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Equal Temperament Frequency Calculator

Calculates the frequency of a note based on its distance in semitones from a reference pitch.

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Frequency

Formula first

Overview

This equation defines the mathematical relationship between musical pitches in the 12-tone equal temperament system, where an octave is divided into twelve logarithmically equal intervals. It allows for the precise calculation of any note's frequency based on a fixed reference pitch, ensuring tonal consistency across all musical keys.

Symbols

Variables

f = Frequency, = Reference, n = Semitones

Frequency
Hz
Reference
Hz
Semitones
Variable

Apply it well

When To Use

When to use: Use this formula when designing digital synthesizers, developing audio software, or calculating intervals for instrument tuning. It assumes a standard Western scale where the frequency ratio of an octave is 2:1 and each semitone is equal to the 12th root of 2.

Why it matters: This mathematical model solved the historical problem of 'wolf intervals' found in older tuning systems, allowing musicians to play in any key without retuning. It is the fundamental standard for MIDI technology, electronic oscillators, and modern acoustic piano tuning.

Avoid these traps

Common Mistakes

  • Using the wrong reference frequency.
  • Convert units and scales before substituting, especially when the inputs mix Hz.
  • Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

One free problem

Practice Problem

A synthesizer is set to a reference pitch of 440 Hz (A4). What is the frequency of the note C5, which is exactly 3 semitones above the reference?

Reference440 Hz
Semitones3

Solve for:

Hint: Divide the semitones by 12 to find the exponent of 2, then multiply the result by the reference frequency.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Equal temperament
  2. Britannica: Equal temperament
  3. The Science of Sound by Thomas D. Rossing, F. Richard Moore, Paul A. Wheeler
  4. NIST Special Publication 811: Guide for the Use of the International System of Units (SI)
  5. Wikipedia article 'Equal temperament'
  6. Wikipedia article 'Concert pitch'
  7. Wikipedia article 'Piano key frequencies'
  8. Wikipedia article 'Just intonation'