How this calculator works
Twelve-tone equal temperament makes every semitone the same ratio, the twelfth root of two. Two notes twelve semitones apart are therefore exactly an octave apart, a factor of two in frequency. Seven semitones, a perfect fifth, multiply the frequency by about 1.4983 rather than the simple ratio of 3 to 2 that just intonation uses, which is a difference of about two cents. Shifting a note is integer arithmetic on the position in the chromatic scale: add the semitone count to the note number, wrap into the next octave when it passes B, and read off the new name. Descending shifts work the same way with negative numbers.
Formula
frequency ratio = 2^(semitones / 12). A shift of n semitones multiplies the frequency by that ratio. Note positions use MIDI numbers, where C4 is 60 and A4 is 69; add n to the starting number and convert 60 to C4, 61 to C#4, and so on.
Worked example
Seven semitones give a ratio of 2^(7/12) = 1.4983, the equal-tempered perfect fifth. Twelve semitones give exactly 2, the octave. Starting on C4 and moving up seven semitones lands on G4, at 261.6256 x 1.4983 = 392.00 Hz. Moving down three semitones from A4 lands on F#4, because the note numbers run 69, 68, 67, 66.
Advertisement
Frequently asked questions
Why are equal-tempered ratios not whole numbers?
A circle of twelve identical semitones cannot be built from exact 3:2 and 5:4 ratios, because those ratios do not stack into an exact octave. Equal temperament spreads the error evenly so that every key sounds acceptable, at the cost of slightly impure fifths and thirds compared with just intonation.
How many cents are in a semitone?
One hundred. A cent is one hundredth of a semitone, so two notes a quarter tone apart differ by 50 cents. Tuners use cents because small tuning errors are easier to read than fractions of a semitone.
Does an octave shift change the note name?
The letter stays the same and the octave number changes. Moving up twelve semitones from C4 gives C5, which is the same pitch class one octave higher. Twelve semitones away is the same ratio of 2 regardless of the starting note.
What about instruments with different tuning?
Some ensembles tune to 442 or 443 Hz instead of 440, and some music uses just intonation or historical temperaments. The semitone ratios change slightly under those systems. This page assumes twelve-tone equal temperament with A4 at 440 Hz.