz (an equal-tempered E5).

Common Mistakes

Using log base 10 instead of log base 2 - the factor must pair 1200 with log₂.

Forgetting that cents are signed: f₂ below f₁ gives a negative value (a descending interval).

Assuming an equal-tempered fifth is 3:2 - it is 700 cents, about 2 cents flat of the pure 701.96.

Treating cents as linear in frequency; doubling Hz is always 1200 cents regardless of register.

Comparing readings from an instrument that has not stabilised in pitch - warm up before measuring.

Tips & FAQ

How many cents can people hear? Trained ears notice mistuning of about 5–10 cents on sustained tones; beating between close pitches is audible even finer. Why are cents better than Hz for tuning? A 1 Hz error at 100 Hz is a big interval but trivial at 1000 Hz; cents normalise this so a fixed value means the same musical amount everywhere. What is the syntonic comma? The gap between four pure fifths and two octaves plus a pure major third, about 21.5 cents - the historical headache that temperaments exist to manage. Can I build microtonal scales with this? Yes - pick a step size in cents (e.g. 1200÷19 ≈ 63.16 for 19-EDO) and use reverse mode to generate every pitch from a reference.

Cents & Interval Calculator

Frequently Asked Questions

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