Semitone Frequency Calculator - Transpose Pitch by Semitones & Cents

What is a semitone frequency calculator?

A semitone frequency calculator works with musical intervals and sound frequencies, measured in hertz (Hz). It can find the interval between two frequencies in semitones and cents. It can also find the frequency that lies a given number of semitones above or below a starting frequency. Musicians, audio engineers, and instrument builders use it to compare pitches, check tuning accuracy, and convert between frequency and musical notation.

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 an equal-tempered semitone. Cents let musicians measure pitch differences too small to hear as a distinct note change, such as slight detuning.

How to calculate semitones from frequency

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

The formula for the number of semitones ($n$) between two frequencies is the standard equal-temperament relation $n = 12\log_2(f_n/440\ \text{Hz})$, with any starting frequency in place of 440 Hz:

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

Here $f_1$ is the starting frequency in Hz, $f_2$ is the ending frequency in Hz, and $\log_2$ is the base-2 logarithm.

The cents formula

Because one semitone equals 100 cents, the cents formula multiplies the same ratio by 1200 instead of 12:

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

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 of those pairs 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.

The equal temperament ratio

In equal temperament, 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 in a row and the frequency doubles, which is one octave:

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

How to use the calculator

The calculator has two modes.

Interval between two frequencies

1. Enter the first frequency in Hz. This can be a reference pitch, such as concert A at 440 Hz, or any frequency being measured.
2. Enter the second frequency in Hz, the one being compared against the first.
3. The calculator shows the interval as both semitones and cents, rounded to two decimal places.

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

Frequency from semitones

1. Enter a base frequency in Hz.
2. Enter a number of semitones (negative values go down), and optionally a fine offset in cents.
3. The calculator returns $f = f_\text{base} \times 2^{(n + c/100)/12}$ and the nearest equal-temperament note name, such as A4, with A4 set to 440 Hz.

Worked example

Middle C has a frequency of about 261.63 Hz. The G above it is about 392.00 Hz. Applying the formula:

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

The calculator shows 7.00 semitones and 699.99 cents, because both note frequencies are rounded. Seven semitones is a perfect fifth, one of the most common intervals in music, so the result matches what musicians expect for this pair of notes.

A second example: measuring a small pitch error

Cents are useful for measuring 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. How far off is the resonance from that note?

• Cents = 1200 × log₂(147 / 146.83) ≈ 2.0 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 below lists standard interval names in equal temperament, along 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 produce intervals that sound smooth together, with no audible beating, but they differ from equal temperament by an amount that depends on the interval. A just major third (5:4) is narrower than the equal-tempered major third, by about 14 cents.

Pythagorean tuning builds a scale entirely from stacked perfect fifths (ratio 3:2). Its fifths are pure and beatless, but the process leaves its major thirds 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 works well for music that stays in a small set of closely related keys.

Microtonal systems divide the octave into more than 12 steps. These are used 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 that 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 semitones from frequency?

Divide the higher frequency by the lower one, take the base-2 logarithm of that ratio, then multiply by 12: semitones = 12 × log₂(f₂/f₁). Multiply the result by 100 to get cents instead.

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; it depends on the starting pitch, because a semitone is a ratio (about 1.059463), not a fixed frequency gap. 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, regardless of the starting pitch.

How do MIDI note numbers relate to semitones?

Each MIDI note number is one semitone apart from its neighbors. A4 (440 Hz) is MIDI note 69, so middle C, nine semitones lower, is MIDI note 60. Frequency can be found from a MIDI note number with: frequency = 440 × 2^((n − 69)/12).

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

Equal temperament divides the octave into 12 identical steps so that music can move freely between keys, but this compromise pulls most intervals slightly away from the pure whole-number ratios found in 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 precise is the calculator's output?

The calculator works in full floating-point precision and displays frequencies, semitones and cents to two decimal places. The result is only as precise as the frequencies entered.

References

1. University of New South Wales, School of Physics (Music Acoustics). Note names, MIDI numbers and frequencies. Notes and frequencies — n = 12·log₂(f/440 Hz), cents = 1200·log₂(f/440 Hz), A4 = MIDI 69, f = 2^((m−69)/12) × 440 Hz.
2. University of New South Wales, School of Physics (Music Acoustics). Tartini tones and temperament: an introduction for musicians. Tartini tones and temperament — the equal-tempered semitone is divided into 100 cents; a just major third has ratio 5:4 and is narrower than the equal-tempered third.