What this calculator does
A semitone is the smallest standard step in Western music, the distance between one piano key and the very next one, black or white. Because modern instruments are tuned to 12-tone equal temperament, every semitone represents exactly the same frequency ratio, 2 to the power of one-twelfth, no matter which note you start from. This calculator takes a starting frequency and a number of semitones and works out the resulting pitch.
That fixed ratio is what makes transposition and tuning predictable. Move up 12 semitones and the frequency exactly doubles, an octave. Move up 7 and you land on a perfect fifth, close to one and a half times the original frequency. Enter a negative number to move down instead of up.
The formula
Multiply the starting frequency by 2 raised to the power of the number of semitones divided by 12. Twelve equal semitone steps make one octave, and one octave is always a doubling of frequency, so each individual semitone must be the twelfth root of 2, roughly 1.0595, applied repeatedly.
| Term | Meaning |
|---|---|
| Semitone | The smallest interval used in standard Western tuning, one twelfth of an octave. |
| Equal temperament | The tuning system where every semitone has the identical frequency ratio, letting instruments play in any key. |
| Cents | A finer unit for pitch, 100 cents to a semitone, used for tuning precision smaller than a full step. |
| Ratio | How many times higher (or lower) the resulting frequency is compared with the starting one. |
The inputs explained
| Field | What to enter |
|---|---|
| Starting frequency (Hz) | The starting pitch in hertz. 440 Hz is concert pitch A4, the standard tuning reference. |
| Semitones to shift (+ up, − down) | How many semitones to shift. Use a positive number to go up in pitch and a negative number to go down. |
When to use it
Transposing a reference pitch
If a tuner or instrument is set to a non-standard reference frequency, shifting it by a known number of semitones finds the equivalent pitch at standard tuning, or the other way around.
Building a semitone chart
Running the same starting frequency through 1 to 12 semitones lays out the exact frequency of every note in an octave, useful for building tuning tables or checking a synthesiser’s output.
Working out an interval by ear
If you can count how many semitones separate two notes, this calculator confirms the exact frequency ratio between them rather than relying on a rough estimate.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is the semitone chart above A4 (440 Hz)?
Common semitone shifts above concert pitch A4, through to a full octave.
| Semitones up | Resulting frequency | Frequency ratio |
|---|---|---|
| +1 | 466.16 Hz | 1.059× |
| +2 | 493.88 Hz | 1.122× |
| +3 | 523.25 Hz | 1.189× |
| +4 | 554.37 Hz | 1.260× |
| +5 | 587.33 Hz | 1.335× |
| +7 | 659.26 Hz | 1.498× |
| +9 | 739.99 Hz | 1.682× |
| +12 | 880.00 Hz | 2.000× |
How does an octave shift affect different starting notes?
The same one-octave jump applied to a few common reference pitches.
| Starting frequency | Resulting frequency | Frequency ratio |
|---|---|---|
| 110 Hz | 220.00 Hz | 2.000× |
| 220 Hz | 440.00 Hz | 2.000× |
| 261.63 Hz | 523.26 Hz | 2.000× |
| 440 Hz | 880.00 Hz | 2.000× |
| 523.25 Hz | 1,046.50 Hz | 2.000× |
Questions
Why is the ratio 2 to the power of 1/12?
An octave is a doubling of frequency, and equal temperament splits that octave into 12 equal steps. For 12 identical multiplicative steps to compound into a doubling, each step must be the twelfth root of 2, since that number multiplied by itself 12 times equals exactly 2.
Does this work for any starting frequency?
Yes. The semitone ratio is fixed regardless of pitch, so the same formula applies whether the starting frequency is a low bass note or a high treble one.
What is the difference between a semitone and a cent?
A cent is a finer subdivision, with 100 cents making up one semitone. Cents are used when tuning needs to be more precise than a full semitone step, such as fine-tuning an instrument by ear.
How do I shift down instead of up?
Enter a negative number of semitones. The same formula applies, and the resulting frequency will be lower than the starting one.
For the wider relationship between frequency, wave speed and wavelength, see the wave speed calculator. To find how fast two close frequencies pulse against each other, try the beat frequency calculator.