Semitone Frequency Calculator - Transpose Pitch by Semitones & Cents

What Is a Semitone Frequency Calculator?

A semitone frequency calculator finds the pitch, in hertz (Hz), that lies a given number of semitones above or below a starting frequency. A semitone is the smallest step in standard Western music, the distance between one piano key and the very next one. This tool also works the other way: given two frequencies, it finds the interval between them in semitones and cents.

How to Calculate Frequency From Semitones

Musical pitch follows equal temperament, a tuning system where each octave splits into 12 equal semitones. Because pitch is heard on a ratio scale, not a straight line, each semitone step multiplies the frequency by a fixed factor rather than adding a fixed number of hertz. That factor is the twelfth root of 2, about 1.05946.

The formula for the target frequency is:

f = f₀ × 2^(n / 12)

• f is the target frequency in Hz.
• f₀ is the base frequency in Hz.
• n is the number of semitones, positive for a step up and negative for a step down.

This tool adds an optional fine-tuning term in cents, where 100 cents equal one semitone:

f = f₀ × 2^((n + c / 100) / 12)

Here c is the cents offset. Setting c to 0 gives the plain semitone formula above.

Finding the Nearest Note Name

Once the target frequency is known, the calculator reports the nearest note name and octave using equal temperament with A4 (the A above middle C) tuned to 440 Hz, the standard concert pitch. It first converts the frequency to a MIDI note number:

midi = 69 + 12 × log₂(f / 440)

The result is rounded to the nearest whole number, since 69 is the MIDI number of A4. The note name comes from the pattern C, C#, D, D#, E, F, F#, G, G#, A, A#, B, repeating every 12 numbers, and the octave comes from dividing the MIDI number by 12.

Worked Example

Start at A4, 440 Hz, and go up 7 semitones with no cents offset.

1. n = 7, f₀ = 440.
2. f = 440 × 2^(7/12).
3. 2^(7/12) ≈ 1.498307.
4. f ≈ 440 × 1.498307 ≈ 659.26 Hz.

The nearest note to 659.26 Hz is E5, seven semitones above A4, which is a musical interval called a perfect fifth.

A second example uses a full octave: 12 semitones above A4 gives 440 × 2^(12/12) = 440 × 2 = 880 Hz, the note A5. Doubling the frequency always raises the pitch by exactly one octave.

How to Calculate the Interval Between Two Frequencies

The reverse question, given two frequencies, asks how many semitones separate them. The formula is:

semitones = 12 × log₂(f₂ / f₁)

cents = semitones × 100

Here f₁ is the first (reference) frequency and f₂ is the second frequency. A positive result means f₂ is higher than f₁; a negative result means it is lower.

For example, 440 Hz to 880 Hz gives 12 × log₂(880/440) = 12 × log₂(2) = 12 semitones, or exactly 1200 cents, confirming the octave relationship from the worked example above.

Cents as a Finer Unit

A cent is one hundredth of a semitone. Cents let musicians and audio engineers describe pitch differences too small to be a full semitone step, such as the slight detuning between two instruments or a tuning error. Most listeners can notice a pitch difference of around 5 to 10 cents, though this varies by person and by the sound involved.

Frequently Asked Questions

What is a semitone? A semitone is the smallest interval in standard Western music, equal to the distance from one piano key to the next key, black or white.

Why does going up an octave double the frequency? An octave equals 12 semitones. Because each semitone multiplies frequency by 2^(1/12), 12 semitones multiply it by (2^(1/12))^12 = 2, which is exactly double.

What reference pitch does this calculator use? It uses A4 = 440 Hz, the most common concert pitch standard, with equal temperament tuning across all 12 notes.

Can the semitone value be negative? Yes. A negative value lowers the pitch. For example, -12 semitones from 440 Hz gives 220 Hz, one octave below A4.

What is the difference between semitones and cents in this tool? Semitones are whole scale steps. The cents field is an optional fine offset added on top, where 100 cents equal one semitone, useful for pitches that fall between two notes.

Why might the nearest note not match the frequency exactly? Equal temperament only produces exact note frequencies at whole-semitone steps from A4. Any frequency that is not exactly on one of those steps, including one with a cents offset, is rounded to its closest note.