Semitone Frequency Calculator - Transpose Pitch by Semitones & Cents

What is a semitone frequency calculator?

A semitone frequency calculator converts between musical intervals and sound frequencies measured in hertz (Hz). It works in two directions. It can take a starting frequency and move it up or down by a given number of semitones and cents, reporting the new frequency and the closest musical note. It can also take two frequencies and report the interval between them in semitones and cents.

A semitone is the smallest interval in standard Western music, the gap between one piano key and the very next one, black or white. An octave contains 12 semitones. A cent is a smaller unit still: one hundredth of a semitone, so an octave holds 1200 cents. Cents let musicians measure pitch differences too small to hear as a distinct note change, such as slight detuning.

How to calculate the frequency of a semitone step

The calculator uses equal temperament, the standard tuning for modern pianos, guitars, and most Western instruments. In equal temperament the octave is split into 12 equal steps, so every semitone is exactly the same size.

To move a starting frequency $f_0$ up by $n$ semitones and $c$ cents:

$f = f_0 \times 2^{(n + c/100)/12}$

A positive $n$ raises the pitch and a negative $n$ lowers it. The cents field is a fine offset; 100 cents equal one semitone, so entering 100 cents has the same effect as adding 1 to the semitone count.

The equal temperament ratio

Each semitone step multiplies the frequency by:

$r = 2^{1/12} \approx 1.059463$

Multiply a frequency by 1.059463 and it rises by one semitone. Do that 12 times and the frequency doubles, which is one octave:

$(2^{1/12})^{12} = 2$

The nearest note

Alongside the frequency, the calculator names the closest equal-tempered note, using A4 = 440 Hz as the reference. It converts the frequency to a MIDI note number, rounds it to the nearest whole number, then reads off the note name and octave:

$m = \operatorname{round}\left(69 + 12 \times \log_2\left(\frac{f}{440}\right)\right)$

Note names use scientific pitch notation, where the number is the octave: A4 is concert A at 440 Hz, and C4 is middle C. Because the value is rounded, a frequency between two notes is reported as whichever note is closer.

How to calculate semitones between two frequencies

The second mode runs the same relationship backwards. The number of semitones $n$ between two frequencies is:

$n = 12 \times \log_2\left(\frac{f_2}{f_1}\right)$

Here $f_1$ is the starting frequency in Hz and $f_2$ is the ending frequency. Because one semitone equals 100 cents, the cents version multiplies the same ratio by 1200:

$c = 1200 \times \log_2\left(\frac{f_2}{f_1}\right)$

A positive result means the second frequency is higher than the first. A negative result means it is lower.

Why the formula uses a logarithm

Human hearing judges pitch by frequency ratio, not by the raw gap in Hz. Doubling a frequency always sounds like the same jump, one octave, no matter where you start.

• 110 Hz to 220 Hz is one octave (12 semitones)
• 220 Hz to 440 Hz is one octave (12 semitones)
• 440 Hz to 880 Hz is one octave (12 semitones)

Each pair has a different gap in Hz but the same ratio, so a logarithm is needed to turn that ratio into a steady, linear scale of semitones.

How to use the calculator

Pick a calculation mode first.

Frequency from semitones. Enter the base frequency in Hz, the number of semitones to move (negative to go down), and an optional cents offset. The calculator returns the new frequency in Hz and the nearest note.

Interval between frequencies. Enter two frequencies in Hz. The calculator returns the interval from the first to the second, in semitones and in cents.

Every frequency entered must be greater than zero. A zero or negative frequency has no pitch, so the calculator reports an error instead of a result.

Worked example

Moving a pitch up by semitones

Middle C has a frequency of about 261.63 Hz. Raising it by 7 semitones:

• Frequency = 261.63 × 2^(7/12) ≈ 261.63 × 1.059463^7 ≈ 392.00 Hz.
• Nearest note: G4.

Seven semitones is a perfect fifth, one of the most common intervals in music, so G above middle C is the expected answer.

Adding a cents offset

Starting from concert A at 440 Hz with 0 semitones and a 50-cent offset:

• Frequency = 440 × 2^(50/1200) ≈ 452.89 Hz.

Fifty cents is half a semitone, a quarter tone. It sits midway between A4 and A#4, so the nearest-note reading is only an approximation for a pitch like this.

Measuring the interval between two frequencies

Feeding the same two notes into the second mode, with 261.63 Hz first and 392.00 Hz second:

• Semitones = 12 × log₂(392.00 / 261.63) ≈ 7.00.
• Cents = 1200 × log₂(392.00 / 261.63) ≈ 699.99.

The result is a fraction of a cent short of a clean 700 because 261.63 Hz and 392.00 Hz are themselves rounded values.

