Semitone Frequency Ratio Calculator

Calculator inputs

Formula used

For standard twelve-tone equal temperament, N equals 12. One octave doubles frequency, and twelve equal semitone steps divide that octave logarithmically.

How to use this calculator

1. Select forward or reverse calculation mode.
2. Enter the source frequency and its unit.
3. Enter semitones, or enter a second frequency.
4. Choose octave divisions and reference tuning.
5. Select precision, then calculate the results.
6. Export the calculation as CSV or PDF.

Semitone interval reference

Understanding semitone frequency ratios

A semitone frequency ratio describes pitch spacing mathematically. Equal temperament divides each octave into equal logarithmic steps. Standard Western tuning uses twelve steps per octave. One octave always represents a frequency ratio of two. Therefore, each semitone uses the twelfth root of two. That constant is approximately 1.059463094. Multiplying a frequency by this value raises pitch one semitone.

The relationship works in both directions. A positive interval raises the starting frequency. A negative interval lowers it by the corresponding ratio. Twelve positive semitones double a frequency exactly. Twelve negative semitones divide the frequency by two. Seven semitones produce the familiar equal-tempered perfect fifth ratio. Five semitones produce an equal-tempered perfect fourth.

Frequency, cents, and octaves

Cents provide a finer logarithmic pitch measurement. One semitone equals one hundred cents in twelve-tone equal temperament. An octave contains twelve hundred cents. These units help compare measured frequencies with theoretical tuning values. They are especially useful for acoustics, instrument tuning, audio engineering, and psychoacoustics.

Using custom equal temperaments

This calculator also supports custom octave divisions. Replace twelve with another division count to explore alternative equal temperaments. For example, twenty-four divisions create quarter-tone steps. Each step then uses the twenty-fourth root of two. The same exponential formula still applies.

Reference tuning affects note names and MIDI estimates. A4 normally equals 440 Hz, but orchestras sometimes use other references. Changing A4 preserves interval ratios while shifting note frequencies. Reverse mode can compare any two positive frequencies. It reports semitones, cents, octaves, and the frequency multiplier. These results make logarithmic pitch relationships easier to study and verify. The calculator can also support laboratory demonstrations involving oscillators and wave generators. Comparing measured tones with theoretical ratios helps students connect exponential mathematics with physical frequency, resonance, and audible pitch. Use sufficient decimal precision for scientific comparisons and measurements.

Example data