Equal Temperament Frequency
Reference for equal-tempered tuning frequency = A4 × 2^(n/12).
Calculate the frequency of any note in 12-tone equal temperament with A4 = 440 Hz.
The Formula
In 12-tone equal temperament, the frequency of a note n semitones above (or below) a reference frequency f₀ is the reference multiplied by 2 raised to the power of n / 12. Each semitone is the twelfth root of 2, approximately 1.05946.
Reference Frequency
The international standard reference is A above middle C at 440 Hz. Many orchestras tune slightly higher (442-445 Hz) for added brightness, particularly in central Europe.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| f(n) | Frequency of target note | Hz |
| f₀ | Reference frequency (commonly A4 = 440 Hz) | Hz |
| n | Semitones above reference (negative for below) | integer |
Example — Middle C
Find the frequency of middle C (C4) given A4 = 440 Hz.
A4 to C4 is 9 semitones down: n = −9
f = 440 × 2^(−9/12) = 440 × 2^(−0.75)
2^(−0.75) ≈ 0.5946
f = 440 × 0.5946
C4 ≈ 261.63 Hz
Example — Octave Up
Find A5 (one octave above A4).
One octave = 12 semitones: n = 12
f = 440 × 2^(12/12) = 440 × 2
A5 = 880 Hz
An octave is exactly a 2:1 frequency ratio in any tuning system. Equal temperament gets octaves right; what it changes is how the steps between octaves are spaced.
Semitone Ratios
| Interval | Semitones | Equal Temperament Ratio | Just Intonation Ratio | Cents Difference |
|---|---|---|---|---|
| Unison | 0 | 1.000 | 1/1 | 0 |
| Minor 2nd | 1 | 1.05946 | 16/15 | +12 |
| Major 2nd | 2 | 1.12246 | 9/8 | −4 |
| Minor 3rd | 3 | 1.18921 | 6/5 | +16 |
| Major 3rd | 4 | 1.25992 | 5/4 | −14 |
| Perfect 4th | 5 | 1.33484 | 4/3 | +2 |
| Tritone | 6 | 1.41421 | varies | varies |
| Perfect 5th | 7 | 1.49831 | 3/2 | −2 |
| Minor 6th | 8 | 1.58740 | 8/5 | +14 |
| Major 6th | 9 | 1.68179 | 5/3 | −16 |
| Minor 7th | 10 | 1.78180 | 16/9 | +4 |
| Major 7th | 11 | 1.88775 | 15/8 | −12 |
| Octave | 12 | 2.00000 | 2/1 | 0 |
The "cents difference" column shows how far equal temperament deviates from pure just intonation. Cents are a logarithmic unit where 100 cents equals one semitone and 1200 cents equals one octave.
Cents Formula
Cents measure pitch differences logarithmically. Trained musicians can distinguish pitch differences of 5-10 cents; differences below 3 cents are inaudible to most listeners. Equal temperament's largest deviation from just intonation is the major third at 14 cents flat — audibly different in a slow choral or string passage, less noticeable in fast music.
Example — Calculating Cents Between Two Notes
A choir sings what they think is a perfect fifth above A4 (440 Hz) at 658 Hz. How many cents off from equal-tempered E5?
Equal-tempered E5 = 440 × 2^(7/12) ≈ 659.26 Hz
cents = 1200 × log₂(658 / 659.26) = 1200 × (−0.0019)
≈ −3.3 cents flat (essentially in tune)
Why Equal Temperament?
Equal temperament is a compromise. Just intonation produces purer-sounding intervals within one key but makes music in distant keys sound badly out of tune. Equal temperament makes every key equally usable at the cost of every interval (except the octave) being slightly off from pure. This trade is what enabled the development of Western music as we know it — composers like Bach explicitly wrote music to demonstrate the freedom of well-tempered tuning.
When to Use It
- Programming digital instruments and synthesizers
- Tuning calculations for guitar fret positions
- Audio signal processing — pitch shifting, harmonization
- MIDI note-number to frequency conversion (f = 440 × 2^((m−69)/12))
- Music theory analysis and transposition algorithms
- Building tuning forks and calibration references
Alternative Tunings
Microtonal music uses divisions of the octave other than 12 — 19, 22, 24, 31, and 53 tones per octave are common in academic and experimental music. Some traditions (Indian classical, gamelan, Arabic music) use entirely different tuning systems based on small-integer frequency ratios. The equal temperament formula generalizes: for N-tone equal temperament, f(n) = f₀ × 2^(n / N).