Measuring a small pitch error

Cents are useful for tiny mismatches. Suppose a room has an unwanted resonance at 147 Hz, and an engineer is working on a track built around the note D, whose standard fundamental frequency is 146.83 Hz.

• Cents = 1200 × log₂(147 / 146.83) ≈ 2.00 cents.

The resonance sits about 2 cents sharp of D, a gap too small for most listeners to hear as a different pitch, but still measurable and worth noting when treating a room acoustically.

Common musical intervals

The table lists standard interval names in equal temperament, with their size in semitones, cents, frequency ratio, and the resulting pitch when starting from A440 (440 Hz).

These figures apply to equal temperament only. Other tuning systems place the same interval names at slightly different frequencies.

Alternative tuning systems

Equal temperament is the modern standard, but it is not the only way to tune music, and it is not perfectly in tune with the physics of sound.

Just intonation builds intervals from simple whole-number ratios, such as 3:2 for a perfect fifth or 5:4 for a major third. These ratios sound perfectly smooth together, with no audible beating, but they differ from equal temperament by up to about 16 cents depending on the interval. A just major third (5:4) is about 14 cents flatter than the equal-tempered major third.

Pythagorean tuning builds a scale entirely from stacked perfect fifths (ratio 3:2). Its fifths are pure and beatless, but its major thirds come out about 8 cents sharper than equal-tempered major thirds. Compared with the pure 5:4 major third of just intonation, the Pythagorean third is sharper still, by close to 22 cents, a gap known as the syntonic comma. Pythagorean tuning was common in medieval and early Renaissance music.

Meantone temperament narrows the perfect fifths slightly so that major thirds come out pure. It was widely used from the Renaissance into the early Baroque period and suits music that stays in a small set of closely related keys.

Microtonal systems divide the octave into more than 12 steps, common choices being 19, 24, 31, or 53 steps. They appear in experimental and some non-Western music.

Practical uses

• Music production: pitching samples, tuning synthesizer oscillators, and checking that layered sounds line up in tune.
• Audio engineering: relating equalizer bands or room resonances to musical notes, as in the room-resonance example above.
• Instrument building: guitar makers place frets using the constant 17.817, derived from $2^{1/12}$, so each fret raises the pitch by one semitone. Organ builders and bell founders use the same math to plan pipe lengths and tune bells.
• Research and education: studying how humans perceive pitch, analyzing animal calls, or teaching logarithms and sound physics.

Frequently asked questions

How do you calculate the frequency a number of semitones away?

Multiply the starting frequency by 2 raised to the power of the semitone count divided by 12: f = f₀ × 2^(n/12). To include a cents offset, add the cents divided by 100 to the semitone count before dividing by 12.

How do you calculate semitones from two frequencies?

Divide the second frequency by the first, take the base-2 logarithm of that ratio, then multiply by 12: n = 12 × log₂(f₂/f₁). Multiply by 1200 instead of 12 to get the answer in cents.

What is the difference between a semitone and a cent?

A semitone is the standard interval between adjacent notes in Western music, such as neighboring piano keys. A cent is 1/100 of a semitone, so there are 1200 cents in an octave. Cents give a finer scale for measuring small pitch differences that fall between semitones.

How many Hz is one semitone?

There is no fixed number of Hz per semitone. A semitone is a ratio, about 1.059463, not a fixed frequency gap, so its size in Hz depends on the starting pitch. One semitone above 440 Hz is 466.16 Hz, a rise of 26.16 Hz. One semitone above 220 Hz is 233.08 Hz, a rise of only 13.08 Hz.

What frequency is 12 semitones above 440 Hz?

880 Hz. Twelve semitones make one octave, and one octave always means doubling the frequency, whatever the starting pitch.

How do MIDI note numbers relate to semitones?

Each MIDI note number is one semitone from its neighbors. Middle C is MIDI note 60 and A440 is MIDI note 69. Frequency follows from a MIDI note number with f = 440 × 2^((m − 69)/12), which is the formula the calculator inverts to name the nearest note.

Why do some intervals in equal temperament sound slightly out of tune?

Equal temperament divides the octave into 12 identical steps so music can move freely between keys, but this compromise pulls most intervals slightly away from the pure whole-number ratios of just intonation. The equal-tempered major third, for example, is about 14 cents sharper than the pure 5:4 major third. Trained listeners can sometimes hear this, though most people do not notice it in normal listening.

How accurate does a frequency measurement need to be for tuning?

For everyday musical purposes, accuracy within about 5 cents is generally enough, close to the limit of average pitch discrimination. Professional instrument tuning usually aims for about 2 cents, and some trained musicians can detect differences as small as 1 cent under good conditions.